Benjamin Leontief Alpers, from his book "Dictators, Democracy, and American Public Culture":

"From the standpoint of the US government, the description of Nazism as an embodiment of the divide-and-conquer principle sent the perfect message: the defeat of Nazism depended first and foremost on unity."

It is, consequently, a common theme within the American popular imagination that the Nazi Party practiced "divide and conquer" - not only in the sense of setting their domestic political opponents against each other but also in the sense of geopolitics. Besides "Don't be a Sucker!", "Divide and Conquer", the third film in Frank Capra's "Why We Fight" war-time film series, directly paints this image, as does the 1944 "Weapons of War". Martin Niemöller's now famous quote beginning in "First they came for the socialists..." also reinforces this image.

But such an understanding of Nazi strategy ignores the other potent political tactics employed by the Nazis, including populism, nationalism, coalition building, propaganda tactics, and the ability to make themselves palatable to major German constituencies. It also ignores the mistakes by their opponents, particularly the Communists, the Nazis' main opponents, and the German right, which ultimately entered into a coalition with the Nazis for their own gain. And of course, it ignores Germany's military, economic, and geopolitical success in the early years of Nazi rule, which rallied German society behind the Nazis and enabled them to pursue their more radical and sinister ambitions.

The rise of the Nazi Party is a complex topic that would take entire books to properly explain. But to begin to address your question: "divide and conquer" is probably not the best description of how the Nazis rose to power. As Oded Heilbronner writes in "German or Nazi Antisemitism?":

The origins of Nazism, and hence Nazi antisemitism, lie in the crisis of Weimar society, which was reflected in a profound radicalization and politicization of that society. The radical populism which so typified the antisemitic peripheries in the period of the Second Reich gained a central position in the Weimar period as a result of the war, the revolution and the events of the early 1920s. Nazism was a general mart for all the social movements that had existed on the fringes of Wilhelmine society, had risen to prominence during the war and had become influential during the 1920s. But what was even more significant was that, in addition to the extremist currents that had infiltrated the party, Nazism in the period before 1933 also represented central streams of Wilhelmine society: national liberals, social Conservatism, Catholics and the socialist left. Democratic, conservative, liberal and Marxist ideas could be found within the party, together with calls for social and political reform under an authoritarian or populist democratic regime. ...

Most historians agree today that during the early years of the Third Reich ‘the war against the Jews’ was not the main goal of the new regime. ... The main task was to remove Jews from their positions in the state and the economy, but Jews could continue to live in Germany. As late as 1936, a Jewish funeral in a village in the southern Rhine region in western Germany could be accompanied by the heads of the local Nazi Party who had come to pay their last respects to an anonymous Jew. As Ulrich Herbert has recently claimed, the German people did not fanatically support anti-Jewish policy until the late 1930s. As late as 1938, the heads of the SS could oppose Kristallnacht on the grounds that the disturbances might be too violent. But, as the Second World War reached its climax and the Germans were mired in the mud and snow of Russia, and the reverses of 1942 showed the Nazi leadership that victory was turning into defeat, the heads of the Nazi state took it on themselves to order the ‘Final Solution of the Jewish Question’ as well as cruel and vindictive actions against other social and ethnic groups.

From the above description, the Nazis began as opportunistic populists. They took advantage of major economic and political crises in Germany, which were the result of World War I, reparations, etc., to offer an alternative to the "establishment." They promised jobs and bread, economic and geopolitical revitalization, and a restoration of traditional German values. Yes, they also attacked other groups, particularly the Weimar establishment, the Communists, and at times the Jews, but more importantly they were careful to present themselves and their policies as being beneficial for the vast majority of German society, so as to maximize their base of support. From "The Nazi Rise to Power":

Hitler and other Nazi speakers carefully tailored their speeches to each audience. For example, when speaking to businessmen, the Nazis downplayed antisemitism and instead emphasized anti-communism and the return of German colonies lost through the Treaty of Versailles. When addressed to soldiers, veterans, or other nationalist interest groups, Nazi propaganda emphasized military buildup and return of other territories lost after Versailles. Nazi speakers assured farmers in the northern state of Schleswig-Holstein that a Nazi government would prop up falling agricultural prices. Pensioners all over Germany were told that both the amounts and the buying power of their monthly checks would remain stable.

To insert a bit of common sense - it would have been impossible for the Nazis to consolidate German society through only "divide and destroy." They had to build their own support base, too, or else why would people vote for them, as they ultimately did in 1932, when the Nazis became and remained the biggest political party in Germany in two separate elections? The Nazis' path to power might have involved scapegoating and attacking the existing establishment as well as their opponents, but in the end, they were successful because they were able to present themselves, to the German public, as an appealing alternative to both the Weimar elites and, particularly, the Communists.