I am taking a course in representation theory of finite groups, and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and $\mathrm{SL}(2,9)$ with a view towards generalizing them for arbitrary primes. I followed Liebeck's book Representations and Characters of Groups and though I find it an extremely beautiful and inspiring book I would be glad if I had more resources on these. (Specifically, what is the general strategy of finding subgroup structure aside from determining the trivial ones; also how does the fact that 7 is a prime and 9 a prime power change the scenario?
Also, how does representing these groups over $\mathbb{C}$ help us to visualize the groups better than what they are? And what is so special about these groups? And like the special linear groups over $\mathbb{R}$ or $\mathbb{C}$ is there any geometric interpretation of these groups?
Thanks in advance.
EDIT: I can see subgroups of order $p,\ p{-}1$ and $p{+}1$ inside $\mathrm{SL}(2,p)$:
The existence of a subgroup of order $p$ is guaranteed by Sylow's theorem, the matrix $\begin{pmatrix} a & 0 \\ 0 & a^{-1} \end{pmatrix}$ where $a$ is a nonzero element of the field gives us a subgroup with cardinality $p{-}1$, and subgroups of order $p{+}1$ can be constructed from action of $\mathrm{GL}(2,p)$ on a two dimensional vector space over the finite field.
Are there any more obvious possible subgroups? And is there anything special about $\mathrm{SL}(2,7)$?