Let G be an almost simple group whose socle is an alternating, classical, or sporadic group, and let H be a non-parabolic maximal subgroup of G. We prove that any maximal subgroup M of H can be generated by at most 7
Nearby in the stack
elements, and that this bound is sharp when the socle of
G
is alternating or classical; this improves the bound of
12
due to Burness, Liebeck and Shalev. When the socle is sporadic, at most
5
generators suffice, and this is again best possible. The proof relies upon a detailed structural analysis of
H
and
M
, especially when
H
is a subgroup of a wreath product. In particular, we determine the chief factors of
M
and subsequently bound its number of generators using the theory of crowns. We also correct the classification of maximal subgroups
H
of almost simple groups requiring more than three generators, established by Lucchini, Marion and Tracey. Their bound of five generators remains valid, but their classification is missing several pairs