  In this work, we consider the following inhomogeneous nonlinear Schr\"odinger
equation \begin{align*}
  i\partial_t u+\Delta u +|x|^{-b}|u|^{p-1}u=0,\quad (x, t)\in
\mathbb{R}^N\times\mathbb{R} \end{align*} with $0<b<\mbox{min}\{2, N\}$ and
$1+\frac{4-2b}{N}<p<2^{\ast}$. We assume that $u_0 \in H^{1}(\mathbb{R}^N)$ and
satisfies \begin{align*}
  E(u_0)^{s_c}M(u_0)^{1-s_c}<E(Q)^{s_c}M(Q)^{1-s_c}, \end{align*} and
\begin{align*}
  \|\nabla u_0\|_{L^2}^{s_c}\|u_0\|_{L^2}^{1-s_c}>\|\nabla
Q\|_{L^2}^{s_c}\|Q\|_{L^2}^{1-s_c}, \end{align*} where $Q$ is the ground state
solution of the corresponding elliptic equation. We prove that when
$1+\frac{4-2b}{N}<p\leq1+\frac{4}{N}$, then the corresponding solution blows up
in finite time. Besides, when $1+\frac{4}{N}<p<2^{\ast}$, we prove the finite
or infinite time blow-up and obtain a precise lower bound of blow-up rate.
