Definition. Whenever $M$ is a countable model of $\mathrm{ZFC}$ (not necessarily well-founded) write $\mathrm{Inn}(M)$ for the poset of inner models of $M$ ordered by inclusion.
Question. For which countably infinite posets $P$ does there exist a model $M$ of $\mathrm{ZFC}$ such that $P \cong \mathrm{Inn}(M)$?
Partial answers and/or references appreciated. Also: if we need to assume that $M$ is well-founded (or even transitive) to get a useful answer, then so be it.