On finite groups with given maximal subgroups

We prove that a finite group whose every maximal subgroup is simple or nilpotent is a Schmidt group. A group whose every maximal subgroup is simple or supersoluble can be nonsoluble, and in this case we prove that its chief series has the form $1\subset K\subseteq G$, $K\simeq PSL_2(p)$ for a suitable prime $p$, $|G:K|\le2$.