We can say something reasonably general about finite abelian groups.

Throughout, we'll let $\mathcal{L}(G)$ be the lattice of subgroups of a finite group $G$ and $\overline{\mathcal{L}(G)}$ be the proper part of $\mathcal{L}(G)$ (i.e., $\mathcal{L}(G)$ without the top and bottom elements). Recall that the *order complex* $\Delta(P)$ of a poset $P$ is the (abstract) simplicial complex whose faces are the chains of $P$.

The following result is due to Kratzer and Thévanaz (Corollaire 4.10). Translation due to John Shareshian.

**Theorem:**
*Let $G$ be a finite solvable group with chief series
$$1 = G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \cdots \triangleleft G_r = G.$$
For $1 \leq i < r$, let $m_i$ be the number of complements to $G_i/G_{i-1}$ in $G/G_{i-1}$. Then $\Delta(\overline{\mathcal{L}(G)})$ has the homotopy type of a wedge of $$m = \displaystyle\prod_{i=1}^{r-1}m_i$$ $(r-2)$-spheres.*

On the other hand, we have one of the foundational results of poset topology, Philip Hall's Theorem. See, e.g., Wachs' notes on poset topology.

**Philip Hall's Theorem:** *For any poset $P$ with top element $\hat{1}$ and bottom element $\hat{0}$,
$$\mu_P(\hat{0},\hat{1}) = \tilde{\chi}(\Delta(\overline{P})),$$
where $\tilde{\chi}$ is the reduced Euler characteristic and $\overline{P}$ is $P\setminus \{\hat{0},\hat{1}\}$.*

In particular, when $G$ is abelian (and hence solvable), we can compute the Möbius function $\mu_{\mathcal{L}(G)}(K,H)$ by applying Kratzer and Thévanaz's result to the interval $[K,H]$ in $\mathcal{L}(G)$.