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This material is covered in detail in Dickson's "Linear Groups with an exposition of the Galois Field Theory", chapter XXII and Huppert's "Endliche Gruppen", chapter II, paragraph 8. Since I don't speak german and Dickson's treatment often requires deciphering, I was wondering if there is a "modern" account of this somewhere.

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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".

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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.

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    Thanks. I keep wondering why no one has ever translated Huppert's book though.2012-06-26
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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.

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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).