Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
66 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×GL(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | GL(3,𝔽2) | C4 | 1 | 1 | yes | characteristic, direct product factor |
| 3 | C4×S4 | — | 7 | 2 | no | maximal, Hall |
| 4 | C4×ΓL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 5 | C2×S4 | — | 7 | 2 | no | — |
| 6 | C4×A4 | — | 7 | 2 | no | — |
| 7 | SL(2,ℤ/4) | — | 7 | 2 | no | — |
| 8 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 9 | C4×D4 | — | 21 | 1 | no | Sylow |
| 10 | C4×C7 | — | 8 | 1 | no | — |
| 11 | C2×A4 | — | 7 | 2 | no | — |
| 12 | S4 | — | 7 | 2 | no | — |
| 13 | S4 | — | 7 | 2 | no | — |
| 14 | C4×D3 | — | 28 | 1 | no | — |
| 15 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 16 | C2×D4 | — | 21 | 1 | no | — |
| 17 | C22×C4 | — | 7 | 2 | no | — |
| 18 | C4⋊C4 | — | 21 | 1 | no | — |
| 19 | C22⋊C4 | — | 21 | 2 | no | — |
| 20 | C42 | — | 21 | 1 | no | — |
| 21 | C2×C7 | — | 8 | 1 | no | — |
| 22 | C2×D3 | — | 28 | 1 | no | — |
| 23 | A4 | — | 7 | 2 | no | — |
| 24 | C3×C4 | — | 28 | 1 | no | — |
| 25 | Dic3 | — | 28 | 1 | no | — |
| 26 | C23 | — | 7 | 2 | no | — |
| 27 | D4 | — | 21 | 1 | no | — |
| 28 | D4 | — | 21 | 2 | no | — |
| 29 | D4 | — | 21 | 1 | no | — |
| 30 | C2×C4 | — | 21 | 1 | no | — |
| 31 | C2×C4 | — | 21 | 1 | no | — |
| 32 | C2×C4 | — | 21 | 2 | no | — |
| 33 | C2×C4 | — | 21 | 1 | no | — |
| 34 | C7 | — | 8 | 1 | no | Sylow |
| 35 | C2×C3 | — | 28 | 1 | no | — |
| 36 | D3 | — | 28 | 1 | no | — |
| 37 | D3 | — | 28 | 1 | no | — |
| 38 | C22 | — | 7 | 2 | no | — |
| 39 | C22 | — | 21 | 2 | no | — |
| 40 | C22 | — | 21 | 1 | no | — |
| 41 | C4 | — | 21 | 1 | no | — |
| 42 | C4 | — | 21 | 1 | no | — |
| 43 | C4 | — | 21 | 1 | no | — |
| 44 | C4 | — | 42 | 1 | no | — |
| 45 | C4 | GL(3,𝔽2) | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 46 | C3 | — | 28 | 1 | no | Sylow |
| 47 | C2 | — | 21 | 1 | no | — |
| 48 | C2 | — | 21 | 1 | no | — |
| 49 | C2 | C2×GL(3,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C4 | GL(3,𝔽2) |
| commutator, hypocenter | GL(3,𝔽2) | C4 |
| Frattini, cosocle | C2 | C2×GL(3,𝔽2) |
| socle, prime radical | C2×GL(3,𝔽2) | C2 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. GL(3,𝔽2) | C4 | direct (N × H) |
| 45. C4 | GL(3,𝔽2) | 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 |
| C4×S4 | 7 |
| C4×ΓL(1,𝔽8) | 8 |