Are Lesbian and Gay the only subsets of homosexuality?

Yes.

Homosexuality is when person is attracted by same gender.

And since there are only two genders there's only gay and lesbian. However, there's other sexualities like heterosexuality, bisexuality, and asexuality.

I'd say pansexuality would be another subset because it includes being attracted to intersex people (hermaphrodites), and transsexuals. But of course, "homo" means "same," so... if there was a term for trans men attracted to other trans men, or trans women attracted to other trans women, then that would be true "homo"sexuality. But then again, trans men *are* real men, and trans women are real women... I mean, I'm one of the former and I consider myself a very real, normal dude...

Ah; I rambled...! Short answer is "Yes!" :)

no...

there's the george washington haircut subset...

I think so... If you're bisexual you are bisexual not homosexual and if you are pansexual you are pansexual not homosexual.

Uhm, sorta, but there's pan-sexual or bi-sexual.