Lower bounds on the 2-class number of the 2-Hilbert class field of imaginary quadratic number fields with elementary 2-class group of rank 3

Let k be an imaginary quadratic number field with C-k,C-2, the 2-Sylow subgroup of its ideal class group, isomorphic to Z/2Z x Z/2Z x Z/2Z. We formulate lower bounds on the 2-class number of the 2-Hilbert Class Field, k(1), of these fields by examining the capitulation of ideals of k in its seven unramified quadratic extensions. We describe fields Ic for which \C-k1,C-2\ greater than or equal to 8 and \C-k1,C-2\greater than or equal to 16.