Quaternion Group has Normal Subgroup without Complement

Theorem
Let $Q$ denote the quaternion group.

There exists a normal subgroup of $Q$ which has no complement.

Proof
From Subgroups of Quaternion Group:

From Quaternion Group is Hamiltonian, all these subgroups are normal.

For two subgroups to be complementary, they need to have an intersection which is trivial.

However, apart from $\set e$ itself, all these subgroups contain $a^2$.

Hence none of these subgroups has a complement.