  We discuss the procedure for gauging on-site $\mathbb{Z}_2$ global symmetries
of three-dimensional lattice Hamiltonians that permute quasi-particles and
provide general arguments demonstrating the non-Abelian character of the
resultant gauged theories. We then apply this general procedure to lattice
models of several well known fracton phases: two copies of the X-Cube model,
two copies of Haah's cubic code, and the checkerboard model. Where the former
two models possess an on-site $\mathbb{Z}_2$ layer exchange symmetry, that of
the latter is generated by the Hadamard gate. For each of these models, upon
gauging, we find non-Abelian subdimensional excitations, including non-Abelian
fractons, as well as non-Abelian looplike excitations and Abelian fully mobile
pointlike excitations. By showing that the looplike excitations braid
non-trivially with the subdimensional excitations, we thus discover a novel
gapped quantum order in 3D, which we term a "panoptic" fracton order. This
points to the existence of parent states in 3D from which both topological
quantum field theories and fracton states may descend via quasi-particle
condensation. The gauged cubic code model represents the first example of a
gapped 3D phase supporting (inextricably) non-Abelian fractons that are created
at the corners of fractal operators.

Strongly Correlated Electrons;Quantum Physics