Maximal subgroups of small index of finite almost simple groups

We prove in this paper that every almost simple group R with socle isomorphic to a simple group S possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index 1(S) of a maximal subgroup of S or a conjugacy class of core-free maximal subgroups with a fixed index v(S) <= 1(S)(2), depending only on S. We also prove that the number of subgroups of the outer automorphism group of S is bounded by log(3) 1(S) and 1(S)(2) < vertical bar S vertical bar.