Cyclic normal is not join-closed

Statement
It is possible to have a group $$G$$ with two cyclic normal subgroups (i.e., subgroups that are both cyclic and normal) whose join is not a cyclic normal subgroup.

Since the join is always a normal subgroup, the particular thing that fails is that the join is cyclic.

Related facts

 * Abelian normal is not join-closed
 * Abelian characteristic is not join-closed
 * Normality is strongly join-closed

Example of the quaternion group
In the quaternion group, the subgroup generated by $$i$$ and the subgroup generated by $$j$$ are both cyclic normal subgroups, but their join, which is the whole group, is not cyclic.

Example of a prime-cube order group for odd prime
In the non-Abelian group of order $$p^3$$ obtained as a semidirect product of a cyclic group of order $$p^2$$ and a cyclic group of order $$p$$, there are many different cyclic normal subgroups of order $$p^2$$ (in fact, there are $$p$$ of them). The join of any two of these is the whole group, which is not cyclic.