Gamma World (pre-4E anyway) had an array of separate factions that made up many of the opposing forces in the game--ranging from techno-phobic tribes to bands of scavengers who will make use of anything. The idea being that while there are birds of a feather that stick together (within a species) there are factions that are diverse. Which sort of sets up a more realistic, pluralistic society and makes for a more believable world overall.
But I sort of like the species vs. species aspect portrayed in Kamandi, Thundarr, or even Planet of the Apes. I suppose there's a simplicity that's appealing to me--especially when you consider the animal-centric mutants that populate those worlds.
So I'm thinking that before the alien overlords arrive, species/races tend to keep to themselves. Once earthlings (humans/mutants/otherwise) see that their planet may be wiped out, factions begin to pop up and segregated groups would need to set aside their differences and pool resources.
However, some are still looking to come out on top, so they're only going to align themselves with others they think can aid them. Other factions might be more desperate and be open to anyone who wants to join the cause. And there will be of course, those who want to remain true to their species without aligning with anyone--or those who just won't be accepted by anyone.
But then again, the overlords are supremely outfitted and it's going to take a lot more than a few fractured factions to repel the invaders. In the meantime, they'll have to form organically through gameplay of some sort. I won't be writing up orgs for them to join--it'll have to be something the PCs choose to incorporate!