Finite Group DatabasebetaFinite Group DatabasebetaOrder 972.
516 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C3×H(3,𝔽3) | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#1094 | C3 | 1 | 12 | yes | maximal, complemented |
| 3 | #66437 | C3 | 1 | 27 | yes | maximal, complemented |
| 4 | 3−1+4 | — | 4 | 1 | no | maximal, Sylow |
| 5 | C2×C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 6 | C22×C33 | C32 | 1 | 4 | yes | — |
| 7 | C22×H(3,𝔽3) | C32 | 1 | 9 | yes | — |
| 8 | #1094 | C32 | 1 | 9 | yes | — |
| 9 | #1094 | C32 | 1 | 72 | yes | complemented |
| 10 | C3×C22⋊C9 | C32 | 1 | 36 | yes | complemented |
| 11 | #737 | — | 4 | 27 | no | — |
| 12 | C3×3−1+2 | — | 4 | 12 | no | — |
| 13 | C3×H(3,𝔽3) | A4 | 1 | 1 | yes | characteristic |
| 14 | C2×C33 | — | 3 | 4 | no | — |
| 15 | C2×H(3,𝔽3) | — | 3 | 9 | no | — |
| 16 | C22×C32 | — | 9 | 4 | no | — |
| 17 | C22×C32 | C33 | 1 | 12 | yes | — |
| 18 | C22×C32 | C33 | 1 | 1 | yes | characteristic |
| 19 | C22⋊C9 | C33 | 1 | 27 | yes | — |
| 20 | C33 | C3×A4 | 1 | 4 | yes | — |
| 21 | 3−1+2 | — | 4 | 9 | no | — |
| 22 | 3−1+2 | — | 4 | 72 | no | — |
| 23 | H(3,𝔽3) | C3×A4 | 1 | 9 | yes | — |
| 24 | C3×C9 | — | 4 | 36 | no | — |
| 25 | C2×C32 | — | 3 | 1 | no | — |
| 26 | C2×C32 | — | 3 | 12 | no | — |
| 27 | C2×C32 | — | 9 | 12 | no | — |
| 28 | C22×C3 | — | 3 | 1 | no | — |
| 29 | C22×C3 | — | 3 | 12 | no | — |
| 30 | C22×C3 | C34 | 1 | 1 | yes | characteristic |
| 31 | C32 | — | 9 | 4 | no | — |
| 32 | C32 | C32×A4 | 1 | 12 | yes | — |
| 33 | C32 | C32×A4 | 1 | 1 | yes | characteristic |
| 34 | C9 | — | 4 | 27 | no | — |
| 35 | C2×C3 | — | 3 | 3 | no | — |
| 36 | C2×C3 | — | 3 | 1 | no | — |
| 37 | C2×C3 | — | 9 | 12 | no | — |
| 38 | C22 | 3−1+4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 39 | C3 | — | 3 | 1 | no | — |
| 40 | C3 | — | 3 | 12 | no | — |
| 41 | C3 | C33×A4 | 1 | 1 | yes | characteristic |
| 42 | C2 | — | 3 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C33×A4 |
| commutator, socle, cosocle | C22×C3 | C34 |
| Fitting, prime radical | C22×C3×H(3,𝔽3) | C3 |
| hypercenter | C3×H(3,𝔽3) | A4 |
| hypocenter | C22 | 3−1+4 |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C3 ◁ C3×H(3,𝔽3) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#1094 | C3 | semidirect |
| 3. #66437 | C3 | semidirect |
| 9. #1094 | C32 | semidirect |
| 10. C3×C22⋊C9 | C32 | semidirect |
| 38. C22 | 3−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×H(3,𝔽3) | 3 |
| C3×#1094 | 3 |
| #66437 | 3 |
| 3−1+4 | 4 |