Galois realizability of a central C4-extension of D8

Given a finite group G, what are the necessary and sufficient conditions on a field K for it to have an extension field with Galois group G over K? A large body of research has accumulated concerning this question, particularly in regard to small 2-groups. Specifically, conditions have been determined for all but one of the groups of order 32. In this work, we solve the problem for this last group. With these new results, complete conditions are now known for all 2-groups of order less than 64.