In this paper, we prove t h a t the Lie group of typeE8hw a unique conjugacy class of subgroups isomorphic to the finite simple groupL(2 61) of linear fractional transformations over t h e field of 6 1 elements. This result settles the last open case of a conjecture made by B. Kostant in 1983 [Iiost] concerning the occurrence of the finite simple groupin a complex simple Lie group with Coxeter numberh. It also settles a case left open in the classifications of finite simple subgroups ofobtained by the first two authors in 1987.