“Race” is, in origin, a mythology drawn around culture; it is either an actual ethnicity that is an in-group or an imaginary one ascribed on the basis of external appearance as an out-group. And the mere act of creating that distinction by an in-power group can create a shared experience reifying the ascribed ethnicity into a real one over time.
But, no, race is not apart from culture, but a product and aspect of it.