[en] We study the energetics of vortices and vortex lattices produced by rotation in the cyclic phase of F=2 spinor condensates. In addition to the familiar triangular lattice predicted by Tkachenko for ⁴He, many more complex lattices appear in this system as a result of the spin degree of freedom. In particular, we predict a magnetic-field-driven transition from a triangular lattice to a honeycomb lattice. Other transitions and lattice geometries are driven at constant field by changes in the temperature-dependent ratio of charge and spin stiffnesses, including a transition through an aperiodic vortex structure. Finally, we compute the renormalization of the ratio of the spin and charge stiffnesses from thermal fluctuations using a nonlinear sigma model analysis