Do Noroviruses include many families?
You are mixing different things here: Virus are grouped based on their genetic material (dsDNA, ssDNA ssRNA and so on) into several groups (see here for details), then grouped into families, subgrouped into a genus and then into species. 

So it is: Group > Family > Genus > Species.

What you have here are are different positions in this classification.
The names you mention belong all to the same tree:

Group: IV ((+)ssRNA-Viruses) > Family (Caliciviridae) > Genus (Norovirus) > Species (Norwalk Virus).

The Astrovirus belongs into the same group, but then form an own family.
To answer your question: No, since the Norovirus forms its own genus in the caliciviridae family.