Finite Groups with Few Almost Maximal Subgroups

Let $G$ be a finite group. L. Morgana, C. E. Praeger, A.E. Zalesski \cite {MOR} introduced the notion of almost maximal subgroup of $G$. A subgroup $H$ of $G$ is said to be almost maximal in $G$ if $N_G(H)$ is maximal in $G$. In this paper, let $AM(G)$ denote the number of conjugacy classes of all almost maximal subgroups of $G$. First, we prove that $G$ is Dedekindian if and only if $AM(G)=0$. Second, we show that the finite groups with $AM(G)\leq 1$ are supersolvable. Furthermore, we characterize the structure of finite groups with $AM(G)=1$. Final, we investigate a lower bound of $AM(G)$ for certain non-solvable groups.