Large maximal subgroups of almost simple classical groups

Abstract A proper subgroup 𝐻 of a finite group 𝐺 is called large if | H | 3 ≥ | G | \lvert H\rvert^{3}\geq\lvert G\rvert . This paper characterises all the large maximal subgroups of almost simple groups whose socle is a finite simple classical group. For such a socle G 0 G_{0} and a core-free subgroup H 0 H_{0} , define Q ⁢ ( G 0 , H 0 ) \mathcal{Q}(G_{0},H_{0}) as the set of almost simple groups 𝐺 with socle G 0 G_{0} that have a large maximal subgroup 𝐻 satisfying H 0 = H ∩ G 0 H_{0}=H\cap G_{0} . We establish necessary and sufficient conditions on the pair ( G 0 , H 0 ) (G_{0},H_{0}) for the set Q ⁢ ( G 0 , H 0 ) \mathcal{Q}(G_{0},H_{0}) to be nonempty.