Class numbers and nilpotent subgroups of $\mathrm{P}\mathrm{G}\mathrm{L}(2,q)$

We show that for certain odd prime powers $q$, the number of conjugacy classes of $\mathrm{P}\mathrm{G}\mathrm{L}(2,q)$ is greater than the order of its largest nilpotent subgroup. This answers negatively a question of Liebeck and Pyber.