We consider the ground-state structure of a rotating Bose-Einstein condensate with spin-orbit coupling which is confined in a harmonic plus qurtic potential. Combined effects of spin-orbit coupling and rotation on the ground-state structure of such a system are investigated in detail. Results show that a large number of novel ground-state structures, such as stripe, two rows, snakeskin piebald, and so on, can be produced under the combination of anisotropic spin-orbit-coupling and rotation.