Finite Group DatabasebetaFinite Group DatabasebetaOrder 378.
26 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #4427 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C3⋊AGL(1,𝔽7) | — | 3 | 1 | no | maximal |
| 3 | D63 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×ΓL(1,𝔽8) | D3 | 1 | 1 | yes | characteristic |
| 5 | C7×C9 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C7⋊C9 | — | 2 | 1 | no | — |
| 7 | AGL(1,ℤ/9) | — | 7 | 1 | no | maximal, Hall |
| 8 | D21 | — | 3 | 1 | no | — |
| 9 | AGL(1,𝔽7) | — | 9 | 1 | no | — |
| 10 | 3−1+2 | — | 7 | 1 | no | Sylow |
| 11 | C3×C7 | C3×D3 | 1 | 1 | yes | characteristic |
| 12 | ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 13 | C3×D3 | — | 21 | 1 | no | — |
| 14 | D9 | — | 7 | 1 | no | — |
| 15 | D7 | — | 9 | 1 | no | Hall |
| 16 | C32 | — | 7 | 1 | no | — |
| 17 | C9 | — | 14 | 1 | no | — |
| 18 | C9 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 19 | C7 | AGL(1,ℤ/9) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 20 | C2×C3 | — | 63 | 1 | no | — |
| 21 | D3 | — | 21 | 1 | no | — |
| 22 | C3 | — | 21 | 1 | no | — |
| 23 | C3 | C3⋊AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 24 | C2 | — | 63 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C9 | C2×C3 |
| Frattini | C3 | C3⋊AGL(1,𝔽7) |
| socle | C3×C7 | C3×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #4427 | C2 | semidirect |
| 3. D63 | C3 | semidirect |
| 5. C7×C9 | C2×C3 | semidirect |
| 18. C9 | AGL(1,𝔽7) | semidirect |
| 19. C7 | AGL(1,ℤ/9) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #4427 | 2 |
| C3⋊AGL(1,𝔽7) | 3 |
| D63 | 3 |
| AGL(1,ℤ/9) | 7 |