TITLE: is it true that finite groups with unique maximal subgroups are cyclic?
QUESTION [7 upvotes]: Assume that $G$ is a finite group. For any prime $p\mid |G|$, $G$ has a unique maximal subgroup of index $p$, is it true that $G$ is  a cyclic group or direct product of quaternion of order 8?

REPLY [11 votes]: The group $G = A_4 \times C_2$ has unique subgroups of index $2$ and $3$, namely $A_4$ and $C_2^3$.
Another example is $A_6 \times C_{15}$.