Finite Group DatabasebetaFinite Group DatabasebetaOrder 378.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C7×C3⋊D3 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×D3×C7 | — | 3 | 1 | no | maximal |
| 4 | C32×C7 | — | 2 | 1 | no | — |
| 5 | C32×C7 | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C32×C7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | (C3⋊D3)⋊C3 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 8 | C2×C3×C7 | — | 9 | 1 | no | — |
| 9 | D3×C7 | — | 3 | 1 | no | — |
| 10 | D3×C7 | — | 9 | 1 | no | — |
| 11 | H(3,𝔽3) | C2×C7 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C3×C7 | — | 3 | 1 | no | — |
| 13 | C3×C7 | — | 3 | 1 | no | — |
| 14 | C3×C7 | — | 6 | 1 | no | — |
| 15 | C3×C7 | C3×D3 | 1 | 1 | yes | characteristic |
| 16 | C3⋊D3 | C3×C7 | 1 | 1 | yes | characteristic, complemented |
| 17 | C3×D3 | — | 3 | 1 | no | — |
| 18 | C2×C7 | — | 9 | 1 | no | Hall |
| 19 | C32 | — | 2 | 1 | no | — |
| 20 | C32 | D3×C7 | 1 | 1 | yes | characteristic, complemented |
| 21 | C32 | C2×C3×C7 | 1 | 1 | yes | characteristic, complemented |
| 22 | C7 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 23 | C2×C3 | — | 9 | 1 | no | — |
| 24 | D3 | — | 3 | 1 | no | — |
| 25 | D3 | — | 9 | 1 | no | — |
| 26 | C3 | — | 3 | 1 | no | — |
| 27 | C3 | — | 3 | 1 | no | — |
| 28 | C3 | — | 6 | 1 | no | — |
| 29 | C3 | C3×D3×C7 | 1 | 1 | yes | characteristic |
| 30 | C2 | — | 9 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C32 | C2×C3×C7 |
| Frattini | C3 | C3×D3×C7 |
| Fitting | C7×H(3,𝔽3) | C2 |
| socle | C3×C7 | C3×D3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×H(3,𝔽3) | C2 | semidirect |
| 2. C7×C3⋊D3 | C3 | semidirect |
| 5. C32×C7 | D3 | semidirect |
| 6. C32×C7 | C2×C3 | semidirect |
| 7. (C3⋊D3)⋊C3 | C7 | direct (N × H) |
| 11. H(3,𝔽3) | C2×C7 | semidirect |
| 16. C3⋊D3 | C3×C7 | semidirect |
| 20. C32 | D3×C7 | semidirect |
| 21. C32 | C2×C3×C7 | semidirect |
| 22. C7 | (C3⋊D3)⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×H(3,𝔽3) | 2 |
| C7×C3⋊D3 | 3 |
| C3×D3×C7 | 3 |
| (C3⋊D3)⋊C3 | 7 |