With any integral lattice \(\Lambda\) in \(n\)-dimensional euclidean space, one can associate an elementary-abelian 2-group whose elements represent parts of the dual lattice that are similar to \(\Lambda\). The rank of this group is at most the number of primes dividing the determinant of \(\Lambda\); in case of equality, \(\Lambda\) is called strongly modular. In a subsequent paper by Rains and the reviewer it is shown that the rank also is at most \(2[n/2]\) and that lattices attaining this bound exist for any \(n\). The present paper contains a detailed study of the case \(n=2\). In this case a lattice is strongly modular essentially when it has order 4 in the appropriate class group. This allows the author to determine when and how many such lattices for a given determinant exist.