Suppose $G$ is a group. Let’s define the cyclicality index of $G$ using the following recurrent relation:
$$CI(G) = \begin{cases} 1 & \quad G \text{ is cyclic} \\ \max_{H < G} CI(H) + 1 & \quad G \text{ is non-cyclic} \end{cases}$$
The finite groups $G$, such that $CI(G) = 1$ are exactly the cyclic groups.
The finite groups $G$, such that $Cl(G) = 2$ are of the following three classes:
1) $C_p × C_p$, where $p$ is a prime
2) $Q_8$
3) $\langle a,b \mid a^p = b^{q^m} = 1, b^{−1}ab = a^{r}\rangle$, where $p$ and $q$ are distinct primes and $r ≡ 1 \pmod p$, $r^q ≡1 \pmod p$.
But what can be told about finite groups $G$, such that $CI(G) = 3$?
By the classification of finite abelian groups, one can see, that if $G$ is abelian, it should be either $C_p \times C_p \times C_p$ or $C_p \times C_{pq}$ or $C_p \times C_{p^2}$, where $p$ and $q$ are distinct primes.
Non-abelian case remains obscure to me, however I know the examples of such groups: $D_{pq}$ and $D_{p^2}$, where $p$ and $q$ are distinct primes. However, they are definitely not alone...
So, my question is: Is there a way to fully classify all finite groups $G$, such that $CI(G) = 3$? (In a way similar to the classification of groups $G$, such that $CI(G) = 2$)
