Finite Group DatabasebetaFinite Group DatabasebetaOrder 1008.
50 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×GL(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | SO(3,𝔽7) | — | 3 | 1 | no | maximal |
| 3 | GL(3,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 4 | C3⋊AGL(1,𝔽7) | — | 8 | 1 | no | maximal |
| 5 | C3×S4 | — | 14 | 1 | no | — |
| 6 | C3×ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 7 | #213 | — | 21 | 1 | no | maximal |
| 8 | D21 | — | 8 | 1 | no | — |
| 9 | AGL(1,𝔽7) | — | 24 | 1 | no | — |
| 10 | C3×A4 | — | 14 | 1 | no | — |
| 11 | D32 | — | 28 | 1 | no | maximal |
| 12 | S4 | — | 14 | 1 | no | — |
| 13 | C3×D4 | — | 21 | 1 | no | — |
| 14 | D12 | — | 21 | 1 | no | — |
| 15 | C3⋊C8 | — | 21 | 1 | no | — |
| 16 | C3×C7 | — | 8 | 1 | no | — |
| 17 | ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 18 | ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 19 | C3⋊D3 | — | 28 | 1 | no | — |
| 20 | C3×D3 | — | 28 | 1 | no | — |
| 21 | C3×D3 | — | 28 | 1 | no | — |
| 22 | D8 | — | 63 | 1 | no | Sylow |
| 23 | D7 | — | 24 | 1 | no | — |
| 24 | C22×C3 | — | 14 | 1 | no | — |
| 25 | C2×D3 | — | 42 | 1 | no | — |
| 26 | C2×D3 | — | 84 | 1 | no | — |
| 27 | A4 | — | 14 | 1 | no | — |
| 28 | A4 | — | 28 | 1 | no | — |
| 29 | C3×C4 | — | 21 | 1 | no | — |
| 30 | C32 | — | 28 | 1 | no | Sylow |
| 31 | D4 | — | 21 | 1 | no | — |
| 32 | D4 | — | 63 | 1 | no | — |
| 33 | C8 | — | 63 | 1 | no | — |
| 34 | C7 | — | 8 | 1 | no | Sylow |
| 35 | C2×C3 | — | 21 | 1 | no | — |
| 36 | C2×C3 | — | 84 | 1 | no | — |
| 37 | D3 | — | 28 | 1 | no | — |
| 38 | D3 | — | 28 | 1 | no | — |
| 39 | D3 | — | 84 | 1 | no | — |
| 40 | D3 | — | 168 | 1 | no | — |
| 41 | C22 | — | 14 | 1 | no | — |
| 42 | C22 | — | 126 | 1 | no | — |
| 43 | C4 | — | 21 | 1 | no | — |
| 44 | C3 | — | 28 | 1 | no | — |
| 45 | C3 | — | 56 | 1 | no | — |
| 46 | C3 | SO(3,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 47 | C2 | — | 21 | 1 | no | — |
| 48 | C2 | — | 84 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C3×GL(3,𝔽2) | C2 |
| Fitting | C3 | SO(3,𝔽7) |
| LCS | G ▷ C3×GL(3,𝔽2) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×GL(3,𝔽2) | C2 | semidirect |
| 3. GL(3,𝔽2) | D3 | semidirect |
| 46. C3 | SO(3,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×GL(3,𝔽2) | 2 |
| SO(3,𝔽7) | 3 |
| C3⋊AGL(1,𝔽7) | 8 |
| #213 | 21 |
| D32 | 28 |