Here all the extensions are finite.
In local class field theory, one learns that the abelian extensions $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$, where $\zeta$ is a $p^{n}$-th root of unity, are in bijective correspondence with the subgroups $(p)\times (1+p^{n}\mathbb{Z}_{p})$ of the multiplicative group $\mathbb{Q}_{p}^{\times}$.
Do we know similar descriptions for other 'types' of extensions of $\mathbb{Q}_{p}$?
For example, we can play around and wonder what kind of abelian extensions $L_{n}/\mathbb{Q}_{p}$ correspond to the subgroups \begin{equation} (p^{2})\times (1+p^{n}\mathbb{Z}_{p}) \end{equation} and we can ask this for many other subgroups. Also, I wonder if we can detect ramification looking at the these subgroups.
For example, it is known that the $p^{n}$-th cyclotomic extensions I mentioned above are totally ramified, and just so happens that the "$(p)$"-part of the corresponding subgroup has exponent $1$.
I did this for a couple of examples, and it looks like an extension corresponding to some subgroup of the form \begin{equation} (p^{f})\times H, \end{equation} for some subgroup $H$ of $\mathbb{Z}_{p}^{\times}$, has inertia degree $f$. Is this true in general? Thanks a lot.