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".
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.