I am looking for a reference on the space $\mathcal{Sub}(G)$ of closed subgroups of a profinite group $G$, which naturally has the structure of a profinite topological space:

Because the collection of subgroups $\mathcal{Sub}(G_i)$ of a finite group $G_i$ is finite, and the closed subgroups of a profinite group $G = \mathrm{lim}G_i$ can be described as inverse limits of subgroups of the $G_i$, we have that $\mathcal{Sub}(G) = \mathrm{lim}\mathcal{Sub}(G_i)$ is a profinite space.

Could anyone point me to some good references for this space associated to a profinite group $G$?

I would like to see any sort of facts known about this space, but in particular the implications that various finiteness conditions on $G$ (e.g. topologically finitely generated, first countable, $\mathrm{cd}(G)\leq 1$, finitely many quotients of a given order, etc.) have for the space $\mathcal{Sub}(G)$. It would also be great to see descriptions of the quotient space $\mathcal{Sub}(G)/\sim$ by the action of conjugation by $G$ or by subgroups of $G$.

I would even be interested to see references for the closed subgroup space of any compact topological group $G$.

Thank you.