  Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large
if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we
determine all the large maximal subgroups of finite simple groups, and we
establish an analogous result for simple algebraic groups (in this context,
largeness is defined in terms of dimension). An application to triple
factorisations of simple groups (both finite and algebraic) is discussed.
