Reference: P3 Events
As reference for when Souji encounters other P3 guys. Will expand more on it later, but for now...the basic and stuff.
The Kirijo Group is both a business and a yakuza/mafia group, as it wasn't out of the ordinary for yakuza groups to engage in legitimate dealings. They maintain close ties with Nyx's Family and branch, though they were getting distant since Takeharu Kirijo was in power.
When Minato Arisato moves to Iwatodai, he caught the attention of Mitsuru Kirijo and was recruited to be a part of her bodyguards and team. He was also elected as the leader for the more physical aspects, while Mitsuru deals with the business side of things.
After a while of dealing with punks, handling disputes and generally being badass, Ryoji Mochizuki transfer into school and become ~*bestest buddies*~ with them. Well, Minato and Junpei, at least. Aigis consider him as 'dangerous', but doesn't accurately recalls why.
Then, it's revealed that Ryoji's the future boss of Nyx's Family and her son.
And that Minato, somehow, is also a candidate for the title.
Succession crisis ensure.
All while Strega, a sub-group of Nyx fanatics, want Ryoji to lead and to rule the underworld without the softening Kirijo's touch.
Afterwards, when everything's all settled down and all, Nyx approved of Minato and recommends them to be co-bosses. YAY!
...
Then Minato died.
Yeah.
So, Aigis was selected as his substitute since no one would expected a ROBOT! Yukari is unamused, but The Answer didn't happened, since Nyx is human.
So Ryoji (Thanatos) and Aigis (Pallas Athena) are co-bosses of Nyx's Family and leads Mitsuru's bodyguards. ALL IS WELL.