Integer vector subsets invariant under rational matrix

Question

Let $A$ be an $n\times n$ matrix with rational coefficients. Define $H$ to be the maximal subset of $\mathbb{Z}^n$ such that $AH\subset H$.

Question: How to describe $H$ in terms of $A$ (its Jordan normal form or maybe rational normal form)? Is there a special term for $H$ that I can look for?