  There are only 10 Euclidean forms, that is flat closed three dimensional
manifolds: six are orientable $\mathcal{G}_1,\dots,\mathcal{G}_6$ and four are
non-orientable $\mathcal{B}_1,\dots,\mathcal{B}_4$. In the present paper we
investigate the manifold $\mathcal{G}_6$, also known as Hantzsche-Wendt
manifold; this is the unique Euclidean $3$-form with finite first homology
group $H_1(\mathcal{G}_6) = \mathbb{Z}^2_4$.
  The aim of this paper is to describe all types of $n$-fold coverings over
$\mathcal{G}_{6}$ and calculate the numbers of non-equivalent coverings of each
type. We classify subgroups in the fundamental group $\pi_1(\mathcal{G}_{6})$
up to isomorphism. Given index $n$, we calculate the numbers of subgroups and
the numbers of conjugacy classes of subgroups for each isomorphism type and
provide the Dirichlet generating series for the above sequences.
