The short answer to this question is a simple no. The terms "subculture" and "counterculture" are separated in sociology for just this reason. Let me explain. A subculture is defined as a smaller division of an overarching society. A subculture is, however, noticeably different from the rest of the population.
There is an understood meaning of what it is to be part of a subculture, both on the part of the subculture and the overarching society. A classic example of an American subculture would be the Amish. Another way to define a subculture would be to say a subculture is any smaller division of society that is exclusive. This means that the subculture is still part of society, but not everyone is included. For example, police officers, teachers, and doctors are all subcultures.
One rule of thumb that I teach my students is that if a group of people in a society have their own unique and understood lingo, then they are probably a subculture. For example, police codes that are used on radios are probably only really understood by members of the subculture. The important thing to remember is that a subculture is perfectly content with the overarching society. They are simply different from the norm in that society.
A counterculture, on the other hand, has the same designation of being understood to be different from the norm. However, a counterculture runs counter to some aspect of the overarching society. In short, a counterculture wants to make a change to the overarching society. They wish to take their different beliefs and impose them onto the overarching society. They are not content with society's status quo; they have an axe to grind, so to speak.
Neo Nazis would be an example of a counterculture. They want to alter society to suit their desires. The difference is that a subculture does not try to change society. The Amish do not try to convert anyone—they just do their own thing.