A bstract Motivated by an analogous result for K3 models, we classify all groups of symmetries of non-linear sigma models on a torus T _(4)that preserve the N $$ mathcal{N} $$ = (4 , 4) superconformal algebra. The resulting symmetry groups are isomorphic to certain subgroups of the Weyl group of E ~(8), that plays a role similar to the Conway group for the case of K3 models. Our analysis heavily relies on the triality automorphism of the T-duality group SO(4 , 4 , ? ). As a byproduct of our results, we discover new explicit descriptions of K3 models as asymmetric orbifolds of torus CFTs.
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