The maximal subgroups of odd index in finite simple groups were classified by Liebeck and Saxl and independently by Kantor. 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 by Maslova.

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.