I've come across the phrase "by the classification of subgroups of $GL_2(F_q)$" in multiple papers, but never with a reference. Here $F_q$ is a finite field of size $q$. Does anyone know a good reference for this (ideally for someone who is not a group theorist)?

At the textbook level, take a look at Lang's Algebra, Chapter XVIII, Section 12, in my version is on page 712. Seems to be pretty thorough. There is also section 2 of Serre's 1972 ``Propriétés galoisiennes des points d'ordre fini des courbes elliptiques''. 


Dickson is responsible for the classification of subgroups of $SL_2(\mathbb{F}_q)$ (and once you've got this the subgroups of $GL_2(\mathbb{F}_q)$ are easy). You can find a full proof in Suzuki's "Group Theory Part I". It's Chapter 6, Section 3 of that book. If you email me I'll even send you a copy :) 


This classification was wellknown already in the beginning of XXth century; perhaps this explains why people do not bother to give a reference (the result was due to E.H.Moore and, independently, Wiman). Many years ago I read the paper by Howard H. Mitchell, Determination of the ordinary and modular ternary linear groups. Trans. Amer. Math. Soc. 12 (1911), no. 2, 207–242, which addresses the analogous question for $GL_3(\mathbb{F}_q)$, and also contains the detailed information on the subgroups of $GL_2(\mathbb{F}_q)$. (It's of course not so easy to read, as the terminology was quite different...) 

