TITLE: Classification of finite minimal non-cyclic group
QUESTION [8 upvotes]: I am looking for a complete classification of minimal finite non-cyclic groups. Is there any paper or book?

REPLY [6 votes]: A theorem by G.A.Miller and H.C.Moreno states:

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)