Finite Groups Whose Maximal Subgroups are 2-Nilpotent or Normal

Abstract We describe the structure of those finite groups whose maximal subgroups are either 2-nilpotent or normal. Among other properties, we prove that if such a group G does not have any non-trivial quotient that is a 2-group, then G is solvable. Also, if G is a solvable group satisfying the above conditions, then the 2-length of G is less than or equal to 2. If, on the contrary, G is not solvable, then G has exactly one non-abelian principal factor and the unique simple group involved is one of the groups PSL₂ (p^2ᵃ) PSL 2 (p 2 a), where p is an odd prime and a 1 a ≥ 1, or p is a prime satisfying p 1 p ≡ ± 1 (mod~ 8) (mod 8) and a=0 a = 0.