We present wall-resolved large-eddy simulation (LES) of flow with free-stream velocity U∞ over a cylinder of diameter D rotating at constant angular velocity Ω , with the focus on the lift crisis, which takes place at relatively high Reynolds number ReD=U∞D/ν , where ν is the kinematic viscosity of the fluid. Two sets of LES are performed within the ( ReD , α )-plane with α=ΩD/(2U∞) the dimensionless cylinder rotation speed. One set, at ReD=5000 , is used as a reference flow and does not exhibit a lift crisis. Our main LES varies α in 0⩽α⩽2.0 at fixed ReD=6×104 . For α in the range α=0.48−0.6 we find a lift crisis. This range is in agreement with experiment although the LES shows a deeper local minimum in the lift coefficient than the measured value. Diagnostics that include instantaneous surface portraits of the surface skin-friction vector field Cf , spanwise-averaged flow-streamline plots, and a statistical analysis of local, near-surface flow reversal show that, on the leeward-bottom cylinder surface, the flow experiences large-scale reorganization as α increases through the lift crisis. At α=0.48 the primary-flow features comprise a shear layer separating from that side of the cylinder that moves with the free stream and a pattern of oscillatory but largely attached flow zones surrounded by scattered patches of local flow separation/reattachment on the lee and underside of the cylinder surface. Large-scale, unsteady vortex shedding is observed. At α=0.6 the flow has transitioned to a more ordered state where the small-scale separation/reattachment cells concentrate into a relatively narrow zone with largely attached flow elsewhere. This induces a low-pressure region which produces a sudden decrease in lift and hence the lift crisis. Through this process, the boundary layer does not show classical turbulence behaviour. As α is further increased at constant ReD , the localized separation zone dissipates with corresponding attached flow on most of the cylinder surface. The lift coefficient then resumes its increasing trend. A logarithmic region is found within the boundary layer at α=1.0 .