We classify Haag-dual Poincar\'e covariant subsystems $\mathcal{B}\subset \mathcal{F}$ of a graded-local net $\mathcal{F}$ on 4D Minkowski spacetime which satisfies standard assumptions and has trivial superselection structure. The result applies to the canonical field net $\mathcal{F}_\mathcal{A}$ of a net $\mathcal{A}$ of local observables satisfying natural assumptions. As a consequence, provided that it has no nontrivial internal symmetries, such an observable net $\mathcal{A}$ is generated by (the abstract versions of) the local energy-momentum tensor density and the observable local gauge currents which appear in the algebraic formulation of the quantum Noether theorem. Moreover, for a net $\mathcal{A}$ of local observables as above, we also classify the Poincar\'e covariant local extensions $\mathcal{B} \supset \mathcal{A}$ which preserve the dynamics.