Does $\Bbb Q_p^×$ have index $n$ subgroup?

Question

Let $n$ be arbitrary positive integer. Does $\Bbb Q_p^×$ have index $n$ subgroup?

If I could prove this, from local class field theory, I can say $\Bbb Q_p$ has arbitrary degree of extension.

I know $\Bbb Q_p^×$ is isomorphic to $\Bbb Z×\Bbb Z_p×\Bbb Z/（p-1）\Bbb Z$ using formal logarithm.

Reference is also appreciated.

Thank you in advance.