Meaning, were they "taken over" by the Nazi state, or did the two exist in some sort of mutually beneficial relationship?
It is an understatement to say that the position of large industrial firms within the Third Reich is a controversial one. Both during the Third Reich's existence and after, it became an article of faith for some that Hitler owed his existence to powerful industrialists and entrenched business interests. Although this assertion was quite common among the global left, it was not just limited to left wing politics. Concerns like Krupp and Flick were staple villains in American wartime propaganda and JCS 1067, the initial American plan for postwar Germany called for decartelization of German industry and prosecution of industrialists. Yet, many industrialists often fought back charges that they were enablers of Hitler and contended that they had little choice but to follow state orders lest they lose control of their companies. While such accounts were incredibly self-serving, they were not entirely false or complete fabrications. The Third Reich had a poorly thought-out and contradictory approach towards capitalism that made its corporate policy quite capricious.
Goebbels would famously declare that the National Socialism was not against capital and warned Germans not "to confuse capital with capitalism." This inconsistency lay at the heart of Nazi economics and its approach to the distribution of wealth. National Socialist discourse on economics often valorized Germans who through their wits and scientific minds had managed to amass vast fortunes with their wares. The Krupp family was one such hero in the Nazi imagination, but there were others as well. In this formulation, German businessmen left monuments to German racial superiority in the form of massive concerns that brooked no rivals. But the flip side of the coin was that while there was proper German capital, the NSDAP had little reluctance in attacking what they saw as the predatory and exploitative aspects of capitalism. As an economic process, the Goebbels and the like excoriated business practices that left Germans vulnerable and enriched a few at the expense of many.
The problem for German industrialists and businessmen was that the new regime was far from clear as to what exactly constituted good and bad forms of capitalism. Firms like IG Farben or Krupp certainly used ruthless business practices and the cartelization of German heavy industry discouraged competition. The Third Reich certainly swung the axe against German department stores, which had a clear public perception of being owned by Jews. Financial concerns such as banks or insurance companies also felt the pressure of the state during the 1930s as their output was not in tons of steel or German goods, but largely invisible numbers on ledgers. Although the state came to a modus vivendi with German financial concerns, especially as they could both smooth out Aryanization of Jewish property and provided a conduit for laundering Nazi plunder, the relationship between the Third Reich and finance was an uneasy one with the state controlling the tempo of the relationship.
The dangers to business concerns of losing control were somewhat real and apparent. The example of Hugo Junkers was instructive and quite chilling example for those tempted to resist. Even though Junkers was a world-leading innovator in aviation, he was doubly cursed in the eyes of the Nazis for both being a prominent supporter of the SPD and a public advocate for pacifism (privately, it was more complicated, Junkers did sell illicit military aircraft to the Reichswehr and this was part of the leverage the Nazis used against him). Junkers's view of aviation is that developments in aviation technology were a better substitute for violent national competition. Such views had no place in the Third Reich even though its leadership was incredibly air-minded. Soon after Hitler's seizure of power, the state placed Junkers under protective custody and forced him to sign over control of the company to the state. The Junkers case was exceptional, but nothing in Nazi discourse suggested that it was a major break in their policies. Other firms and concerns saw that they had to accede to various demands of the state such as the Aryanization of management and gearing up production for military purposes. Going along with state directives meant retaining some control over their own companies, and this was especially true for vulnerable concerns like finance or foreign-owned companies. In the latter cases, the non-German owners and shareholders of German concerns found themselves increasingly locked out of day to day management on these companies and new laws prevented the remittance of profits outside of the Reich. Companies like Ford, IBM, or GM may have had paper control over some German concerns or subsidiaries, but by 1939/40, they were de facto Germanized.
But if the some firms feared the Nazi stick, the carrot it offered was quite lucrative. German heavy industry was among the most cartelized and concentrated in the world, and the onset of the Depression only encouraged this trend as form of protection. A good portion of heavy industry could be used for rearmament and the state's priority of rearmament meant there was little in the way to check cartel expansion. The new turn in politics also suited many industrialists' own political beliefs. Most of Germany's hyperrich were quite conservative in outlook, and although they were late-comers to the Nazi cause- most preferred to support the mainstream conservative DNVP- Hitler's politics did align with their own political interests to an alarming degree. The anti-labor and anticommunism of industry's leadership was one of the forms of ideological common ground industry had with the new regime, and the replacement of labor unions with the National Socialist Deutsche Arbeitsfront (DAF) and destruction of both the KPD and SPD broke the back of German organized labor. Nazi organizations like the RAD allowed for managers to assign shopfloor malcontents to punitive labor as a disincentive to wildcat striking. But beyond politics, the plunder economy of the Third Reich offered industrialists the chance to pick plum Jewish, and later European property that would enhance the personal and companies' wealth.
Yet even active participation in Nazi economic came with costs that some industrialists began to regret over time. While organized labor disappeared, the DAF represented a type of cronyism and corruption that ate into overhead far more than organized labor did. The endemic corruption of the NSDAP also meant that plunder was rarely evenly distributed. Steel and coal concerns tried to use the 1940 conquests to create massive concerns, but found them running into pressure not only from other German firms scrambling for their share, but friction from a state that did not want to give too much away. Rearmament also skewed profits and ordinary business models to where they were unrecognizable. Profits from state orders was tightly regulated and predicated upon coordination with rival firms and state invective against war profiteering was a dangerous reminder of the limits of the power of money. Additionally, rearmament wrecked havoc on the domestic economy as production weakened the domestic market as Germans worked more and found less to buy. The war heightened some of these problems, especially as the state started using slave and dragooned labor to make up for shortfalls. This led to the SS, which ran the camp system, to become a player in German business. Firms had to pay the SS for the costs of this labor and this added a further level of corruption to the system.
