Answer:
420
Step-by-step explanation
First, we have 6*5 permutations of doges and 8*7 permutations of cats.
So we have 6*5*8*7 permutations of 2 doges and 2 cats.
However, since order doesn't matter we divide that by 2*1*2*1.
Therefore 6*5*8*7/4 = 420