Finite Group DatabasebetaFinite Group DatabasebetaOrder 972.
879 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×H(5,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊#1084 | — | 3 | 40 | no | maximal |
| 3 | H(5,𝔽3) | C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | C2×C3×H(3,𝔽3) | D3 | 1 | 40 | yes | — |
| 5 | C3×C3⋊Dic3 | — | 9 | 40 | no | — |
| 6 | #1084 | — | 9 | 90 | no | — |
| 7 | C3×H(3,𝔽3) | Dic3 | 1 | 40 | yes | complemented |
| 8 | C2×C33 | C3⋊D3 | 1 | 40 | yes | — |
| 9 | C2×H(3,𝔽3) | C3⋊D3 | 1 | 90 | yes | — |
| 10 | C3⋊Dic3 | — | 9 | 40 | no | — |
| 11 | C3×Dic3 | — | 27 | 40 | no | — |
| 12 | C33 | C3⋊Dic3 | 1 | 40 | yes | complemented |
| 13 | H(3,𝔽3) | C3⋊Dic3 | 1 | 90 | yes | complemented |
| 14 | C2×C32 | — | 9 | 40 | no | — |
| 15 | C2×C32 | C3⋊(C3⋊D3) | 1 | 40 | yes | — |
| 16 | C3×C4 | — | 81 | 1 | no | — |
| 17 | Dic3 | — | 27 | 40 | no | — |
| 18 | C32 | — | 9 | 40 | no | — |
| 19 | C32 | C3⋊(C3⋊Dic3) | 1 | 40 | yes | — |
| 20 | C2×C3 | — | 3 | 40 | no | — |
| 21 | C2×C3 | C3⋊(C3⋊(C3⋊D3)) | 1 | 1 | yes | characteristic |
| 22 | C4 | — | 81 | 1 | no | Sylow |
| 23 | C3 | — | 3 | 40 | no | — |
| 24 | C3 | C3⋊(C3⋊(C3⋊Dic3)) | 1 | 1 | yes | characteristic |
| 25 | C2 | #92404 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | C3⋊(C3⋊(C3⋊D3)) |
| commutator, hypocenter | H(5,𝔽3) | C4 |
| Fitting, prime radical, cosocle | C2×H(5,𝔽3) | C2 |
| LCS | G ▷ H(5,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. H(5,𝔽3) | C4 | semidirect |
| 7. C3×H(3,𝔽3) | Dic3 | semidirect |
| 12. C33 | C3⋊Dic3 | semidirect |
| 13. H(3,𝔽3) | C3⋊Dic3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽3) | 2 |
| C3⋊#1084 | 3 |