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 $G = \mathrm{PSL}(3,q)$



Maximal subgroups of odd index in $\mathrm{PSL}(3,q)$
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 ${\rm SL}\_3(q)$.


