Finite non-elementary abelian p-groups whose number of subgroups is maximal

Assume G is a direct product of M _p (1, 1, 1) and an elementary abelian p-group, where M _p (1, 1, 1) = 〈a, b | a ^p = b ^p = c ^p =1, a,b]=c,c,a] = c,b]=1〉. When p is odd, we prove that G is the group whose number of subgroups is maximal except for elementary abelian p-groups. Moreover, the counting formula for the groups is given.