TITLE: Group of order 12
QUESTION [8 upvotes]: Is it true or false that a group of order 12 always has a normal 2-sylow subgroup? 
I have a hunch it is false..

REPLY [4 votes]: The 5 groups of order 12 are
$C_{12}$, $C_6 \times C_2$ in the abelian case, and
$A_4$ (group of all even permutations of length 4),
$D_6$ (group of all symmetries of the regular hexagon),
$C_3 \rtimes C_4$ in the nonabelian case.