Can anyone clarify for me how many cuboctahedra there are in the 24 cell?

You can easilly see one of them inside this projection.

I believe I can trace 8 of them, but I am not sure if there are more.

I would appreciate it if you can also explain how to calculate this effectively.

This quote from wikipedia seems to imply to me that there might be 12 (or 24?) cuboctahedra -

"The cell-first parallel projection of the 24-cell into 3-dimensional space has a cuboctahedral envelope."

https://en.wikipedia.org/wiki/24-cell