Dihedral Group D4 is not Internal Group Product/Proof 2

Proof
The proper subgroups of $D_4$ are of order no greater than $4$.

From Group of Order less than 6 is Abelian, all such proper subgroups are abelian.

From External Direct Product of Abelian Groups is Abelian Group, the group direct product of $2$ of these proper subgroups is in turn abelian.

From Internal and External Group Direct Products are Isomorphic, the internal group product of $2$ of these proper subgroups is in turn abelian.

But $D_4$ is not abelian.

Hence it cannot be the internal group product of any $2$ of these proper subgroups.