Very, very little research has been done on who the prisoners with the green triangles actually were and why they were imprisoned in a camp. One of the very few historical monographs on the subject comes from Dagmar Lieske and deals with criminal prisoners in Sachsenhausen (Unbequeme Opfer? "Berufsverbrecher" als Häftlinge im KZ Sachsenhausen).

Lieske shows just how heterogeneous the group of "criminal" prisoners in Sachenhausen was: The criminal category (meaning those with a green triangle) included both so-called "habitual" or "professional criminals" as well as "preventive prisoners". Given that these categories served as catch-all categories for a lot of people, it is not surprising that Lieske shows that among the prisoners in Sachsenhausen who wore a green triangle, there were people with prior criminal convictions, people transferred to Sachsenhausen from another prison after their sentence had run its course, homeless persons, people who went on strike, prostitutes, homosexuals, so-called "gypsies" and POWs who had committed a crime. Furthermore, the distinction between "criminal" and so-called "asocial prisoners" existed in Sachsenhausen only after 1938. Before that point in time, no distinction between the two groups was drawn and it therefore included both people sentenced for a crime as well as alcoholics, men who didn't pay alimony, and those "unwilling to work".

All in all, Lieske was able to identify about 9000 prisoners of Sachsenhausen who were classified with a green triangle, meaning both "professional criminals" and "preventive prisoners". Of those, e.g., more than a thousand were imprisoned because of sexual crimes, in the vast majority, meaning repeated convictions because of homosexual activity (about 800 of all, about 200 were mainly pimps and prostitutes with repeat convictions, and 53 people sentenced because of other sexual crimes).

Furthermore, most of those classified as "professional criminals" were far from what we imagine when we hear that term. They were not professional hitmen or members of organized crime (though some of them were also imprisoned in the camps). Rather, most "professional criminals" were petty criminals with repeat convictions as Lieske shows in her book, most of them sentenced and then transferred to a concentration camp after petty property crimes.

Also, the point about the prisoners with the green triangles being close to the camp administration and Kapos is largely a post-war narrative that has very little to do with the actual reality of the camps. Because they were not included in the solidarity communities of communist and socialist and other political prisoners, there was a distinct tendency after the war to demonize these priosners in post-war narratives.

So to really hammer this important point home: People imprisoned in Concentration Camps as criminals were in the vast majority people sentenced to harsh punishments in accordance with Nazi law and often transferred to a camp after they had served their sentence. In terms of what crimes they committed, the vast majority were homosexuals sentenced under §175 of the German Criminal Law on the one hand, petty criminals with several convictions on the other.

It is important to keep this in mind when talking about the criminal prisoners in the camps because it keeps the issue in perspective in that rather than talking about hardened criminals or mafiosi, criminal prisoners in the camps were people who ran afoul of the Nazi justice system that was in its essential character, unjust.

Essentially, we also have very few sources about prisoners with green triangles, f.ex. there are virtually no post-war biographies and that has to do with what happened after the war. So what happened when the Allies arrived?

The same thing that happened to homosexual and "asocial" prisoners: They were often re-imprisoned:

The Handbook for Military Government in Germany Prior to Defeat or Surrender (published in 1944) specified that after liberation, one of the first duties of the Allied troops was to separate the victims of Nazi persecution into different and predominantly national categories, a huge and in practice incomplete feat that not only lead Jews to protest (for they wanted to be grouped in one category rather than their national category) and that lead the predominantly German category of victims of social persecution (asocials, homosexuals, criminals) to be grouped in the "criminal" category because their arrest and imprisonment was actually based upon laws.

The procedure as laid out in the Handbook for this group of victims was explicit: "Ordinary criminals with a prison sentence still to serve will be transferred to civil prisons." Meaning that if somebody convicted under §175 or any other law in effect under Nazi rule, which held a provision for imprisonment for up to 10 years, and imprisoned in a Concentration Camp could be imprisoned by the Allies if they believed that the person had not served their sentence in full. For those who had "served their sentence", freedom was guaranteed but especially for homosexuals, the fear of re-imprisonment remained. I go into the details on homosexuals more here.

Similarly, while former criminal prisoners had been imprisoned often unjustly and even if they had been classified as "having served their sentence", they still retained a criminal record based on their sentence in a camp, whether they had been sentenced or not. This in turn also prevented them from receiving money in reparation for what had been done to them or from applying for jobs etc. after the war.

In a certain sense, they continued to be victimized both by the Allies as well as the two German states which arose from the ruins of Nazi Germany. As I write in another answer on the subject of victims post-war:

Far from the "happily ever after" ending, victims of Nazi persecution continued to suffer mentally, psychologically, physically, and financially from the time of their victimization by the Nazis. While liberation meant immediate safety from violent death, it was from a liberation of the suffering and its consequences the Nazis inflicted on their victims. A suffering that resulted not only in grand political changes but affected millions of people who had been subjected to Nazi violence in the course of their whole lives.