What subalgebras does the exceptional Lie algebra $\mathfrak{g}_2$ have?

Question

I am currently working on my bachelors thesis, which is about classifying the (maximal) subgroups of the compact real form $G_2$. So far, using Borel and de Siebenthal theory, I have identified $\mathfrak{su}(3)$ and $\mathfrak{su}(2) \times \mathfrak{su}(2)$ as subalgebras of $\mathfrak{g}_2$. I am pretty sure that $\mathfrak{su}(2) = \mathfrak{so}(3) $ itself is also a subalgebra of $\mathfrak{g}_2$. My question is, if there are any more? Are there any more subalgebras of $\mathfrak{g}_2$ or subgroups of $G_2$? I dont need a proof, I just want to know how much more work is ahead of me, and I can't really find any complete lists for this topic. Thanks a lot!