  There are well known algorithms to compute the class group of the maximal
order $\mathcal{O}_K$ of a number field $K$ and the group of invertible ideal
classes of a non-maximal order $R$. In this paper we explain how to compute
also the isomorphism classes of non-invertible ideals of an order $R$ in a
finite product of number fields $K$. In particular we also extend the
above-mentioned algorithms to this more general setting. Moreover, we
generalize a theorem of Latimer and MacDuffee providing a bijection between the
conjugacy classes of integral matrices with given minimal and characteristic
polynomials and the isomorphism classes of lattices in certain
$\mathbb{Q}$-algebras, which under certain assumptions can be explicitly
described in terms of ideal classes.
