Let $p$ be a prime number. By a **Cartan subgroup** of $GL_n(\mathbb{F}_p)$ I mean an absolutely semisimple maximal abelian subgroup.

When $n=2$, it is well-known* that, for $G \subset GL_n(\mathbb{F}_p)$ of order prime to $p$, either $G$ is contained in a Cartan subgroup, or it is contained in the normalizer of a Cartan subgroup, or its image in $PGL_n(\mathbb{F}_p)$ is isomorphic to $A_4$, $A_5$, or $S_4$.

I would like to know if there is a similar result for larger even values of $n$;

where a subgroup of order prime to $p$ would either be contained in the normaliser of a Cartan, or its projective image be one in a finite list of groups.

*See e.g. section 2 of Serre's "Propriétés galoisiennes..." paper.

`$n$`

will get arbitrarily hard for list-making. – Jim Humphreys Apr 26 '12 at 21:34`$n=2$`

will occur for larger`$n$`

. The subgroup structure rapidly gets much more complicated. – Jim Humphreys Apr 26 '12 at 21:36