Reference for the subgroup structure of $\rm{PSL}_2(q)$

There are some notes by Oliver King containing a statement of the full classification in modern terms. However, this expository paper does not derive the result. A standard reference for the subgroup structure of classical groups is the book by Kleidman and Liebeck, but I don't recall that they cover Dickson's full list. They focus on maximal subgroups. The exposition there is rather, shall we say, "efficient".

Suzuki's Group Theory (I) 3.§6 page 392-418 is modern and very clear. The main theorem is on page 404, which coincidentally is the error code from google books for its page scan.

Two other modern references for the maximal subgroups of ${\rm PSL}(2,q)$ are: Bray, Holt and Roney-Dougal, The maximal subgroups of the low-dimensional finite classical groups, London Mathematical Society Lecture Note Series, vol. 407, 2013, and Michael Giudici, Maximal subgroups of almost simple groups with socle ${\rm PSL}(2,q)$, arXiv:math/0703685.

It's also covered in Gorenstein's Finite Groups (ironically enough, also in section 8 of chapter 2, just like Huppert, but I think this is coincidence).