(Not religious disclaimer) Not sure why people misunderstand that he somehow rules 'hell'. "There's a special place in hell for mutineers and betrayers". That is where Lucifer among many others who betrayed in their time reside.

Judecca, named after the apostle who betrayed Jesus (Judas Iscariot), is the innermost zone of the ninth and final circle of hell. The term also hints at a manifestation of Christian prejudice--which Dante certainly shares--against Judaism and Jews in the Middle Ages: it alludes to the names--Iudeca, Judaica--for the area within certain cities (e.g., Venice) where Jews were forced to live, apart from the Christian population. Together with Judas in this region of hell are others who, by betraying their masters or benefactors, committed crimes with great historical and societal consequences. Completely covered by the ice--like "straw in glass"--the shades are locked in various postures with no mobility or sound whatsoever (Inf. 34.10-15).

Reference: http://danteworlds.laits.utexas.edu/circle9.html

If I remember correctly, Lucifer (Satan) is among these frozen beings eternally being devoured by the 'dragon'.

Can't say that's much a mantle of 'ruling or punishing' anyone.