Embedding to $L^\alpha(0,T;L^\beta(\Omega))$

Question

Good day!

Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.

Consider the space $W = \{ y \in L^2(0,T;V) \colon dy/dt \in L^2(0,T;V') \}$.

It is well-known that $W \subset C([0,T];H)$ where $H = L^2(\Omega)$.

My question: for what $\alpha$ and $\beta$ we may assert that $W \subset L^\alpha(0,T;L^\beta(\Omega))$?

For example, $W \subset L^\infty(0,T;L^2(\Omega))$ and $W \subset L^2(0,T;L^6(\Omega))$ because $H^1(\Omega) \subset L^6(\Omega)$.

May we assert that $W \subset L^8(0,T;L^{24/5}(\Omega))$?

Thank you!