Being more direct the SGDL do not usually use their powers or try to conquer the world.
Belphemon as already explained seems to like his punishment and prefers to be asleep, but by habit he wakes up every 1000 years to cause destruction and then returns to sleep.
Lilithmon says she prefers to stay in the Digital World, so it does not help Barbamon in her plan to conquer all worlds. (She even claims in N0 that she has opposing preferences of other GDLs).
Beelzebumon is the furthest away from the SGDL (He states it in N0) and usually does not cause injustice. Beelzebumon even helped Taiga and Rina in the fight against Barbamon. If I'm not mistaken, he mentioned Decode already had partners, but I'm not sure.
Leviamon despite his description of villain, prefers to just sleep and eat. In fact he helps the protagonist in Decode because he wanted to eat, and at N0 he just stood on the side of Demon and Barbamon because they had promised food to him. In Xros Wars Leviamon even helps the guardians of the digital world in the fight against the Bagra Army (which had the help of Lilithmon, who apparently was in it for loving Bagramon).
Lucemon has three personalities. He can be neutral and just observe everything that happens in the worlds and then judge those beings who call themselves good or evil. He can be good and rule the world with benevolence or he can be evil. If Lucemon is harmed in an "unfair" way he can go crazy and want to cause destruction (This is what happened in Frontier and Decode). In Next 0rder Lucemon explains this as being a moment when he gets "crazy".
Demon and Barbamon are the only ones who are "pure evil". In Next 0rder they even complain that despite the group being the "Seven Great Demon Lords" only the two seemed to care.
@SomebodyData. The Digital World System is the Host Computer, usually it is
Yggdrasil (Digimon).