Finite groups with three classes of maximal subgroups

Using the classification of finite simple groups, a description is obtained of finite groups having exactly three classes of conjugate maximal subgroups. If such a group is not solvable, then its factor group modulo its Frattini subgroup is isomorphic to $\mathrm{PSL}(2,7)$ or $\mathrm{PSL}(2,2^p)$, where $p$ is a prime. To prove this result, it was necessary to describe finite groups having at most two classes of conjugate nonnormal maximal subgroups.
Bibliography: 28 titles.