Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
70 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×GL(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | SO(3,𝔽7) | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | GL(3,𝔽2) | C22 | 1 | 1 | yes | characteristic, complemented |
| 4 | C2×AGL(1,𝔽7) | — | 8 | 1 | no | maximal |
| 5 | C2×S4 | — | 14 | 1 | no | — |
| 6 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 7 | AGL(1,𝔽7) | — | 8 | 2 | no | — |
| 8 | C2×D8 | — | 21 | 1 | no | maximal, Sylow |
| 9 | C2×D7 | — | 8 | 1 | no | — |
| 10 | C22×D3 | — | 28 | 1 | no | maximal |
| 11 | C2×A4 | — | 14 | 1 | no | — |
| 12 | S4 | — | 14 | 1 | no | — |
| 13 | S4 | — | 14 | 1 | no | — |
| 14 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 15 | C2×D4 | — | 21 | 1 | no | — |
| 16 | C2×D4 | — | 21 | 1 | no | — |
| 17 | D8 | — | 21 | 2 | no | — |
| 18 | D8 | — | 21 | 2 | no | — |
| 19 | C2×C8 | — | 21 | 1 | no | — |
| 20 | C2×C7 | — | 8 | 1 | no | — |
| 21 | D7 | — | 8 | 2 | no | — |
| 22 | C22×C3 | — | 28 | 1 | no | — |
| 23 | C2×D3 | — | 28 | 1 | no | — |
| 24 | C2×D3 | — | 28 | 1 | no | — |
| 25 | C2×D3 | — | 28 | 2 | no | — |
| 26 | C2×D3 | — | 28 | 2 | no | — |
| 27 | A4 | — | 14 | 1 | no | — |
| 28 | C23 | — | 14 | 1 | no | — |
| 29 | C23 | — | 42 | 1 | no | — |
| 30 | D4 | — | 21 | 1 | no | — |
| 31 | D4 | — | 21 | 1 | no | — |
| 32 | D4 | — | 21 | 2 | no | — |
| 33 | D4 | — | 42 | 1 | no | — |
| 34 | D4 | — | 42 | 1 | no | — |
| 35 | C2×C4 | — | 21 | 1 | no | — |
| 36 | C8 | — | 21 | 2 | no | — |
| 37 | C7 | — | 8 | 1 | no | Sylow |
| 38 | C2×C3 | — | 28 | 2 | no | — |
| 39 | C2×C3 | — | 28 | 1 | no | — |
| 40 | D3 | — | 28 | 1 | no | — |
| 41 | D3 | — | 28 | 1 | no | — |
| 42 | D3 | — | 28 | 2 | no | — |
| 43 | C22 | — | 14 | 1 | no | — |
| 44 | C22 | — | 21 | 1 | no | — |
| 45 | C22 | — | 28 | 1 | no | — |
| 46 | C22 | — | 42 | 1 | no | — |
| 47 | C22 | — | 42 | 2 | no | — |
| 48 | C22 | — | 84 | 1 | no | — |
| 49 | C4 | — | 21 | 1 | no | — |
| 50 | C4 | — | 21 | 1 | no | — |
| 51 | C3 | — | 28 | 1 | no | Sylow |
| 52 | C2 | — | 21 | 1 | no | — |
| 53 | C2 | — | 21 | 1 | no | — |
| 54 | C2 | — | 28 | 2 | no | — |
| 55 | C2 | SO(3,𝔽7) | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2 | SO(3,𝔽7) |
| commutator, cosocle, hypocenter | GL(3,𝔽2) | C22 |
| Frattini | C1 | G |
| socle | C2×GL(3,𝔽2) | C2 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×GL(3,𝔽2) | C2 | semidirect |
| 2. SO(3,𝔽7) | C2 | direct (N × H) |
| 3. GL(3,𝔽2) | C22 | semidirect |
| 55. C2 | SO(3,𝔽7) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×GL(3,𝔽2) | 2 |
| SO(3,𝔽7) | 2 |
| C2×AGL(1,𝔽7) | 8 |
| C2×D8 | 21 |
| C22×D3 | 28 |