Direct factor is not upper join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor) not satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property).
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Statement

We can have a subgroup $H$ of a group $G$ , and intermediate subgroups $K_{1}$ and $K_{2}$ such that $H$ is a direct factor of $K_{1}$ as well as a direct factor of $K_{2}$ , but $H$ is not a direct factor of the join of subgroups $⟨ K_{1}, K_{2}$ .

Proof

Example of the dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Consider the dihedral group of order eight:

$G : = ⟨ a, x ∣ a^{4} = x^{2} = e, x a x = a^{- 1} ⟩$ .

Let:

$H = ⟨ a^{2} ⟩, K_{1} = ⟨ a^{2}, x ⟩, K_{2} = ⟨ a^{2}, a x ⟩$ .

Then:

$H$ is a direct factor of $K_{1}$ , which is a Klein four-group and is the internal direct product of $H$ and $⟨ x ⟩$ .
$H$ is a direct factor of $K_{2}$ , which is a Klein four-group and is the internal direct product of $H$ and $⟨ a x ⟩$ .
$H$ is not a direct factor of $⟨ K_{1}, K_{2} ⟩ = G$ . In fact, since $H$ is the center of $G$ , it intersects every nontrivial normal subgroup nontrivially (nilpotent implies center is normality-large).