On the number of non-cyclic subgroups of finite p-groups

Let $G$ be a finite $p$-group and $\delta (G)$ denote the number of all non-cyclic subgroups of $G$. In this paper, an upper bound for $\delta (G)$ is obtained. Furthermore, we prove that $\delta (G)\le \delta (M_p(1,1,1)\times C_p^{n-3})$ (if $p=2$, then $\delta (G)\le \delta (D_8\times C_2^{n-3})$), for any non-elementary abelian $p$-group $G$ of order $p^n$.