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Let $p$ be a prime number, and denote by $\mathbb{F}_p$ the field with $p$ elements.

Is there a classification of the maximal subgroups of $G = \mathrm{SL}_3(\mathbb{F}_p)$ ?

I am interested in the indices of these subgroups in $G$ and in their ranks (minimal cardinality of a generating set).

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    $\begingroup$ Yes, by Mirchell in 1911 for odd $q$ and by Hartley in 1925 for even $q$. There a table in my recent book with Bray and Roney-Dougal on maximal subgroups of classical groups in low dimensions. $\endgroup$
    – Derek Holt
    Commented Sep 28, 2015 at 9:36
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    $\begingroup$ @DerekHolt I would gladly accept an answer of yours explaining these things. I could not find a full version of your book available online, and I am not sure where exactly in the book can I find the result. Also, the 1911 paper of Hartley is out of my reach. Thanks a lot! $\endgroup$
    – Pablo
    Commented Sep 28, 2015 at 10:12

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The original references are:

H.H. Mitchell. Determination of the ordinary and modular ternary linear groups. Trans. Amer. Math. Soc. 12 (1911), 207-242.

R.W. Hartley. Determination of the ternary collineation groups whose coefficients lie in the $\mathrm{GF}(2^n)$. Ann. of Math. 27 (1925/6), 140-158.

Our book is

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, John N. Bray, Derek F. Holt, Colva M. Roney-Dougal, London Mathematical Society Lecture Notes 407, CUP, 2013.

I hope you won't find the complete book online, because that might adversely affect its sales! If you send me an e-mail, I can send you the table for ${\rm SL}(3,q)$ from the book.

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    $\begingroup$ Sad to say, it IS available. I won't say there, to safeguard your sales, but it is not hard to find. $\endgroup$
    – Igor Rivin
    Commented Sep 28, 2015 at 11:23
  • $\begingroup$ @DerekHolt Sadly enough, I am still having trouble with extracting the information I need from the tables - It seems that I will have to go through the whole book if I want to follow the notation. All I want is to know (at least roughly) the ranks of the maximal subgroups. I am really glad to see that a classification exists (this is my first question) but I would also like to have an answer for my second question: "I am interested in the indices of these subgroups in $G$ and in their ranks (minimal cardinality of a generating set)." $\endgroup$
    – Pablo
    Commented Sep 28, 2015 at 12:46
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    $\begingroup$ Chapter 8 containing the tables was designed to be readable independently of the rest of the book (and we suspected that most readers would probably only look at Chapter 8), so you only need look at Chapter 8. All of the subgroups are 2-generated except for $(q-1)^2:S_3$, which required $3$ generators when $q \equiv 1 \bmod 3$. Calculating the indices in $G$ should be routine. I am, happy to answer specific questions on particular subgroups. $\endgroup$
    – Derek Holt
    Commented Sep 28, 2015 at 14:45
  • $\begingroup$ @DerekHolt This is great! My questions regarding maximal subgroups are now settled. If I want to find a (not necessarily maximal) subgroup $H \leq \mathrm{SL}_3(\mathbb{F}_p)$ with highest rank possible. Which one should I take? $\endgroup$
    – Pablo
    Commented Sep 28, 2015 at 15:03
  • $\begingroup$ @DerekHolt as I have indicated twice in my last comment, I am now interested in arbitrary (NOT maximal) subgroups. I appreciate your help a lot. Thanks! $\endgroup$
    – Pablo
    Commented Sep 28, 2015 at 15:44

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