What Are the Subgroups and Normal Subgroups of D4?

[Trigger]

Calculate all the subgroups of D_4.
Which of them are normal subgroups? (It can
be shown that any subgroup containing half
the elements of a group G is a normal
subgroup, and if a has order 2 then {e,a} is
a normal subgroup iff a commutes with all
elements of G.)

{e,R^2} happens to be a normal subgroup.
Give the Cayley table of D_4/{e,R^2}.

 

[Trigger]

I have made out a Cayley Table for D4, but do not know how to calculate the subgroups. Please help!

 

[matt grime]

What the heck is R?

Let D_4 be e,t,t^2,t^3,s,st,st^2,st^3 t the rotation s the reflection (see how explaining notation can help?)

Let H be a proper subgroup. if H contains t, then it contains all its powers, and has order 4. It can contain no other elements as its order must divide 8, and hence would contain all elements.

If H doesn't contain t then... do some thinking and use the definitions of things.