I'm looking for a reference describing the closed connected Lie subgroups of $SU(3)$. I know they are $SU(2)\times U(1)$, $SU(2)$, $SO(3)$ and several abelian subgroups based on this mathoverflow post. However the references given there only point to general information about Lie groups, and don't directly list the Lie subgroups of $SU(3)$.

Does anyone know of any direct reference for this fact?

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    $\begingroup$ If you assume in addition that your subgroup is connected then the answers given in the MO post give you the complete list. Things are harder for disconnected subgroups. You may have to search for classification of finite subgroups first. $\endgroup$ Sep 18, 2013 at 20:23
  • $\begingroup$ @studiosus Thanks for the input - I'm looking for a reference for the connected subgroups only. The disconnected subgroups are described in the following reference: W. M. Fairbairn, T. Fulton, and W. H. Klink. Finite and Disconnected Subgroups of SU3 and their Application to the Elementary-Particle Spectrum. Journal of Mathematical Physics, 5(8):1038, 1964. $\endgroup$ Sep 18, 2013 at 20:28
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    $\begingroup$ The subgroups are closed and hence Lie subgroups. Furthermore, since $SU(3)$ is compact so too are its closed subgroups. Thus you are asking for a classification of the compact Lie groups which embed into $SU(3)$. The connected + simply connected ones are totally classified, and are products of $SU(n), Sp(n),Spin(n), G_2, F_4, E_6, E_7$, and $E_8$. Hence you need only look for the non-simply connected ones. $\endgroup$ Sep 18, 2013 at 20:41
  • $\begingroup$ @TylerHolden Thanks for your comment - do you know any direct reference for $SU(3)$ specifically? $\endgroup$ Sep 18, 2013 at 20:57
  • $\begingroup$ To give some motivation: I'm using this fact in a quantum computing paper, so the audience is not necessarily familiar with Lie theory. It would be helpful to have a direct reference so that the readers don't have to trust me that it's "well-known." $\endgroup$ Sep 18, 2013 at 20:59

1 Answer 1


The three maximal (proper) closed subgroups of $ SU_2 $ are known. The first is $ N(T) $ the normalizer of the maximal torus. The two other maximal closed subgroups are finite: the binary icosahedral group $ 2.A_5 \cong SL(2,5) $ of order $ 120 $ and the binary octahedral group $ 2.S_4 \cong SL(2,3):2 \cong SmallGroup(48,28) $ of order $ 48 $. For details see

What are the finite subgroups of $SU_2(C)$?

What are the maximal closed subgroups of $ SU_3 $? https://arxiv.org/pdf/math/0605784.pdf 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 $$ N(T)=S(U_1 \times U_1 \times U_1)\rtimes S_3 $$ 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 https://en.wikipedia.org/wiki/Valentiner_group) and $$ <\zeta_3I> \times GL_3(\mathbb{F}_2) $$ of order $ 3(168)=504 $. Both these two groups are central extensions by $ \zeta_3 I $ of a finite simple group. A third finite maximal closed subgroup is the complex reflection group with Shephard-Todd number 25


and order $ 3(216)=648 $ which happens to be a a central extension again by $ \zeta_3 I $ of the Hessian group https://en.wikipedia.org/wiki/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, which I denote $ S(Cl_1(3)) $.)

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

To summarize, the maximal closed subgroups of $ SU_3 $ are Type I (normalizer of a maximal connected subgroup) \begin{align*} & U_2 \cong S(U_2\times U_1)\\ & \zeta_3 \times SO_3(\mathbb{R}) \\ \end{align*} Type II (finite maximal closed subgroup) \begin{align*} & \zeta_3 \times GL(3,2) \\ & 3.A_6\\ & S(Cl_1(3)) \end{align*} Type III (normalizer of a positive dimensional non maximal connected subgroup) $$ N(T)=S(U_1 \times U_1 \times U_1)\rtimes S_3 $$ the 1 component is not maximal connected because it is contained in the rank 2 subgroup $ U_2 $ above.

Again, $ S(Cl_1(3)) $ is the determinant one subgroup of the single qutrit Clifford group i.e. Shephard-Todd-25.

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 https://math.stackexchange.com/a/4477296/758507 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 $ \zeta_3 \times GL_3(\mathbb{F}_2) $ and the commutator subgroup of the qutrit Clifford group, which has size $ 3(72)=216 $. These are again unitary $ t $-designs for $ t=2 $.


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