  In a projective plane PG(2,K) defined over an algebraically closed field K of
characteristic 0, we give a complete classification of 3-nets realizing a
finite group. An infinite family, due to Yuzvinsky, arises from plane cubics
and comprises 3-nets realizing cyclic and direct products of two cyclic groups.
Another known infinite family, due to Pereira and Yuzvinsky, comprises 3-nets
realizing dihedral groups. We prove that there is no further infinite family.
Urzua's 3-nets realizing the quaternion group of order 8 are the unique
sporadic examples. If p is larger than the order of the group, the above
classification holds true in characteristic p>0 apart from three possible
exceptions Alt_4, Sym_4 and Alt_5.
