What is the name of the culture that exists within the dominant culture?

The dominant culture is (and also gets referred to as) "majority culture", that is the leading culture of the majority people in some place.

The "other", smaller culture is refereed to as:
-subculture
-subordinate culture
-non-dominant culture
-minority culture

An informal structure in a bureaucracy would be something like so-called "office politics". where there's a formal structure set up for the organization, but then an informal subculture as well that has a way of organizing the bureaucracy further. this can be positive because it sets up an added degree of organization, albeit informal, within the bureaucracy.