If the ring of integers corresponds to finite cyclic groups, what infinite rings do the other finite groups correspond to?

If the ring of integers corresponds to finite cyclic groups, what infinite
rings do the other finite groups correspond to?

Consider $\mathbb{Z}_n$, where $n$ is any positive integer. Then there is
a subgroup of $\mathbb{Z}_n$ for each divisor $d$ of $n$, so that the
number of subgroups equals the number of divisors of $n$. If
$|subg(\mathbb{Z}_n)| < |div(n)|$, then there's a non-trivial divisor that
doesn't generate a proper nontrivial subgroup, etc. So the subgroups of
$\mathbb{Z_n}$ correspond to divisors of $n$ in the ring $\mathbb{Z}$.
What about other finite groups, is there a ring such that divisors in the
ring of the finite group order $|G|$ correspond to subgroups of $G$?
Thanks.