I don't know about criminal organisations in, say, Eastern Germany, so I'm just going to talk about the USSR itself.

The simple answer is the Vory v Zakone. The Vory v Zakone (it means something like "Thieves with a code") were originally a prison gang in the gulags who became a USSR-wide mafia (it should be noted that Vory generally refers to senior criminals, not the people at the bottom. Similar to the Mafia's "made men"). When you think of "the Russian Mafia" this is what you're thinking of. Elaborate initiation rituals, symbolic tattoos, oaths to never betray each other to the authorities, etc. If you've seen American TV programmes with Russian criminals sporting elaborate tattoos showing they are members of the Russian Mafia, that's where the tattoos come from (though, while they were somewhat elaborate, and did have meaning, such as showing how many people the Mafioso had killed, they were generally less elaborate than on TV, of course).

Here's some good examples of Vory tattoos: https://video-images.vice.com/articles/58477f6cace19e023cedaa42/lede/1490708414431-russian-criminal-tattoo-fuel-damon-murray-interview-876-body-image-1414835904.jpeg?crop=0.8743xw:1xh;0.0619xw,0xh&resize=1200:*

Note the common use of Orthodox and Soviet symbols and personages.

However... the issue when we discuss organised crime is most people (including the police, who have incentives to do so because it makes prosecution easier) have an exaggerated view of criminal organisations. If I say "what's the Japanese Mafia?" you'll say "The Yakuza." But there's no organisation called the Yakuza. There are multiple criminal organisations, with a shared culture, like the Yamaguchi-Gumi, Sumoyoshi-Kai, etc, which together are called "Yakuza".

Similarly, everyone knows that Italy has traditionally had a big problem with the Mafia. Except the Mafia has been pretty weak for a couple of decades. And in Naples, the Mafia was never a big problem: the Camorra was the problem. Today, the Mafia is a much less serious issue for police than the 'Ndrangheta. But because the Mafia was most famous, the word "Mafia" has morphed from being a term to describe the Sicilian criminal organisation to one that is used to describe any criminal gang. So when you see news articles like "Politician in South-East Italy revealed to have Mafia links" you'll think of the Sicilian Mafia, even though it's probably the 'Ndrangheta.

This is also the case with the Vory. It's better not to think of them so much as a USSR-wide organisation, but rather as a USSR-wide criminal culture. There was no "boss" of the Vory. If you go on google right now and type in "Boss of Bosses" it'll tell you that most law enforcement agencies believe that today Semion Mogilevich is the "boss of bosses" of the Russian Mafia. This is, frankly, incredibly misleading (I've read the internal FBI report on him). There is no "boss of bosses" and never has been.

Rather, across the USSR, different cities had local criminal gangs that had links to the prison gangs of the Gulags, and had similar initiation rituals (the "Crowning" ceremony where two other Vory declared you were now a Vory, for example), tattoos, etc. Vory were heads of gangs, and if they travelled to another city the local criminals would recognise them as a big cheese in the Underworld, and possibly work together on a job. But there was no pan-USSR organisation taking orders from a central leader.

I should also note that the position of the Vory within Russian criminal culture changed radically after the collapse of the USSR, but you were only asking before the collapse.

The other thing to remember is that not all criminal organisations look like criminal organisations.

In the Brezhnev era, most cities in the USSR were in the grip of powerful criminal gangs. Groups of criminals worked together to defraud the people and the government. They blackmailed those who opposed them, smuggled illegal merchandise, ran brothels, etc. They didn't burgle houses or demand protection racket payments like the Vory; frankly, their crimes were much more serious and insidious.

The name of these groups? Soviet bureaucrats.

I don't mean this as a value judgement, that the Soviet state was illegitimate and criminal. I mean huge numbers of Soviet bureucrats worked with their fellows to commit crimes. Corruption was so common that when two prosecutors tried to prosecute senior Uzbek officials for defrauding the government, they realised that they would have to arrest the entire bureaucratic class of Uzbekistan. And this went all the way to the top. Brezhnev knew how common corruption was, and he just didn't care, because he was involved himself.