First, I'd like to understand what the compact open subgroups of $H(\mathbb{Q}_p)$ are, where $H$ is an inner form of $GL_n$ over $\mathbb{Q}_p$.

Second, I'd like to know where I can read about this for other reductive groups.

Any pointers would be greatly appreciated. Thanks!

allthe compact open subgroups, but only the interesting ones; in which case what you want is the Moy–Prasad theory, as first expounded in ams.org/mathscinet-getitem?mr=1253198 and ams.org/mathscinet-getitem?mr=1371680 . A much more user-friendly introduction appears in Joe Rabinoff's lovely senior thesis. $\endgroup$allcompact subgroups), see Richard Pink, Compact subgroups of linear algebraic groups, J. Algebra 206 (1998), no. 2, 438-504. $\endgroup$6more comments