Is there any easier way to find out every proper subgroups of Dihedral group 4?

Question

There are 8 elements in Dihedral group 4

When finding proper subgroups of this, do I have to list every element and draw a table to find out subgroups ?

Answer 1

By Lagrange's theorem we need to really find subgroups of order 2 and 4. As identity element has to be there finding a subgroup of order 2 is the same as finding an element $s$ of order 2. Rotation by $pi/2$ and all reflections give one each.

To get a subgroup of order 4, there are two types: cyclic and non-cyclic. There is just one cyclic group, rotations by angles, $pi/2,\pi,3\pi/2, 2\pi$.

For the non-cyclic you need to find two elements $s_1,s_2$ of order 2 such that $s_1s_2=s_2s_1$, then $\{id, s_1,s_2,s_1s_2\}$ will be a subgroup. This is the only case you have to look up the table.