Order of intersection of two subgroups

Question

Let $G$ be a group and $H$, $K$ be subgroups of $G$. In general, $\vert H \cap K \vert$ is a common divisor of $\vert H \vert$ and $\vert K \vert$. When $G$ is a cyclic group, the order of $H \cap K$ is $\gcd(\vert H \vert, \vert K \vert)$. When it is not cyclic, the same rule does not apply. What if one subgroup is cyclic and the other is noncyclic? Is there a way to know the exact order like that of two cyclic subgroups?