Drawing subgroup diagram of Dihedral group $D4$

Question

I am reading Fraleigh p. 80 in A First Course in Abstract Algebra, and in the book i see the elements and subgroup diagram of the dihedral group $D_4$. Here are they:

Here is how i try to draw the subgroup diagram: p₀ must be included in every subgroup since it is identity element. Then i look at p₁, and try to find the subgroups including p₁, since p₁ is included, the inverse of it must be included also, and p₁op₁ must be included also, and so on . I need to check whether the result of these computations make it closed. But this lookslike a very long process. Is there an easy way to do that? For example, by looking at u₁ can we immediately say that it is included or not in a subgrouo without checking all compositions of u₁ with u₁? Or by looking at p₁ can we find < p₁ > easily? Thanks

Answer 1

Personally, I am familiar to the following presentation of $D_4$ (or $D_8$). I added it here maybe you find it easier. In fact, You can feel the elements in this presentation easier.

Answer 2

Geometric interpretation is convenient for $D_4$. It is the invariance group of the square.

$D_4$ has three $\frac{\pi}{2}$ rotations, making up the subgroup $R_0$, $R_1$, $R_2$, $R_4$ (replacing $\rho$ of the textbook of the original query with upper case $R$). It is also clear that $R_0$ and $R_2$ ($\pi$ radians rotations) make up a subgroup.

$M_1$ and $M_2$ can be taken to be reflections in lines joining the opposite sides of the square and $d_1$ and $d_2$ are reflections in the two diagonals. It is obvious that each of these four elements squared is the identity, making four more proper subgroups of order 2.

A little visualisation shows that $R_2$ has the property of exchanging $M_1$ and $M_2$ and separately $d_1$ and $d_2$. This explains the other order 4 subgroups in the figure from the textbook.