Consider, $\text{GL}(n,q)$, which is the group of $n\times n$ invertible matrices with entries from the finite fields $\mathbb{F}_q$, with $q$ elements.(We know $q$ is a prime-power). My question is really a broad one. What is known about maximal subgroups of $\text{GL}(n,q)$. By maximal subgroup, I mean the following: For a group $G$, a subgroup $H\leq G$ is maximal if there doesn't exist any proper subgroup of $G$ which properly contains $H$.
Is there any method by which one can construct maximal subgroups in $\text{GL}(n,q)$. The group $\text{SL}(n,q)$ is the subgroup of matrices with determinant 1. What are all maximal subgroups of $\text{GL}(n,q)$ containing $\text{SL}(n,q)$, and so on. Therefore, What I basically want to know is what all is known, or if there is any very well-known result in this topic.
P.S- I searched a lot in the internet, but don't seem to find much on this or maybe there is no clear answer, I don't know. Any references will be highly appreciated.
Thanks in advance.
The factor group ${\text GL}(n,q)/{\text SL}(n,q)$ is the cyclic group $\Bbb{F}_q^*$ of order $q-1$ because the subgroup ${\text SL} (n,q)$ is the kernel of the $\text{det}$ homomorphism.
So maximal subgroups of ${\text GL}(n,q)$ containing $\text{SL}(n,q)$ are in one-to-one correspondence with the maximal subgroups of that cyclic group and therefore with the prime divisors of $q-1$.
Update 1. There are maximal subgroups which are not above $SL(n, q)$. For example, any maximal subgroup containing all upper triangular nonsingular matrices.
Update 2. See also this text, sections 5-10 for a description of maximal subgroups of ${\text GL}(n,q)$.