Large subgroups of simple groups

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⩾|G||H|3⩾|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.