Let \(\mathcal{K}\) be the set of number fields \(K\) with discriminant \(\pm2^a\) (equivalently, \(K/\mathbb{Q}\) is unramified outside \(\{2, \infty \}\)) and \(9\leq[K : \mathbb{Q}]\leq 15\). Using well known lower bounds for discriminants and a study of higher ramification groups, the author proves that \(\mathcal{K}=\emptyset\). As a corollary he obtains that the following simple groups can not be Galois groups of an extension of \(\mathbb{Q}\) with discriminant \(\pm2^a\): Mathieu groups \(M_{11}\) and \(M_{12}\), \(\text{PSL}(3, 3)\) and the alternating groups \(A_i\), \(8<i<16\).