The optimal systems and symmetry breaking interactions for the (1+2)-dimensional heat equation are systematically studied. The equation is invariant under the nine-dimensional symmetry group H2. The details of the construction for an one-dimensional optimal system is presented. The optimality of one- and two-dimensional systems is established by finding some algebraic invariants under the adjoint actions of the group H2. A list of representatives of all Lie subalgebras of the Lie algebra h2 of the Lie group H2 is given in the form of tables and many of their properties are established. We derive the most general interactions F(t,x,y,u,u x ,u y ) such that the equation u t =u xx +u yy +F(t,x,y,u,u x ,u y ) is invariant under each subgroup.