Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
40 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7⋊(C42⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C7×#126 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C2×C22⋊ΓL(1,𝔽8) | — | 4 | 1 | no | maximal |
| 4 | C7×C4⋊C4 | — | 3 | 1 | no | — |
| 5 | C7×C22⋊C4 | — | 3 | 1 | no | — |
| 6 | C42×C7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | #126⋊C3 | — | 7 | 1 | no | maximal, Hall |
| 8 | C22⋊ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 9 | C23×C7 | A4 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×C7 | — | 3 | 1 | no | — |
| 11 | C2×C4×C7 | — | 3 | 1 | no | — |
| 12 | C42⋊C3 | — | 7 | 1 | no | — |
| 13 | C2×ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 14 | #126 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C22×C7 | — | 6 | 1 | no | — |
| 16 | C22×C7 | C2×A4 | 1 | 1 | yes | characteristic |
| 17 | C4×C7 | — | 6 | 1 | no | — |
| 18 | C4×C7 | — | 6 | 1 | no | — |
| 19 | C2×A4 | — | 28 | 1 | no | — |
| 20 | ΓL(1,𝔽8) | — | 16 | 1 | no | Hall |
| 21 | C4⋊C4 | — | 3 | 1 | no | — |
| 22 | C22⋊C4 | — | 3 | 1 | no | — |
| 23 | C42 | C2×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×C7 | — | 3 | 1 | no | — |
| 25 | C2×C7 | — | 4 | 1 | no | — |
| 26 | A4 | — | 28 | 1 | no | — |
| 27 | C23 | C22⋊ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 28 | C2×C4 | — | 3 | 1 | no | — |
| 29 | C2×C4 | — | 3 | 1 | no | — |
| 30 | C7 | #126⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 31 | C2×C3 | — | 112 | 1 | no | — |
| 32 | C22 | — | 6 | 1 | no | — |
| 33 | C22 | C2×C22⋊ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 34 | C4 | — | 6 | 1 | no | — |
| 35 | C4 | — | 6 | 1 | no | — |
| 36 | C3 | — | 112 | 1 | no | Sylow |
| 37 | C2 | — | 3 | 1 | no | — |
| 38 | C2 | — | 4 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42×C7 | C2×C3 |
| Frattini | C22 | C2×C22⋊ΓL(1,𝔽8) |
| Fitting | C7×#126 | C3 |
| socle | C22×C7 | C2×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7⋊(C42⋊C3) | C2 | semidirect |
| 2. C7×#126 | C3 | semidirect |
| 6. C42×C7 | C2×C3 | semidirect |
| 14. #126 | ΓL(1,𝔽8) | semidirect |
| 23. C42 | C2×ΓL(1,𝔽8) | semidirect |
| 30. C7 | #126⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7⋊(C42⋊C3) | 2 |
| C7×#126 | 3 |
| C2×C22⋊ΓL(1,𝔽8) | 4 |
| #126⋊C3 | 7 |