Suppose $G$ is a finite group with a dihedral maximal subgroup. Suppose that $G$ is not isomorphic to $\operatorname{PSL}(2,q)$ for some any prime-power $q$. Is $G$ always solvable?



Finite groups with a dihedral maximal subgroup
Not necessarily . The Suzuki simple groups ${ \rm Sz}(2^{2n+1})$ also have dihedral maximal subgroups. These are the normalizers of the Hall subgroup of order $2^{2n+1}-1.$


