Finite Unramified Galois Extension

Question

I'm wondering whether finite unramified Galois extensions of p-adic field number fields (i.e. extensions of $\mathbb{Q}_p$ for some prime $p$) are cyclic? The absolute Galois group is isomorphic to the profinite completion of the integers. Hence I think that every finite unramified Galois extension is a finite quotient of the absolute Galois group (as the extension is then a subfield of the maximal unramified extension, which is exactly the one with Galois group the absolute Galois group). Then it should follow that its Galois group is a finite quotient of $\hat{\mathbb{Z}}$. Now, is it true that it must be $\mathbb{Z}/{n \mathbb{Z}}$? And is my argument above correct?