Finite Group DatabasebetaFinite Group DatabasebetaOrder 972.
57 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#6882 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×#733 | C3 | 1 | 3 | yes | maximal |
| 3 | C4×C9⋊C9 | C3 | 1 | 1 | yes | maximal, characteristic |
| 4 | #6882 | C4 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 5 | C2×#733 | C2×C3 | 1 | 3 | yes | — |
| 6 | C2×C9⋊C9 | C2×C3 | 1 | 1 | yes | characteristic |
| 7 | C4×3−1+2 | — | 3 | 3 | no | — |
| 8 | C3×C4×C9 | — | 3 | 1 | no | — |
| 9 | C3×C4×C9 | C32 | 1 | 1 | yes | characteristic |
| 10 | #733 | C3×C4 | 1 | 3 | yes | — |
| 11 | C9⋊C9 | C3×C4 | 1 | 1 | yes | characteristic |
| 12 | C2×3−1+2 | — | 3 | 3 | no | — |
| 13 | C2×C3×C9 | — | 3 | 1 | no | — |
| 14 | C2×C3×C9 | C2×C32 | 1 | 1 | yes | characteristic |
| 15 | C32×C4 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 16 | C4×C9 | — | 3 | 1 | no | — |
| 17 | C4×C9 | — | 9 | 1 | no | — |
| 18 | C4×C9 | — | 9 | 3 | no | — |
| 19 | 3−1+2 | — | 3 | 3 | no | — |
| 20 | C3×C9 | — | 3 | 1 | no | — |
| 21 | C3×C9 | C32×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×C32 | C2×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 23 | C2×C9 | — | 3 | 1 | no | — |
| 24 | C2×C9 | — | 9 | 1 | no | — |
| 25 | C2×C9 | — | 9 | 3 | no | — |
| 26 | C3×C4 | — | 3 | 1 | no | — |
| 27 | C3×C4 | #732 | 1 | 1 | yes | characteristic |
| 28 | C32 | C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 29 | C9 | — | 3 | 1 | no | — |
| 30 | C9 | — | 9 | 1 | no | — |
| 31 | C9 | — | 9 | 3 | no | — |
| 32 | C2×C3 | — | 3 | 1 | no | — |
| 33 | C2×C3 | C2×#732 | 1 | 1 | yes | characteristic |
| 34 | C4 | #6882 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 35 | C3 | — | 3 | 1 | no | — |
| 36 | C3 | C4×#732 | 1 | 1 | yes | characteristic |
| 37 | C2 | C2×#6882 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #732 |
| commutator | C3×C9 | C32×C4 |
| Frattini, cosocle | C2×C3×C9 | C2×C32 |
| socle | C2×C3 | C2×#732 |
| prime radical | C2×C32 | C2×H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C32×C4 ◁ C3×C4×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #6882 | C4 | direct (N × H) |
| 34. C4 | #6882 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6882 | 2 |
| C4×#733 | 3 |
| C4×C9⋊C9 | 3 |