I am taking a course in Analytic Number Theory. The main object of study is arithmetical functions. Moreover, if we look at the arithmetical functions which do not vanish at $1$, then they form a group under Dirichlet multiplication, $*$. An important subgroup of this group is the group of multiplicative functions. My question is: Is there other important subgroups of this group? What do we know about these groups? Is there any "famous" group among these subgroups?
Thanks, in advance.