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> Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers?
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I can only find incomplete lists on wikipedia, mathoverflow or other sources in the internet.



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> Question 2: Is there a computer algebra package that can list those finite subgroups up to some order? Maybe a package of GAP?
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Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$
Finite subgroups of $\operatorname{SO}(4)$ and $\operatorname{O}(4)$ are enumerated in Chapter 4 of the book:


*Conway, John H.; Smith, Derek A.*, **On quaternions and octonions: their geometry, arithmetic, and symmetry**, Natick, MA: A K Peters (ISBN 1-56881-134-9/hbk). xii, 159 p. (2003). [ZBL1098.17001](https://zbmath.org/?q=an:1098.17001).


There may be results for some $n>4$, but since every finite group has faithful orthogonal representations, this gets hopeless at some point.


