Finding subgroups of $G=\langle x,y,z~|~x^2y^2z^2 \rangle$ using covering space theory

Question

Consider the group $G$ having a presentation $G=\langle x,y,z~|~x^2y^2z^2 \rangle$. I am trying to find all subgroups of $G$ of index 6 using covering space theory. It is well-known that the connected sum $X=3\Bbb RP^2$ of three projective planes has fundamental group isomorphic to $G$. Also for each subgroup of $G=\pi_1(X)$, there is a covering space $p:\tilde{X}\to X$ such that $p_*(\pi_1(\tilde{X}))=H$, and if the index $[G:H]$ is $n$, then $p$ is $n$-sheeted. Thus the question reduces to find all $6$-sheeted covering spaces of $X$, but I can't see a way because I've never seen a covering spaces of connected sums. Any hints?

Answer 1

The following is too long for a comment (and is, hence, a CW post).

There is at most $2,362$ subgroups of index at most $6$.

gap> F:=FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> rels:=[(F.1)^2*(F.2)^2*(F.3)^2];
[ f1^2*f2^2*f3^2 ]
gap> G:=F/rels;
<fp group of size infinity on the generators [ f1, f2, f3 ]>
gap> Size(LowIndexSubgroupsFpGroup(G,6));
2362