[Math] Finite groups in which all proper subgroups are cyclic

Question

Is there any classification of finite group in which all proper subgroups are cyclic?

Would you please tell me a reference?

Answer 1

All subgroups of a cyclic group (including the proper ones) are cyclic. And there is the following classification of non-cyclic finite groups, such that all their proper subgroups are cyclic:

A finite group $G$ is a minimal noncyclic group if and only if $G$ is one of the following groups:

1) $C_p × C_p$, where $p$ is a prime

2) $Q_8$

3) $\langle a,b | a^p = b^{q^m} = 1, b^{−1}ab = a^{r}\rangle$, where $p$ and $q$ are distinct primes and $r ≡ 1 \pmod q$, $r^q ≡1 \pmod p$.

This theorem first appeared in "Non-abelian groups in which every subgroup is abelian" by G.A.Miller and H.G.Moreno (1903)