I have been trying to look at the maximal subgroups of $\mathrm{PSL}(2,p)$ for $p > 2$ prime and believe I have a sufficiently good idea of how those isomorphic to $C_p \rtimes C_{\frac{1}{2}(p-1)}$ and $D_{p+1}$ (using the convention that $|D_{2n}| = 2n$ rather than $4n$) look and interact with others, but I am having some issues with the following:
I'm not sure how to properly justify that the intersection of two distinct groups isomorphic to $D_{p-1}$ is either cyclic of order $\frac{p-1}{2}$ or consists only of involutions (I think it may only contain one involution or be trivial, but I don't know how to show this much either)
I have no idea how to go about finding the subgroups that crop up isomorphic to $A_4$, $S_4$ and $A_5$ and I can't try to look at them in GAP to figure it out, since GAP treats $\mathrm{PSL}(2,p)$ as a permutation group and I do not know how to translate this back into matrices.
I have been trying to look at the book on Maximal Subgroups of Low-Dimensional Classical Groups by Bray, Holt and Roney-Dougal but can't seem to find any indication as to what these subgroups actually "look" like.
I have also been trying to look at Aschbacher's classification to attempt to get a better idea of how they work, but the description of $\mathcal{C}_2$ as a stabiliser of a direct sum decomposition has me confused, since I am assuming in this case a subgroup $H$ stabilises a direct sum $V = V_1 \oplus V_2$ if $H V_1 \subseteq V_1$ and $H V_2 \subseteq V_2$ but then I am not sure how a dihedral group could do this (in the way I am thinking of them). This also has me similarly confused with the permutation groups since I am not entirely sure what an extraspecial group in an absolutely irreducible representation might be. I know what all of the individual terms mean, but that's about it!
Any help would be greatly appreciated!