  For a finite group $G$, let $d(G)$ denote the minimal number of elements
required to generate $G$. In this paper, given a finite almost simple group $G$
and any maximal subgroup $H$ of $G$, we determine a precise upper bound for
$d(H)$. In particular, we show that $d(H)\leq 5$, and that $d(H)\geq 4$ if and
only if $H$ occurs in a known list. This improves a result of Burness, Liebeck
and Shalev. The method involves the theory of crowns in finite groups.
