Let $G$ be a finite simple group. Is it possible to find two distinct proper non-trivial subgroups $H_1$ and $H_2$ of $G$ such that $\langle H_1 , H_2\rangle\lneq G$ and for each maximal subgroup $M$ of $G$, if $H_i\nleq M$, then $H_j\leq M$ ($i , j\in \{1,2\}$)?
If it is dificult to answer the question in the class of finite simple groups, could we answer it in certain classes of finite simple groups? (such as alternating groups, minimal simple groups, ...). Many thanks.