I'm trying to learn how to apply local class field theory and I thought about trying to enumerate some low degree abelian extensions of $\mathbb{Q}_p$. The easiest case is the quadratic extensions i.e. the cyclic extensions of degree 2. But here no local class field theory is needed as quadratics are easy to classify by just working directly with the polynomials and finally counting the index of the subgroup of squares.
My question is how to do the cyclic of degree 3 case? I know that local class field theory gives me an isomorphism
$\mathbb{Q}_p^\times/\textrm{Nm}_{K/\mathbb{Q}_p}(K^\times)\to \textrm{Gal}(K/\mathbb{Q}_3)\simeq C_3.$
Hence, to somehow list the abelian extensions of degree 3, I would need to be able to somehow classify the open subgroups of index 3 in $\mathbb{Q}_p^\times$. Is there any simple way to do that or is there an easier approach for solving the original problem? I guess it might be difficult to actually write down each cyclic degree 3 extension, but just knowing the number of them would already help.
Is there any simple method that extends to cyclic or dihedral extensions of degree 4?