Is there a list of all real unital subalgebras of M(2,C)?

Question

Is there a complete classification of all real unital subalgebras of $M(2,\mathbb C)$ up to isomorphism? The list should include $M(2,\mathbb C)$, the quaternions, complex numbers, split-complex numbers, dual numbers, 2x2 real matrices, $\mathbb C \oplus \mathbb C$, the tensor product of the dual numbers with $\mathbb C$, the dual-complex numbers and so on.