It is the case that every finite normal field extension of $\mathbb{Q}$ with abelian Galois group is contained in a cyclotomic extension. What about prime cyclotomic extensions?
Is there a simple characterization of those number fields $K$ that occur as subfields of some $\mathbb{Q}(\zeta_p)$ for $p$ a prime? Clearly $K$ must be normal over $\mathbb{Q}$ with cyclic Galois group. Do all such occur?
Your question is a bit ill-posed. By Galois theory, the subfields of the cyclotomic field $L=\mathbf Q(\zeta_p)$ are the subfields $L^G$ fixed by the subgroups $G$ of $(\mathbf Z/p)^*$. But I guess you ask for more, e.g. for primitive elements of the $L^G$ ? If so, the answer is given by the so called Gaussian periods mod $p$ : let $G$ be a subgroup of $(\mathbf Z/p)^*$; for any $a\in \mathbf Z/p$, define $S(G,a)$ as the sum over $g\in G$ of the elements $\zeta_p^{ag}$. After establising some properties of the Galois action on the Gaussian periods, it is easy to show that $L^G=\mathbf Q(S(G,1))=\mathbf Q(S(G,a))$ for all $a\in(\mathbf Z/p)^*$. The same result is valid on replacing $p$ by an integer $n$ which is square-free. The most accessible ref. for a beginner is the bachelor thesis of D. Tijsma, "Gaussian Periods", Utrecht, 2016, which you can find using Google. The result I cited is lemma 5-12 therein. Weaker results are available for non square-free $n$.