<a href="http://mathoverflow.net/questions/91649/subgroups-of-gl2-q">This question</a> on subgroups of $GL(2,q)$ asked by <a href="http://mathoverflow.net/users/18394/jan">Jan</a>, and especially wonderful answers to it given by <a href="http://mathoverflow.net/users/14450/geoff-robinson">Geoff Robinson</a>, <a href="http://mathoverflow.net/users/10194/ralph">Ralph</a>, and <a href="http://mathoverflow.net/users/18060/will-sawin">Will Sawin</a> showing that "almost no finite groups" inject in $GL(2,q)$ made me wonder (completely recreationally, I have to admit) whether there exists $N$ such that every finite group, or "most finite groups" inject in $GL(N,q)$. 

Probably no such $N$ exists, but the ideas I had when thinking about the $N=2$ case use the specifics of the $2\times2$-situation way too much. Is it true, for instance that, along the lines of Ralph's and Will's answer, an abelian $p$-subgroup of $GL(N,q)$ may only have a bounder number of cyclic factors?