  Multiple-$Q$ states manifest themselves in a variety of noncollinear and
noncoplanar magnetic structures depending on the magnetic interactions and
lattice structures. In particular, cubic-lattice systems can host a plethora of
multiple-$Q$ states, such as magnetic skyrmion and hedgehog lattices. We here
classify momentum-dependent anisotropic exchange interactions in the
cubic-lattice systems based on the magnetic representation analysis. We
construct an effective spin model for centrosymmetric cubic space groups,
$Pm\bar{3}m$ and $Pm\bar{3}$, and noncentrosymmetric ones, $P\bar{4}3m$,
$P432$, and $P23$: The former include the symmetric anisotropic exchange
interaction, while the latter additionally include the Dzyaloshinskii-Moriya
interaction. We demonstrate that the anisotropic exchange interaction becomes
the origin of the multiple-$Q$ states by applying the anisotropic spin model to
the case under $Pm\bar{3}$. We show several multiple-$Q$ instabilities in the
ground state by performing simulated annealing. Our results will be a reference
for not only exploring unknown multiple-$Q$ states but also understanding the
origin of the multiple-$Q$ states observed in both noncentrosymmetric and
centrosymmetric magnets like EuPtSi and SrFeO$_3$.
