Let $K$ be a finite extension of $\mathbb{Q}\_p$. Let $\mathcal{O}$ be its ring of integers and $\mathfrak{m}$ the maximal ideal. Pick a uniformiser $\pi$. The construction using theory of Lubin--Tate formal groups yields an abelian Galois extension $K\_\pi$ of $K$ such that $\mathrm{Gal}(K\_\pi/K)\simeq\mathcal{O}^\times$. Note that it is well-known that $\mathcal{O}^\times\simeq\mu\_{q-1}\times\mathbb{Z}/p^a\times\mathbb{Z}\_p^d$ where $d=[K:\mathbb{Q}\_p]$ and $q$ the number of elements of the residue field of $\mathcal{O}$ and $a$ some integer.



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> There should be Galois sub-extensions of $K$ in $K\_\pi$ corresponding to each copy of $\mathbb{Z}\_p$. Is it known what they are?
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Field extension corresponding to a quotient of units of local fields
The isomorphism you write down involving the $\mathbb{Z}\_p$ is not canonical, so one cannot describe them without fixing additional data.
For $\mathbb{Q}\_p$ you have $a=1$ and this tower is the totally ramified (p-)cyclotomic tower.


