I read from a paperarXiv: cond-mat/0602109 by a theoretical physicist, Prof. Frank Bais, close subgroups of $SO(3)$ is given by ${C_n,D_n,T,O,I,SO(2)\rtimes Z_2}$, where $C_n$ is the cyclic group of order $n$, $D_n$ is dihedral group of order $n$, $T$ is the tetrahedral group, $O$ is the octahedral group, and $I$ is the icosahedral group. However, I have some questions to understand this.
Is $SO(2)$ a subgroup of $SO(3)$? The subgroups listed in the paper mentioned above is about close subgroups. What's the meaning of 'close' here?
If $SO(2)$ is a subgroup of $SO(3)$, can it be parameterized as a $3\times 3$ matrix as \begin{align} \left( \begin{array}{ccc} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{array} \right) \end{align} with a imagined '$z$' axis?
Can the subgroup $SO(2)\rtimes Z_2$ be parameterized as a $3\times 3$ matrix? E.g, \begin{align} \left( \begin{array}{ccc} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & \ 1 \end{array} \right), \left( \begin{array}{ccc} \cos\theta & \sin\theta & 0 \\ \sin\theta & -\cos\theta & 0 \\ 0 & 0 & \ -1 \end{array} \right) \end{align}
Moreover, any suggestions of tutorial references or textbooks that I can find the list of subgroups or the way I can calculate them by myself are also warmly welcome. I am not familiar with continuous groups.