I am trying to find the connected subgroups of the Lie Group $SL_2(\mathbb{C})$ up to conjugation. My thought was to find the sub-algebras of the Lie Algebra $\frak{sl}_2(\mathbb{C})$, and then exponentiate. For the one dimensional sub algebras, it should just be determined by the Jordan form. I am a little stuck with the 2-dimensional sub algebras.
I think that the answer should just be $$\left\{\begin{pmatrix}a & b\\0 & -a \end{pmatrix}: a,b \in \mathbb{C}\right\} $$
but i'm not sure how to prove it. Any advice would be appreciated.