  We show that the complexity of the Lie module $\mathrm{Lie}(n)$ in
characteristic $p$ is bounded above by $m$ where $p^m$ is the largest $p$-power
dividing $n$ and, if $n$ is not a $p$-power, is equal to the maximum of the
complexities of $\Lie(p^i)$ for $1 \leq i \leq m$.
