Abstract. Face 2-colourable triangular embeddings of complete tripartite graphs K_n,n,n correspond to biembeddings of Latin squares of side n. We consider biembeddings that contain any of the five Latin squares derived from the Cayley tables of finite groups of order 8. Up to isomorphism, we determined all such biembeddings.

As regards their number:

1. the Cayley table of cyclic group C₈ is in 701 nonisomorphic biembeddings;
2. the Cayley table of C₄xC₂ is in 1,964 nonisomorphic biembeddings;
3. the Cayley table of C₂xC₂xC₂ is in 49 nonisomorphic biembeddings;
4. the Cayley table of dihedral gropup D₄ is in 227 nonisomorphic biembeddings;
5. the Cayley table of quaternion gropup Q is in 236 nonisomorphic biembeddings