Finite Groups Whose Maximal Subgroups are 2-Nilpotent or Normal

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(p2ᵃ) 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 .