If you consider the symmetry group of the regular icosahedron to be "particularly large" (the group contains several rotations about non-coplanar axes and also orientation-reversing reflections), then the following sequence of refinements of the icosahedral triangulation may be of interest to you. Each of them consists of a family of triangles, their mesh converges to zero, while the triangles remain fairly robust, i.e., not too different from being equilateral and not too different from each other. (There is little hope for finding a triangulation consisting of arbitrarily many triangles that are both small and congruent at the same time.)
Start by projecting the faces of the inscribed regular icosahedron to the sphere; get 20 congruent equilateral triangles. Then partition each triangle by connecting the midpoints of its sides by geodesic arcs on the sphere. Each triangle of the first triangulation is replaced by four smaller triangles; the one in the middle remains equilateral, the other three do not. Keep refining the partition recursively. After the $n$-th partition you will have a triangulation consisting of $20n^4$ triangles, all of approximately the same small diameter, and all not too far from being equilateral. I think the difference between their shapes is not greater than it is after the first partition. The symmetry group for each of these triangulations remains unchanged.