Abstract: I will introduce a stabilizer code model with a qutrit at every edge on a square lattice and with non-invertible plaquette operators. The degeneracy of the ground state is topological as in the toric code, and it also has the usual deconfined excitations consisting of pairs of electric and magnetic charges. However, the model exhibits novel types of confined fractonic excitations composed of a cluster of adjacent faces (defects) with vanishing flux. Their energies grow linearly with their sizes, and these are also immovable by local operations. Deconfined excitations can be either absorbed by these fractonic defects or acquire restricted mobility in presence of the latter. Magnetic monopoles can exist within such a cluster of defects. Furthermore, local operators can annihilate the ground state. All these properties can be captured via a novel type of fusion category in which the product is associative but does not commute, and can be expressed as a sum of (operator) equivalence classes which includes that of the zero operator. I will also introduce many other variants of this model and discuss how these can be included within the quantum field theory paradigm in the continuum limit.