Normal Subgroups and Quotient Groups · Medium problem

$H$ is normal, since $r^3$ is central in $D_6$

$H$ is normal, since $|H|=4$ divides $|D_6|=12$

$H$ is not normal, since $4$ does not divide $12$ evenly

$H$ is not normal: conjugating $s$ by $r$ gives $rsr^{-1} = r^2s \notin H$