In this paper all finite 2-groups G are determined up to isomorphism with the property that G > Ω[sub3](G) and |Ω[sub3](G)| ≤ 2[sup5]. It is also shown that, for each integer n ≥ 2, any finite 2-group G containing exactly one subgroup of order 2[supn+2] and exponent at most 2[supn] satisfies |Ω[subn](G)| = 2[supn+2].