Summary: Four-level quantum systems, known as quartits, and their relation to two-qubit systems are investigated group theoretically. Following the spirit of Klein’s lectures on the icosahedron and their relation to Hopf sphere fibrations, invariants of complex reflection groups occurring in the theory of qubits and quartits are displayed. Then, real gates over octits leading to the Weyl group of $E_{8}$ and its invariants are derived. Even multilevel systems are of interest in the context of solid state nuclear magnetic resonance.