The subdivision graph $S(Σ)$ of a graph $Σ$ is obtained from $Σ$ by `adding a vertex' in the middle of every edge of $\Si$. Various symmetry properties of $§(Σ)$ are studied. We prove that, for a connected graph $Σ$, $S(Σ)$ is locally $s$-arc transitive if and only if $Σ$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. The diameter of $S(Σ)$ is $2d+δ$, where $Σ$ has diameter $d$ and $0\leqslant δ\leqslant 2$, and local $s$-distance transitivity of $§(Σ)$ is defined for $1\leqslant s\leqslant 2d+δ$. In the general case where $s\leqslant 2d-1$ we prove that $S(Σ)$ is locally $s$-distance transitive if and only if $Σ$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. For the remaining values of $s$, namely $2d\leqslant s\leqslant 2d+δ$, we classify the graphs $Σ$ for which $S(Σ)$ is locally $s$-distance transitive in the cases, $s\leqslant 5$ and $s\geqslant 15+δ$. The cases $\max\{2d, 6\}\leqslant s\leqslant \min\{2d+δ, 14+δ\}$ remain open.