  Given a group $G$, its lattice of subgroups $\mathcal{L}(G)$ can be viewed as
a simplicial complex in a natural way. The inclusion of $1_G, G \in
\mathcal{L}(G)$ implies that $\mathcal{L}(G)$ is contractible, and so we study
the topology of the order complex $\widehat{\mathcal{L}(G)} := \mathcal{L}(G)
\setminus \{1_G,G\}$. In this short note we consider the homotopy type of
$\widehat{\mathcal{L}(G)}$ where $G \cong PGL_2(p^{2^n})$, $p \geq 3$, $n \geq
1$ and show that $\widehat{\mathcal{L}(G)}$ is contractible. This is consistent
with a conjecture of Shareshian on the homotopy type of order complexes of
finite groups.
