Find all the subgroups of D4 Which subgroups are normal What

Find all the subgroups of D_4. Which subgroups are normal? What are all the factor groups of D_4 up to isomorphism?

Solution

There is no nice way to find all 10 subgroups, but they are:

(note here that we will use the standard notation r for rotation and s for reflection)

{1}

{1,s}

{1,sr}

{1,sr²}

{1,sr³}

{1,r²}

{1,r,r²,r³}

{1,r²,s,sr²}

{1,r²,sr,sr³}

D4

The normal subgroups are:

{1,r²}

{1,r,r²,r³}

{1,r²,s,sr²}

{1,r²,sr,sr³}

D4