Finite Group DatabasebetaFinite Group DatabasebetaOrder 378.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #4430 | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 2 | C2×C32×C7 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C7⋊C9 | C3 | 1 | 3 | yes | maximal, complemented |
| 4 | C32×C7 | C2×C3 | 1 | 1 | yes | characteristic |
| 5 | C7⋊C9 | C2×C3 | 1 | 3 | yes | complemented |
| 6 | C2×3−1+2 | — | 7 | 1 | no | maximal, Hall |
| 7 | C2×C3×C7 | — | 3 | 1 | no | — |
| 8 | C2×C3×C7 | C32 | 1 | 1 | yes | characteristic |
| 9 | 3−1+2 | — | 7 | 1 | no | Sylow |
| 10 | C3×C7 | — | 3 | 1 | no | — |
| 11 | C3×C7 | C2×C32 | 1 | 1 | yes | characteristic |
| 12 | C2×C32 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 13 | C2×C9 | — | 7 | 3 | no | — |
| 14 | C2×C7 | 3−1+2 | 1 | 1 | yes | characteristic, complemented, Hall |
| 15 | C32 | C2×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 16 | C9 | — | 7 | 3 | no | — |
| 17 | C7 | C2×3−1+2 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 18 | C2×C3 | — | 3 | 1 | no | — |
| 19 | C2×C3 | C3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 20 | C3 | — | 3 | 1 | no | — |
| 21 | C3 | C2×C3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 22 | C2 | #4430 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #4430 | C2 | direct (N × H) |
| 3. C2×C7⋊C9 | C3 | semidirect |
| 5. C7⋊C9 | C2×C3 | semidirect |
| 14. C2×C7 | 3−1+2 | semidirect |
| 17. C7 | C2×3−1+2 | semidirect |
| 22. C2 | #4430 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #4430 | 2 |
| C2×C32×C7 | 3 |
| C2×C7⋊C9 | 3 |
| C2×3−1+2 | 7 |