I'm having trouble understanding what a subdirect product is.
Say $G$ is a subdirect product of $H=\prod H_i$ - this means that the homomorphisms $f_i:G\to H_i$ are surjective, which can be trivially accomplished as $f_i(g)=1$.
So isn't it the case that $G$ is a subdirect product of $H$ for all $G,H$?
$G$ has to be a subalgebra of $H$. If $G$ is a subalgebra of $H$, then "subalgebra inclusion" refers to the mapping that identifies each element of $G$ with itself considered as an element of $H$. So for each pair $g, g' \in G$, if $g \not= g'$, there must be an $i$ with $f_i(g) \not= f_i(g')$.
For an example in the case of abelian groups, the group of integers can be viewed as a subdirect product of the groups of integers modulo $n$: $\mathbb{Z}$ is isomorphic to the subgroup of the product $\prod_{n > 0}\mathbb{Z}/n\mathbb{Z}$ comprising the elements of the form $(j + 1\mathbb{Z}, j + 2\mathbb{Z}, j + 3\mathbb{Z}, \ldots)$.