Abstract: Based on the index properties of maximal subgroups, we constructed a set of maximal subgroups with non-prime-power indices, provided some classifications of the associated second maximal subgroups, and discussed the structure of the group in depth by combining  their strength and  parity of the orders, obtaining several sufficient conditions for a group to belong to the related formations.

Key words: maximal subgroups with non-prime-power index, second maximal subgroup, non-solvable group, formation