lie groups - how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6?

As a follow-up to Jim's answer (which came in as I was typing an inferior answer), let me add that the 7 possible embeddings are given in the $C_3$ entry of Table VI in the paper: Classification of semisimple subalgebras of simple Lie algebras by Lorente and Gruber. It's of course based on Dynkin, but they work out the details up to rank 6.

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Added

The defining vectors for the 7 embeddings are given by: (1,0,0), (1,1,0), (1,1,1), (2,2,0), (3,1,0), (3,1,1) and (5,3,1). Recall that the embedding with defining vector $(a,b,c)$ is one for which the Cartan generator $H$ of the $mathfrak{sl}(2)$ subalgebra is given by $H = a H_1 + b H_2 + c H_3$, where $(H_i)$ is an orthonormal basis of a Cartan subalgebra of $C_3$ containing $H$.