Let $S\subseteq R$ be commutative rings with $1$. It is given that $S$ is local and $R$ is integral over $S$. I need to show that $R$ is semilocal that is $R$ has finitely many maximal ideals.
It is also given that $R$ is Cohen-Macaulay (if you need that). But I don't see how this will be useful. Assume it is also given that both $R$ and $S$ are Noetherian.
I don't know if we can conclude from the above conditions whether $R$ is semilocal or not. But if we include another assumption that: for each ideal $I\subseteq S$, $IR\cap S=I$, then $R$ is semilocal. This assumption was given in my case which I didn't realize first.
Using the above additional hypothesis $S/\mathfrak m\subseteq R/\mathfrak mR$ and the extension is integral. But $S/\mathfrak m$ is a field and hence $R/\mathfrak mR$ is Artinian and so it has finitely many maximal ideals. Since maximal ideals of $R$ must contain $\mathfrak mR$, we conclude that $R$ is semilocal. (Above $\mathfrak m$ is the maximal ideal of $S$.)