1. The problem statement, all variables and given/known data I want to find the Sylow 2-subgroups of the permutation group S4 2. Relevant equations I dont understand why is my application of Sylow's third theorem wrong. 3. The attempt at a solution The order of S4 is 24=233. Thus, there are Sylow 2-subgroups and Sylow 3-subgroups by Sylow's first theorem. By Sylow's third theorem, the number of Sylow 2-subgroups is k=1 mod 2= 1,3,5,.... and must divide the order of the group. Thus, k=1 or 3. But, each permutation (12), (13), (23), (14), (34) together with the identity permutation forms a subgroup of order 2 of S4. Thus, K should not be 3.