The problem of the existence of a decomposition of the complete graph Kn into disjoint copies of K-5\e has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Gamma be a (K-27, K-5\e)-design. I show that | Aut(Gamma)| divides 2k3 for some k = 0 and that Sym(3) not less than or equal to Aut(Gamma). I construct (K-27, K-5\e)-designs by prescribing Z6 as an automorphism group, and show that up to isomorphism there are exactly 24 (K-27, K-5\e)-designs with Z6 as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed Z6. Finally, the existence of (K-5\e)-designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851-864. (C) C 2012 Wiley Periodicals, Inc.