The primary honeycombs consist of the face-transitive uniform honeycombs, as well as their regiments under the symmetry of which they are face-transitive. These include the chon regiment members with o4o3o4x and o3o4x *b3o symmetry, and the octet regiment members with x3o4o *b3o symmetry.
These are the two members of the chon regiment with o3o4x *b3o symmetry or greater. Chon has chon symmetry while cuhsquah has octet symmetry. Chon also has regiment members with x4o3o4x symmetry or even square tiling prismatic honeycomb symmetry, but these are in other categories.
1. Chon - Cubic honeycomb. Cells are cubes. Verf is an oct. Symbol is o4o3o4x, can also be o3o4x *b3o, x4o3o4x, or lots of other things.
2. Cuhsquah - Cubihemisquare honeycomb. Cells are cubes and squats. Verf is a thah. Symbol is (o3o3o4x4/3*a)/2
The octet regiment has 32 typical members. Of those, 11 have the full o3o4x *b3o symmetry and the remainder have o3o3o3x3*a symmetry. In this category are the 11 members with full symmetry.
A render of the vertex figures of full-symmetry octet regiment members using Bowers' POV-Ray code, by creeperman7002 on the Polytope Discord.
3. Octet - Tetrahedral-octahedral honeycomb. Cells are tets and octs. Verf is a co. Symbol is o3o4x *b3o or o3o3o3x3*a.
4. Tehtrah (also called datha) - Tetrahemitriangular honeycomb. Cells are tets and trats. Verf is an oho. Symbol is o3x3o3o3/2*a3*c.
5. Ohtrah - Octahemitriangular honeycomb. Cells are octs and trats. Verf is a cho. Symbol is (o3x3o4o4/3*a3*c)/2.
6. Octutoh - Octahedral-tomotetrahedral honeycomb. Cells are octs and tuts.
7. Tuthtrah - Tomotetrahedral-hemitriangular honeycomb. Cells are trats and tuts.
8. Cohtrah - Cuboctahedral-hemitriangular honeycomb. Cells are trats and coes.
9. Tetcoh - Tetrahedral-cuboctahedral honeycomb. Cells are tets and coes.
10. Tutcoh - Tomotetrahedral-cuboctahedral honeycomb. Cells are tuts and coes.
11. Ohohaht - Octahedral-octahemioctahedral-hemihexagonal honeycomb. Cells are octs, ohoes, and hexats. In this regiment, hexat cells appear in compounds of three. This compound is called "fexat" for "fissal hexagonal tiling."
12. Tohohaht - Tetrahedral-octahemioctahedral-hemihexagonal honeycomb. Cells are tets, ohos, and hexats.
13. Chohaht - Cubohemioctahedral-hemihexagonal honeycomb. Cells are choes and hexats.
The following honeycombs are self conjugate: all.
The following form conjugate pairs: none.