I assume you mean the philosophy of mathematics. In my mind, it is basically just a part of philosophy with lots of (interesting) questions, rather than having lots of sub-branches, as something like ethics does (which is quite diverse).
In the philosophy of mathematics, people talk about things like:
- What would serve as a good foundation of mathematics? For instance, research on how mathematics can be done constructively. (mathematical logic)
- What can one say about undecidable statements in mathematics? (i.e. can they be given a truth value, or not. This is also mathematical logic).
- Can we give a better idea of what mathematical beauty means? (aesthetics)
- Is mathematics invented or discovered? (misc.)
- Why does physics need mathematics? (misc. / philosophy of science)
Unfortunately I fear that I probably haven't told you anything new so I apologize if this is the case.