Before revision:  Our main result is that there are 6,966 combinatorialtriangulations of the hexahedron which can be classified into 174 equivalence classes. Our results are consistent with theoretical results available on the subdivision of the cube .
After revision:  Our main result is that there are 6,966 combinatorialtriangulations of the hexahedron which can be classified into 174 triangulations has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. This result is purely combinatorial. We also demonstrate that a minimum of 114 triangulations do correspond to hexahedron geometrical triangulations. Each hexahedron is valid for finite element computations (strictly positive Jacobian determinant). We exhibit the tetrahedral meshes for these geometries and show in particular triangulations of the hexahedron with 14 tetrahedra .