Final Goal: Given $p$, an odd prime, and $\mathbb F_p$, the corresponding finite field, enumerate the subgroups of $G=\mathrm{PGL}(2, \mathbb{F}_p)$, $H < G$, such that $|G|/|H|$ is odd.
Harder but seemingly doable goal: Given $p$ being an odd prime, enumerate the subgroups of $G=\mathrm{PGL}(2,\mathbb F_p)$.
Speculative idea: Maybe one can translate the result from the classification result of $\mathrm{PGL}(2,\mathbb C)$ to $\mathrm{PGL}(2,\mathbb F_p)$.