Do we know a complete classification of all closed subgroups of $U(2)$.
1) upto isomorphism
2) upto conjugation
If so could someone provide me a reference as to where I might find this or some techniques as to how does one begin such a classification.The only ones I can see apriori are the usual $T^2$ (sitting as diagonal matrices) $S^1$ (sitting as (p,q) circles inside $T^2$) and the various groups that come from lifting subgroups of $SO(3)$ to $SU(2)$ under the usual covering map. I presume there should be a lot more but do we have some sort of classification in the literature?
Or if its easier to somehow classify just the abelian ones it would still be helpful is someone could point me to a source on this classification as well.
Thanks!