# What are the maximal closed subgroups of $\operatorname{SU}_3$?

$$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$$What are the maximal closed subgroups of $$\SU_3$$?

This question is inspired by Lie subgroups of SU(3). Interesting partial answers to that question, treating only the case of connected subgroups, are given by José Figueroa-O'Farrill and Neil Strickland.

Easier example: The three maximal closed subgroups of $$\SU_2$$ are the binary octahedral group, the binary icosahedral group and the normalizer of the maximal torus. Details given here: What are the finite subgroups of $$SU_2(C)$$?

My attempt: The maximal closed subgroups of $$\SU_3$$ are $$U_2\cong S(U_2\times U_1)$$ and $$\operatorname{SO}_3(\mathbb{R}) \times C_3 \cong \langle\operatorname{SO}_3(\mathbb{R}),\zeta_3 I\rangle$$ (I think this is the full normalizer, maybe there is some extra finite order stuff? Weirdly this type of subgroup doesn't seem to show up in table 5 of Antoneli, Forger, and Gaviria - Maximal Subgroups of Compact Lie Groups) and the normalizer of the maximal torus $$N(T)=T^2 \rtimes S_3$$ where here the symmetric group $$S_3=W$$ is the Weyl group of $$\SU_3$$. The finite maximal closed subgroups are $$3 \times \Sigma_{168} \;, \; 3.\Sigma_{216},\;, \; 3.\Sigma_{360}$$ where $$\Sigma_{168} \cong \SL_3(2)$$ is a the simple group of order $$168$$, $$\Sigma_{360} \cong A_6$$ is the simple group of order $$360$$ and $$\Sigma_{216} \cong \mathbb{F}_3^2 \rtimes \SL_2(3)$$ is an affine transformation group, sometimes called the Hessian group. I believe that $$3.\Sigma_{216}$$ is the full automorphism group of an extra special $$3$$-group of order $$27$$ of $$+$$ type, $$\operatorname{Aut}(3^{2+1}_+)$$.

$$\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Cl{Cl}$$Yes the the above is the correct list of maximal closed subgroups of $$\SU_3$$.

Antoneli, Forger, and Gaviria - Maximal Subgroups of Compact Lie Groups classifies all maximal closed subgroups of $$\SU_n$$ whose identity component is not simple (here trivial counts as simple). According to this paper, pages 1013–1018, the maximal closed subgroups of $$\SU_3$$ of this type are the normalizer of the maximal torus $$N(T)=S(U_1 \times U_1 \times U_1): S_3$$ as well as $$S(U_2 \times U_1 )\cong U_2.$$ The maximal closed subgroups with trivial identity component are the finite groups: $$3.A_6$$ of order $$3(360)=1080$$ (known as the Valentiner group) and $$\langle \zeta_3I\rangle \times \GL_3(\mathbb{F}_2)$$ of order $$3(168)=504$$. Both these two groups are central extensions by $$\langle\zeta_3 I\rangle$$ of a finite simple group. But the first is perfect central extension, indeed a Schur cover. While the second is just a direct product. A third finite maximal closed subgroup is the complex reflection group with Shephard–Todd number 25 (see Complex reflection group) and order $$3(216)=648$$ which happens to be a central extension again by $$\langle\zeta_3 I\rangle$$ of the Hessian group of order 216. (Since you are from quantum computing this last subgroup would be known to you as the (determinant-1 subgroup of the) qutrit Clifford group.)

The only maximal closed subgroup of $$\SU_3$$ with nontrivial simple identity component is the direct product $$\langle\zeta_3I\rangle \times \SO_3(\mathbb{R}).$$

To summarize, the maximal closed subgroups of $$\SU_3$$ are \begin{align*} &N(T)\\ & S(U_2\times U_1)\cong U_2\\ & \langle\zeta_3 I\rangle \times \SO_3(\mathbb{R}) \\ & \langle\zeta_3 I\rangle \times \GL_3(\mathbb{F}_2) = 3 \times PerfectGroup(168) \\ & 3.A_6=PerfectGroup(1080)\\ & S(\Cl_1(3))=SmallGroup(648,532) \end{align*} where $$S(\Cl_1(3))$$ is the determinant-1 subgroup of the single qutrit Clifford group. Every closed subgroup of $$\SU_3$$ is contained in one of these $$6$$ maximal groups.

Also since unitary $$t$$ designs are popular in quantum computing it may be of interest to you that $$3.A_6$$ is a unitary 3-design and $$\zeta_3 \times \GL_3(\mathbb{F}_2)$$ and $$S(\Cl_1(3))$$ are both unitary 2-designs.

This is consistent with claim 3 of my answer to Finite maximal closed subgroup of connected Lie group that all maximal $$2$$-design subgroups of $$\SU_n$$ (all $$3$$ designs are also $$2$$ designs) are finite maximal closed subgroups of $$\SU_n$$.

The only other closed subgroups of $$\SU_3$$ which are unitary $$t$$-designs for $$t \geq 2$$ are the $$\GL_3(\mathbb{F}_2)$$ subgroup of $$\langle\zeta_3 I\rangle \times \GL_3(\mathbb{F}_2)$$ and the commutator subgroup of the qutrit Clifford group, SmallGroup(216,88), which has size $$3(72)=216$$. These are again unitary $$t$$-designs for $$t=2$$.

In addition to all 5 of these 2-designs there is one other Lie primitive subgroup (i.e. not contained in any proper positive dimensional closed subgroup): it is a subgroup of the qutrit Clifford group of size $$6(36)=216$$.

• I get a kick out of you telling yourself about the connection between your answer and your other interests ("Since you are from quantum computing this last subgroup would be known to you …"). 😄 Commented Sep 25, 2022 at 22:37
• oh haha wow that's leftover from when I used this as an answer to math.stackexchange.com/questions/497853/… where the OP explicitly stated an interest in quantum computing, but yes it also applies to me here! Commented Sep 25, 2022 at 22:40