I always tried to understand how the *finite reflection groups* of $\Bbb R^d$ (of some fixed dimension $d$) relate to the *point groups* of the same space $\smash{\Bbb R^d}$ (finite subgroup of the orthogonal group $\smash{\mathrm O(\Bbb R^d)}$).

Initially, I was under the impression that each point group is a subgroup of a finite reflection group. This turned out to be wrong, which is obvious in hindsight. Many reflections groups have symmetries in the placement of their mirrors that can be used to enlarge the group.

So let's take these enlarged groups instead. From my geometric understanding, by that I mean the symmetry groups of the uniform polytopes. So I shall call them **uniform point groups**. Most (or all?) uniform polytopes can be generated from a reflection group, and then it has all the symmetries of this group, but might have more.

Question:Is every point group a subgroup of a uniform point group?

Regardles of the answer to that question, I am open for any statement that sheds light on the placement of reflections groups (or easily derived groups thereof) inside the family of general point groups.

**Update** Sep. 2019

There seems to exist a counterexample in dimension four, namely, a point group denoted $\pm[I\times C_n]$ that is supposedly not the subgroup of a symmetry group of a uniform polytope. This was mentioned in this answer by Günter Rote. Currently I am not able to verify the claim. So, any hint is welcome.

samedimension. Edited the question to make this clearer. $\endgroup$ – M. Winter Sep 15 '19 at 16:48isomorphicto a subgroup of the symmetry group of a simplex. But I am not interested in abstract groups, but matrix groups. $\endgroup$ – M. Winter Nov 4 '19 at 13:431more comment