The motivating question of this talk is, how many number fields are there of a given flavor (e.g., with specified degree and Galois type) and bounded discriminant? Tautologically, fields may be grouped into two classes: those which admit interesting subfields and those that don’t. For example, degree $n$ fields whose Galois closure has Galois group $S_n$ do not admit non-trivial subfields, and such fields have been counted for $n \leq 5$. In this talk, we instead focus on fields that do admit interesting subfields, and we propose a general framework to attack the associated counting problems. This approach also permits one to nontrivially bound the average size of the $\ell$-torsion in the class groups of such fields, for example obtaining such results for the class groups of $D_4$ quartic fields. This is joint work with Jiuya Wang and Melanie Matchett Wood.