Divisibility-closedness is not finite-join-closed

Statement
It is possible to have a group $$G$$ and subgroups $$H,K$$ of $$G$$ such that both $$H$$ and $$K$$ are divisibility-closed subgroups of $$G$$ but the join of subgroups $$\langle H, K \rangle$$ is not a divisibility-closed subgroup of $$G$$.

Related facts

 * Powering-invariance is not finite-join-closed
 * Divisibility-closedness is not finite-intersection-closed

Proof
Suppose $$G$$ is the generalized dihedral group corresponding to the additive group of rational numbers. Let $$H$$ and $$K$$ both be subgroups of order two generated by different reflections. Then, the following are true:


 * $$G$$ is divisible by all primes other than 2.
 * $$H$$ and $$K$$ are both divisibility-closed subgroups on account of being finite groups.
 * $$\langle H, K \rangle$$ is isomorphic to the infinite dihedral group. It is not divisible by any primes, and in particular it is not divisibility-closed in $$G$$.