This problem is from Michael Artin Algebra first edition.
6.3.3) List all subgroups of the dihedral group $D_{4}$ and divide them into conjugacy classes.
I am a bit unsure what I am being asked here since, so far, I have been thinking of conjugacy classes as sets of elements of a group and not sets of subgroups of a group.
The subgroups of $D_{4}$ I have found, where $x$ is a rotation of order 4 and $y$ is a reflection, are:
$D_{4}, \{e\}, \{e, x, x^{2}, x^{3}\}, \{e, x^{2}\}, \{e, y\}, \{e, yx\}, \{e, yx^{2}\}, \{e, x^{2}, y, yx^{2}\}, \{e, yx^{3}\}, \{e, x^{2}, yx, yx^{3}\}$
Using the help of Andreas Caranti in the comments, we can say that Artin is interested in the conjugacy classes under the action of $D_{4}$ on its subgroups. By testing the conjugacy action of the elements of $D_{4}$ on each of the following subgroups: $D_{4}, \{e\}, \{e, x, x^{2}, x^{3}\}, \{e, x^{2}\}, \{e, y\}, \{e, yx\}, \{e, yx^{2}\}, \{e, x^{2}, y, yx^{2}\}, \{e, yx^{3}\}, \{e, x^{2}, yx, yx^{3}\}$, we find that $\{\{e, y\},\{e, yx^{2}\}\}$ and $\{\{e, yx\},\{e, yx^{3}\}\}$ are the only conjugacy classes that are not singletons.
For a simple example of how one would check for the conjugacy classes, take $x\in D_{4}$ and the subgroup $\{e, yx^{2}\}$. We have $xyx^{2}x^{-1}=xyx^{2}x^{3}=xyx^{4}x=xyx=yx^{3}x=y$, so that $\{e, yx^{2}\}$ is conjugate to $\{e, y\}$. We can check for the rest of conjugacy classes in a similar way.