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Given a group $G$, its lattice of subgroups $\mathcal{L}(G)$ can be viewed asa 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 studythe 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 $G \cong PGL_2(p^{2^n})$, $p \geq 3$, $n \geq1$ and show that $\widehat{\mathcal{L}(G)}$ is contractible. This is consistentwith a conjecture of Shareshian on the homotopy type of order complexes offinite groups.

The order complex of $PGL_2(p^{2^n})$ is contractible when $p$ is odd