Is there some sort of classification of all minimal non-cyclic groups?

Question

Does there exist some sort of classification of all minimal non-cyclic groups (non-cyclic groups, such that all their proper subgroups are cyclic)

I know the following classes of such 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 $r ≡ 1 \pmod q$, $r^q ≡1 \pmod p$.

(These three classes completely cover the case, when our group is finite: Classification of finite minimal non-cyclic group)

4)$C_{p^{\infty}}$, where $p$ is a prime

5)$(\{ \frac{n}{p^m}| m, n \in \mathbb{Z} \}, +)$, where $p$ is a prime

(These two classes completely cover the case, when our group is infinite abelian: Does there exist an infinite non-abelian group such that all of its proper subgroups become cyclic?)

6)Infinite non-abelian groups, such that all their nontrivial proper subgroups are isomorphic to $C_{p}$ for a fixed prime $p$ (Tarski monster groups)

7)Infinite non-abelian groups, such that all their nontrivial proper subgroups are isomorphic to $C_{\infty}$ (Does there exist an infinite non-abelian group, such that all its nontrivial proper subgroups are isomorphic to $C_\infty$?).

However, I do not know, whether there exists anything, that does not fall into these classes. I only know, that if such groups exist, they have to be infinite non-abelian.

Answer 1

In his book "Geometry of Defining Relations in Groups" A. Ol'shanskii also proves Theorem 28.2 (which is more general than the existence of Tarski monsters):

For every sufficiently large odd number $n$ there is a 2-generator infinite group $G(n)$ such that every maximal (proper) subgroup of $G(n)$ is cyclic of order $n$.

Now, take a non-prime odd number $n$ and apply this theorem. You obtain a 2-generator group where every proper subgroup is cyclic, but orders of these cyclic subgroups are nonconstant (depend on the subgroup).

With more work, using the same methods, one can get examples where every proper subgroup is cyclic but some of the cyclic subgroups are finite and some are infinite.

Lastly, I would not call your list (even expanded to add groups as above) a "classification" since there is a continuum of examples of such groups and their overall structure is very unclear.