Let $q = p^n$ be any prime power. It is possible to list all subgroups of $G = PSL(2,q)$. This is done for example in [Huppert - Endliche Gruppen]. However, the proof is quite long, and I am only interested in listing the maximal subgroups of $G$ - e.g. to convince myself that there is no nontrivial transitive group action of $G$ on less than $q+1$ elements if $q > 11$.
Does anybody know a reference, where only the maximal subgroups of $G$ are derived? Alternatively, can the standard proof of listing all subgroups somehow be shortened if only maximal subgroups are considered?
Possible Attempt: Since $G$ operates primitive on the projective line, the one point stabilzers are maximal subgroups (they are semidirect products of an elementary abelian $p$-group and a cyclic group). Hence, these are precisely the maximal subgroups with at least one fixed point, all other maximal subgroups are fixed-point-free. Maybe this information can be used to shorten the standard proof.