  We demonstrate a general gauging procedure of a pure matter theory on a
lattice with a mixture of subsystem and global symmetries. This mixed symmetry
can be either a semidirect product of a subsystem symmetry and a global
symmetry, or a non-trivial extension of them. We demonstrate this gauging
procedure on a cubic lattice in three dimensions with four examples:
$G=\mathbb{Z}_3^{\text{sub}} \rtimes \mathbb{Z}_2^{\text{glo}}$,
$G=(\mathbb{Z}_2^{\text{sub}} \times \mathbb{Z}_2^{\text{sub}}) \rtimes
\mathbb{Z}_2^{\text{glo}}$, $1\to \mathbb {Z}_2^\text {sub}\to G\to \mathbb
{Z}_2^\text {glo}\to 1$, and $1\to \mathbb {Z}_2^\text {sub}\to G\to K_4^\text
{glo}\to 1$. The former two cases and the last one produce the non-Abelian
fracton orders. Our construction of the gauging procedure provides an
identification of the electric charges of these fracton orders with irreducible
representations of the symmetry. Furthermore, by constraining the local Hilbert
space, the magnetic fluxes with different geometry (tube-like and
plaquette-like) satisfy a subalgebra of the quantum double models (QDMs). This
algebraic structure leads to an identification of the magnetic fluxes to the
conjugacy classes of the symmetry.
