Let us denote by [Formula: see text] the Clifford group (the circuit or operations generated by Hadamard, [Formula: see text] phase and the controlled-NOT gates) and by [Formula: see text] the set of qubit states that can be prepared by circuits from the Clifford group. In other words, a state [Formula: see text] if [Formula: see text] where [Formula: see text]. We will refer to states in [Formula: see text] as Clifford states. This paper studies the set of all three-qubit Clifford states. We prove that [Formula: see text] has 8640 states and if we define two states [Formula: see text] and [Formula: see text] in [Formula: see text] to be equivalent if [Formula: see text], with [Formula: see text] a local transformation in [Formula: see text], then the resulting quotient space has five orbits. More exactly, [Formula: see text] where the orbit [Formula: see text] is made up of states with entanglement entropy [Formula: see text]. For example, the first orbit [Formula: see text] contains the state [Formula: see text] and corresponds to the unentangled Clifford states. We say that [Formula: see text] is a real state if all its amplitudes [Formula: see text] are real numbers. We also say that an operator is real if all the entries of its matrix representation with respect to the computational basis are real numbers. In this paper, we also study the set of real Clifford 3 qubits and the way this set splits when we identify two real Clifford states [Formula: see text] and [Formula: see text] to be equivalent if [Formula: see text] where [Formula: see text] is a local real Clifford operator. An interesting aspect that follows from this study of Clifford states is the existence of two real Clifford states [Formula: see text] and [Formula: see text] that can be connected with a Clifford local transformation but they cannot be connected with a real Clifford local transformation. This is, the equation [Formula: see text] for [Formula: see text], does have a solution in the set of local transformations from [Formula: see text] but it does not have a solution among the local transformations from [Formula: see text] that are real. We go a little deeper and show that the equation [Formula: see text] does not have a solution for any local operation (not necessarily Clifford) whose entries are real numbers. Finally, we show how the CNOT gates act on the set of Clifford states and also in the set of real Clifford states.