$H^1(\Omega')$ is in general not a subspace of $H^1 (\Omega)$ for bounded domains $\Omega' \subset \Omega$

Question

Reading about Sobolev spaces I found the following statement:

$H^1 ({\Omega}')$ is not a subspace of $H^1(\Omega)$ for $\Omega'\subset \Omega$.

$\left(\text{However } \ H^1_0 (\Omega ')\subset H^1_0(\Omega)\right)$

I guess that the reason behind is that you can not control $f\in H^1(\Omega ')$ outside $\Omega'$. But I'm having a hard time thinking a counterexample or a more rigorous explanation.