Suppose I take an infinite direct power $\prod G$ of some (not necessarily finite) group $G$. I want to know about the subgroups of $\prod G$ that are maximal subject to having trivial intersection with $\oplus G$. Is there a general description of such subgroups (in terms of ultrafilters maybe), without delving into the structure of $G$? If not, are there at least some interesting general constructions of 'large' subgroups of $\prod G$ that intersect trivially with $\oplus G$?

One way to construct subgroups intersecting trivially with $\oplus G$ is as follows: take a family $\mathcal{P}$ of partitions of the indexing set $I$ that is closed under coarsest common refinement, such that all the parts in any given $P \in \mathcal{P}$ are infinite. (For instance $I = \mathbb{Z}$ and $\mathcal{P}$ is congruence classes modulo $n$.) Now take all those $g \in \prod G$ for which there is some $P$ in $\mathcal{P}$ for which the $i$-th entry of $g$ is determined by which part of $P$ contains $i$.

Are the groups I just described ever maximal?