A study of muonic 238U has been performed in a combined (p-, 7f) and (#-, 7t') coincidence experiment to investigate the role of non-radiative transitions and their fission probabilities. An augmentation of the outer fission barrier of AEb = (0.6___ 0.1) MeV due to the presence of the muon is deduced. A significant contribution to the prompt fission yield not only results from the (2p~ Is) and (3d-+ l s) non-radiative transitions, but also from other radiationless transitions. Specifically, the measured fission probabilities of the transitions (2p-~ls), (3 d ---, l s), and (3p--*ls) are (1.5• (5.7 • 1.7)%, and (5.3 • 1.9)%, respectively.