What are the left cosets of $ D_{8} $ with respect to the subgroup $H=\langle a^2\rangle$?

Question

Let $D_8$ denote the dihedral group of order $16$, aka the group of symmetries of the regular $8$-gon.

Using Lagrange's theorem there are $16/4$ cosets which I have worked out to be $H$, $aH$, $bH$, $abH$. However, in the solutions for this question it is stated that the only cosets are $H$, $aH$ and $bH$. I am not sure how this makes sense as this is contradicting Lagrange's theorem.

Answer 1

By Lagrange, the order of the left (or right) coset space $D_8/\langle a^2\rangle$ is $|D_8|\big/|\langle a^2\rangle| = 16/4$. Your solutions manual is in error or the question is asking for something else.