Find a subgroup of $GL_3(\mathbb{Z}_8)$ of index 2

Question

On my final exam yesterday there was one "almost bonus" question which I don't really think I did right. I "guessed" that such a subgroup is $GL_3(\mathbb{Z}_4)$. A hint would be appreciated.

The approach might be to find a subgroup whose order is twice as little as that of $GL_3(\mathbb{Z}_8)$ and use Lagrange's theorem. We also know that any subgroup of index 2 is normal, so we could use this fact.

Answer 1

A plan:

1. Think determinant. Why is determinant a homomorphism $f$ from $GL_3(\Bbb{Z}_8)$ to $\Bbb{Z}_8^*$?
2. Show that $f$ is surjective.
3. Find an index two subgroup $H$ of $\Bbb{Z}_8^*$.
4. Figure out what you can do with $f$ and $H$ to get what you want.