In this paper, we first prove the weak intermittency, and in particular the sharp exponential order $Cλ^4t$ of the second moment of the exact solution of the stochastic heat equation with multiplicative noise and periodic boundary condition, where $λ>0$ denotes the level of the noise. In order to inherit numerically these intrinsic properties of the original equation, we introduce a fully discrete scheme, whose spatial direction is based on the finite difference method and temporal direction is based on the $θ$-scheme. We prove that the second moment of numerical solutions of both spatially semi-discrete and fully discrete schemes grows at least as $\exp\{Cλ^2t\}$ and at most as $\exp\{Cλ^4t\}$ for large $t$ under natural conditions, which implies the weak intermittency of these numerical solutions. Moreover, a renewal approach is applied to show that both of the numerical schemes could preserve the sharp exponential order $Cλ^4t$ of the second moment of the exact solution for large spatial partition number.