  We introduce lattice gauge theories which describe three-dimensional, gapped
quantum phases exhibiting the phenomenology of both conventional
three-dimensional topological orders and fracton orders, starting from a finite
group $G$, a choice of an Abelian normal subgroup $N$, and a choice of
foliation structure. These hybrid fracton orders -- examples of which were
introduced in arXiv:2102.09555 -- can also host immobile, point-like
excitations that are non-Abelian, and therefore give rise to a protected
degeneracy. We construct solvable lattice models for these orders which
interpolate between a conventional, three-dimensional $G$ gauge theory and a
pure fracton order, by varying the choice of normal subgroup $N$. We
demonstrate that certain universal data of the topological excitations and
their mobilities are directly related to the choice of $G$ and $N$, and also
present complementary perspectives on these orders: certain orders may be
obtained by gauging a global symmetry which enriches a particular fracton
order, by either fractionalizing on or permuting the excitations with
restricted mobility, while certain hybrid orders can be obtained by condensing
excitations in a stack of initially decoupled, two-dimensional topological
orders.
