In General linear group $GL(n,q)$, every matrix is conjugate with its rational canonical form. Using this fact we have the conjugacy classes of cyclic groups. My question is about the groups which are generated by two elements. I tried to find some results about conjugacy classes but it seems to be hard, any comment is so appreciated.



subgroups of General linear group with two generators
You would have an answer to your question if you could classify pairs of elements in $GL\_n(\mathbf F\_q)$ up to simultaneous conjugacy. This is the notorious matrix pair problem, which is the quintessential wild problem in representation theory (see [When is a classification problem "wild"?](https://mathoverflow.net/questions/10481/when-is-a-classification-problem-wild) , [Are wild problems related to undecidable ones?](https://mathoverflow.net/q/73940/9672) and [How can classifying irreducible representations be a "wild" problem?](https://mathoverflow.net/q/156156/9672)).
So you are unlikely to get a nice answer for general $n$.


