Let $G$ be a finite group with the following property:

For every nontrivial proper subgroups $H$ and $K$ for which $H\cap K=1$, if the number of their minimal subgroups are $m$ and $n$ respectively, then the number of minimal subgroups of the subgroup generated by $H$ and $K$,$\langle H,K\rangle$, is $m+n$.

What are the possible structures for $G$? I found only cyclic groups and direct product product of a generalized quaternion group and a cyclic group of some odd order. Is there any other case?