  We prove part of a conjecture by Johansson, Kahn and Vu \cite{JKV} regarding
threshold functions for the existence of an $H$-factor in a random graph \gnp.
We prove that the conjectured threshold function is correct for any graph $H$
which is not covered by its densest subgraphs. We also demonstrate that the
main result of \cite{JKV} generalises to multigraphs, digraphs, and a
multipartite model.
