.Do only non-abelian and abelian groups
Do "only"?? These are all the kinds of groups there are out there! Every group is either abelian or non-abelian.
have subgroups of its own factor? Or can other type of groups have it subgroups of its own factor as well?
You'll have to try harder next time and explain what do you mean by "its own factor" (it seemed for a moment like you were talking about non-hopfian groups but I greatly doubt it), and why do you think there are "other" types of groups besides abelian and non-abelian.