Egg groups
The 15 breeding groups and how many Pokemon each holds
Sharing a group is not enough: genderless Pokemon breed only with Ditto, Ditto breeds with everything except the Undiscovered group, and two Pokemon of the same single gender never breed.
Egg groups and breeding
This page lists the 15 egg groups and how many species belong to each. Field is the largest, with 278, followed by Water 1 with 114 and Bug with 91. Undiscovered holds the 151 species that never breed, not even with Ditto, and Ditto sits in a group of its own. Each group links to its own page with every member species. Alternate forms are not counted separately, since they breed exactly like their species.
Sharing a group is required, but it is not enough. The 155 genderless species have no possible partner other than Ditto, and 116 of them cannot breed even with it. Ditto breeds with anything that can breed except another Ditto, and two species that are always the same gender never breed together, shared group or not. Every Pokémon's page lists the species it can breed with once these rules are applied.