Divisibility-closedness is not finite-join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., divisibility-closed subgroup) not satisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).
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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 $G$.