We report on the magnetic phase diagram of the distorted kagome-lattice antiferromagnet U3Ru4Al12 determined through measurements of magnetic and elastic properties. For field applied along the [100] and [120] axes of the hexagonal crystal structure, we find pronounced anomalies in the magnetization and elastic moduli that signal the existence of unknown magnetic phases. Our crystal-electric-field (CEF) analysis evidences interlevel quadrupolar interactions between the ground-state singlet and the first excited doublet. These interactions lead to a large softening of the shear elastic modulus C44. The large number of phases and pronounced elastic softening suggest that geometric frustrations and CEF effects play an important role in the physical properties of U3Ru4Al12.