Imaginary quadratic fields k with Cl_2(k) = (2,2^m) and Rank Cl_2(k^1) =   2
  We classify all complex quadratic number fields with 2-class group of type (2,2^m) whose Hilbert 2-class fields have class groups of 2-rank equal to 2. These fields all have 2-class field tower of length 2. We still don't know examples of fields with 2-class field tower of length 3, but the smallest candidate is the field with discriminant -1015. 
Elliot Benjamin, Franz Lemmermeyer, Chip Snyder