New families of maximum distance separable (MDS) codes are constructed from elliptic curves by exploiting their group structures. In contrast to classical constructions based on divisors supported at a single rational point, the proposed approach employs divisors formed by multiple distinct points constituting a maximal subgroup of the curve. The resulting codes achieve parameters approaching the theoretical upper bound $(q + 1 + \lfloor 2\sqrt{q} \rfloor)/2$ and include non Reed-Solomon (RS) MDS codes. The inequivalence of these codes to RS codes is established through an explicit analysis on the rank of the Schur product of their generator matrices. These results extend the known parameter range of elliptic MDS codes and provide additional evidence supporting the tightness of existing upper bounds for algebraic geometry MDS codes.