A regular edge-transitive graph is said to be semisymmetric if it is mot vertex-transitive. By Folkman [J. Combin. Theory 3 （1967）, 215-232], there is no semisymmetric graph of order 2p or 2p^2 for a prime p, and by Malni6 et al. [Discrete Math. 274 （2004）, 18-198], there exists a unique cubic semisymmetrie graph of order 2p3, the so called Gray graph of order 54. In this paper, it is shown that there is no connected cubic semisymmetric graph of order 4p^3 and that there exists a unique cubic semisymmetric graph of order 8p3, which is a Z2 × Z2-covering of the Gray graph.
