Maximal subgroups of odd index in $\mathrm{PSL}(3,q)$

Question

Let $G = \mathrm{PSL}(3,q)$ for $q$ odd. I am trying to understand a question that involves understanding the subgroups that contain a Sylow $2$-subgroup, and in particular, are subgroups of odd index in $G$. I need to find a complete description of the maximal subgroups of odd index in the group $G = \mathrm{PSL}(3,q)$

Answer 1

The subgroups of ${\rm PSL}_3(q)$ for odd $q$ were first enumerated by H.H. Mitchell in 1911. (The case $q$ even was done by R.W. Hartley in 1925/6.)

Table 8.3 of the book "The Maximal Subgroups of Low-Dimensional Finite Classical Groups" by Bray, Holt and Roney-Dougal provides a convenient list. Using that it is not hard to answer your question.

For $q$ odd, the maximal subgroups of ${\rm SL}_3(q)$ of odd index are as follows:

(i) Two classes of maximal parabolic subgroups with structure ${\rm E}_q^2:{\rm GL}_2(q)$ and index $q^2+q+1$, where ${\rm E}_q$ denotes an elementary abelian group of order $q$ (the additive group of the field). These two classes are interchanged by the graph (inverse-transpose) automorphism of ${\rm SL}_3(q)$.

(ii) When $q \equiv 1 \bmod 4$, we have one class of imprimitive subgroups with structure $(q-1)^2:S_3$.

(iii) When $q = q_0^r$ for some odd prime $r$, we have $\gcd(\frac{q-1}{q_0-1},3)$ classes of subgroups with the structure ${\rm SL}_3(q_0).\gcd(\frac{q-1}{q_0-1},3)$. When there are three such classes, they are all conjugate under a diagonal outer automorphism of ${\rm SL}_3(q)$.

Answer 2

The maximal subgroups of odd index in finite simple groups were classified in Liebeck and Saxl - The primitive permutation groups of odd degree and independently in Kantor - Primitive permutation groups of odd degree, and an application to finite projective planes. In some cases the the subgroups listed need extra conditions to guarantee that they actually are of odd index. An explicit if and only if statement is given in Maslova - Classification of maximal subgroups of odd index in finite simple classical groups: Addendum.

Answer 3

A pretty complete and accessible description of this can be found in a survey article of Oliver King. In addition to the link, here's the citation info:

King, Oliver H., The subgroup structure of finite classical groups in terms of geometric configurations., Webb, Bridget S. (ed.), Surveys in combinatorics 2005. Papers from the 20th British combinatorial conference, University of Durham, Durham, UK, July 10–15, 2005. Cambridge: Cambridge University Press (ISBN 0-521-61523-2/pbk). London Mathematical Society Lecture Note Series 327, 29-56 (2005). ZBL1107.20035.