Does the vector space of the infinitely differentiable functions belongs to the space $\nabla H^1(\Omega)$?

Question

Let $\Omega\subset \mathbb R^3$ be a bounded lipschitz domain. Denote by $H^1(\Omega)$ the usual Sobolev space and by $(C^{\infty}(\Omega))^3$ the vector space of the infinitely differentiable functions. Then, do we have the following inclusion $$ (C^{\infty}(\Omega))^3 \subset \nabla H^1(\Omega)? $$ Thanks.