The SS's control over non-German labor gave postwar German businessmen a continent scapegoat to claim that they bore no real responsibility for the excesses of German industry during the war. Such a narrative was useful in the postwar period as industrialists and business leaders presented themselves as decent patriots whose anticommunism led them to be hoodwinked by Hitler. These excuses found a very receptive audience among the American occupation government, especially as the Cold War geared up and it became important to restore West Germany's economy. But this excuse obscures the many times that the interests of corporations and those of the Third Reich aligned. This arrangement was never equal and politics enjoyed primacy over economics, but more than a few business leaders were active participants in the Third Reich's policies.
Sources
Nicosia, Francis R., and Jonathan Huener. Business and Industry in Nazi Germany. New York: Berghahn Books, 2004.
Tooze, J. Adam. The Wages of Destruction: The Making and Breaking of the Nazi Economy. New York: Viking, 2007.
Wiesen, S. Jonathan. West German Industry and the Challenge of the Nazi Past, 1945-1955. Chapel Hill, NC: University of North Carolina Press, 2001.
_. Creating the Nazi Marketplace: Commerce and Consumption in the Third Reich. Cambridge: Cambridge University Press, 2011.
Corporations were exceptionally powerful in Germany at the time. The best example of the complex relationship between government and corporation in Germany is that of IG Farben.
IG Farben was founded in the 20's as a merger of several different pharmaceutical and chemical corporations (the ones you're most likely to have heard of are BASF and Bayer, which are the first and tenth largest chemical corporations in the world). IG Farben and its predecessors were perhaps most noteworthy (in terms of strategic production) for their antibiotics and synthetic dyes prior to the war; during the war they were also Germany's primary source of rubber, since Germany obviously couldn't import the stuff and is far away from where rubber trees grow naturally. IG Farben was also (perhaps) the single largest contributor to Hitler's campaign warchest.
IG Farben was so important to the war effort that it quickly became involved in strategic planning. When German soldiers rolled over much of Europe and the seized properties and factories were doled out, IG Farben essentially "had dibs" on whatever they wanted.
Nowhere is this more obvious than IG Farben's involvement in the Holocaust. There are plenty of mentions of "work camps" in the historical record. Camps like Auschwitz and Birkenau didn't pop up where they did for no particular reason; they were erected to service IG Auschwitz, a sprawling factory complex that produced Buna-S (synthetic rubber). IG Farben also produced experimental vaccines that were administered to prisoners who had a particular illness. The satellite camps of IG Auschwitz were used for tuberculosis vaccines (and probably others).
IG Farben is a bit of an outlier given how utterly intertwined with the German government, but it does show how powerful a corporation could be. They influenced who got the industrial plunder during an occupation, were able to requisition tens of thousands of prisoners or undesirables for forced labor, were able to use the SS as their own personal security details (and executioners), and IG Farben's own war contributions were always contingent on post-war production (IG Auschwitz was supposed to produce plastics after the war ended, for instance; synthetic rubber just so happened to use similar equipment), with the German government sometimes going as far as to guarantee prices so that IG Farben would be able to turn a profit (such an agreement was inked in '34 and good for ten years, but the war interrupted it).
IG Farben wasn't the only such corporation to profit. Another noteworthy one is Krupp. Krupp Industries was best known for its steel products. Krupp produced everything from tank barrels to U-Boats. That tradition continues today; ThyssenKrupp oversees a marine division that includes Blohm und Voss (which also made U-Boats back in WWII and is responsible for most of Germany's warship exports today) and HDW (which runs the old Krupp shipyard and produces Germany's submarines). But Krupp couldn't make all of these things using just domestic German manpower. It, like IG Farben, required huge numbers of slaves to staff its facilities. Figures vary here, but the most reliable one I've seen is 100,000, representing about 40% of the total Krupp workforce (in contrast to closer to 80% at IG Auschwitz). In return for Krupp continuously expanding production and ensuring that frequent bombing campaigns didn't interrupt work, Alfried Krupp was paid exceptionally well. He liquidated some of his government assets and stored them overseas, hedging against Germany's defeat, and was able to ensure his name and company would continue. Even after Nuremberg, which initially stripped Alfried Krupp of his holdings and wealth (what they could find, anyway), he was able to weasel his way into an amnesty and eventually reclaimed much of the industrial empire he lost. By 1960, Alfried Krupp was the wealthiest man in Europe.
Krupp and IG Farben flourished under Hitler's tenure. These were corporations of such power and influence that they were essentially inviolate. It's telling that, after trying the most infamous members of the Nazi government, that the Allies made a point to try and convict Krupp and IG Farben. They were given that kind of attention because, by 1945, there wasn't much to separate those companies from the government.
A follow up question (I hope this isn't a rules violation) but didn't German and Italian fascism have an explicit corporatism/corporatism corporativism component, i.e.: you couldn't have that flavour of fascism without the private/government cooperation. I even recall a Mussolini quote from maybe the 20s? where he stated something to the effect that fascism should be called corporatism.
Edit: fixed an autocorrect error.