This work is a pioneer investigation of semigroups and groups over the Loubéré Magic Squares. By the Loubéré Magic Squares, we understand the magic squares formed by the De La Loubéré Procedure. The set of the Loubéré Magic Squares equipped with the matrix binary operation of addition forms a semigroup if the underlining set so considered is the multi set of natural numbers; and if we consider the multi set of integer numbers as the underlined set of entries of the square, the set of the squares enclosed with the aforementioned operation forms an abelian group. The Loubéré Magic Squares are always recognized with centre piece C and magic sum M(S). We showcase that the set of the centre pieces and the set of the magic sums form respective abelian groups if both are equipped with integer numbers operation of addition. We also explicate that the set of the eigen values of the squares enclosed with the integer addition (operation) forms an abelian group. We reveal that the subelement (a terminology we introduced) Magic Squares of the Loubéré Magic Squares forms a semigroup and the Subelement Magic Squares of the Loubéré Magic Squares Group forms a group, with respect to the matrix binary operation of addition.