For any number field $K$ with non-elementary $3$-class group ${\mathrm{Cl}}_3\left(K\right)\simeq C_{3^e}\times C_3$, $e\ge 2$, the punctured capitulation type $\varkappa \left(K\right)$ of $K$ in its unramified cyclic cubic extensions $L_i$, $1\le i\le 4$, is an orbit under the action of $S_3\times S_3$. By means of Artin’s reciprocity law, the arithmetical invariant $\varkappa \left(K\right)$ is translated to the punctured transfer kernel type $\varkappa \left(G_2\right)$ of the automorphism group $G_2=\mathrm{Gal}({\mathrm{F}}_3^2\left(K\right)/K)$ of the second Hilbert $3$-class field of $K$. A classification of finite $3$-groups $G$ with low order and bicyclic commutator quotient $G/G^{\text{'}}\simeq C_{3^e}\times C_3$, $2\le e\le 6$, according to the algebraic invariant...

We determine the finite groups whose poset of conjugacy classes of subgroups has breaking points. This leads to a new characterization of the generalized quaternion $2$-groups. A generalization of this property is also studied.