Abstract: It is known that the lower bound on the maximum genus of a 2 connected cubic simplicial graph is at least one third of its cycle rank.This bound is tight.This paper has showed that there is a special structure in such a graph that its maximum genus attains the lower bound,and the order of such a graph is zero modulo three.When the order of a 2 connected cubic simplicial graph is not zero modulo three,it is found that its maximum genus is at least one third of its cycle rank plus one.