Classification of the sublattices of a lattice

In 1945-46, C. L. Siegel proved that an $n$-dimensional lattice $\mathrm{\Lambda }$ of determinant $\mathrm{d}\mathrm{e}\mathrm{t}(\mathrm{\Lambda })$ has at most $m^{n^2}$ different sublattices of determinant $m\cdot \mathrm{d}\mathrm{e}\mathrm{t}(\mathrm{\Lambda })$. In 1997, the exact number of the different sublattices of index $m$ was determined by Baake. This paper presents a systematic treatment for counting the sublattices and deduces a formula for the number of the sublattice classes of determinant $m\cdot {\rm det}(\Lambda )$.