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There are only 10 Euclidean forms, that is flat closed three dimensionalmanifolds: six are orientable $\mathcal{G}_1,\dots,\mathcal{G}_6$ and four arenon-orientable $\mathcal{B}_1,\dots,\mathcal{B}_4$. The aim of this paper is todescribe all types of $n$-fold coverings over the non-orientable Euclideanmanifolds $\mathcal{B}_{3}$ and $\mathcal{B}_{4}$, and calculate the numbers ofnon-equivalent coverings of each type. The manifolds $\mathcal{B}_{3}$ and$\mathcal{B}_{4}$ are uniquely determined among non-orientable forms by theirhomology groups $H_1(\mathcal{B}_{3})=\ZZ_2\times \ZZ_2 \times \ZZ$ and$H_1(\mathcal{B}_{4})=\ZZ_4 \times \ZZ$.  We classify subgroups in the fundamental groups $\pi_1(\mathcal{B}_{3})$ and$\pi_1(\mathcal{B}_{4})$ up to isomorphism. Given index $n$, we calculate thenumbers of subgroups and the numbers of conjugacy classes of subgroups for eachisomorphism type and provide the Dirichlet generating functions for the abovesequences.

On the coverings of closed non-orientable Euclidean manifolds $\mathcal{B}_{3}$ and $\mathcal{B}_{4}$