Lattice-complemented does not satisfy intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., lattice-complemented subgroup) not satisfying a subgroup metaproperty (i.e., intermediate subgroup condition).
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Statement

It can happen that $H$ is a lattice-complemented subgroup of $G$, and $H\leq K\leq G$, with $H$ not a lattice-complemented subgroup in $K$.

Proof

Example of the symmetric group

Further information: symmetric group:S4

Let $G$ be the symmetric group on a set $\{1,2,3,4\}$ of size four.

Consider the subgroups:

$H=\langle (1,3)(2,4)\rangle ,\qquad K=\langle (1,2,3,4)\rangle$.

$H$ is a subgroup of order two and $K$ is a subgroup of order four containing $H$. Further:

• $H$ is a lattice-complemented subgroup in $G$: The symmetric group on the subset $\{1,2,3\}$ is a lattice complement to $H$ in $G$.
• $H$ is not a lattice-complemented subgroup in $K$: Indeed, $K$ is a cyclic group of order four and $H$ is a subgroup of order two, and has no lattice complements.