Orthogonal Latin square graphs based on groups of order 8

An orthogonal Latin square graph is a graph whose vertices are Latin squares of the same order, adjacency being synonymous with orthogonality. We are interested in the orthogonal Latin square graph in which each square is orthogonal to the Cayley table M of a group G and is obtained from M by permuting columns. The structure of this graph is completely determined by the structure of Orth(G), the orthomorphism graph of G. The structure of Orth(G), for G = Z2 × Z4, D8, or Q8, has been determined through computer searches only; we will present a theoretical determination of these structures.