$H$ is a normal subgroup of $D_8$

Question

$H = \{e, x^2, y, x^2y \}$.

Is there any elegent way to show that $H$ is a normal subgroup of $D_8$. Of course i can use bruteforce, but i don't really want to simplify all 32 multiplications. Even if i discard neutral element $7\cdot 3 = 21$ is still pretty huge amount of multiplications.

Answer 1

Once you have already proved that $H$ is a subgroup, proving that $H$ is normal is easy because every subgroup of index $2$ is normal.