Conjugacy-closedness is not join-closed

Statement
We can have a situation where $$H, K$$ are conjuacy-closed subgroups of $$G$$ but the join $$\langle H, K \rangle$$ is not conjugacy-closed in $$G$$.

Example of the dihedral group
Let $$G$$ be the dihedral group of order eight:

$$G = \langle a,x \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle$$.

Suppose $$H, K$$ are subgroups given as follows:

$$H = \{ a^2, e \}, \qquad K = \{ x, e \}$$.

Observe that:


 * Since $$H$$ and $$K$$ are both subgroups of order two, they are both conjugacy-closed in $$G$$.
 * The join $$\langle H, K \rangle$$ is an Abelian subgroup of order four. It is clearly not conjugacy-closed, because the elements $$x$$ and $$a^2x$$ in this subgroup are conjugate in $$G$$.