Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
27 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | SL(2,𝔽7) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | ΓL(1,𝔽8)⋊C4 | — | 8 | 1 | no | maximal |
| 3 | #226 | — | 14 | 1 | no | — |
| 4 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 5 | Q32 | — | 21 | 1 | no | maximal, Sylow |
| 6 | Dic7 | — | 8 | 1 | no | — |
| 7 | Dic6 | — | 28 | 1 | no | maximal |
| 8 | SL(2,𝔽3) | — | 14 | 1 | no | — |
| 9 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 10 | Q16 | — | 21 | 1 | no | — |
| 11 | Q16 | — | 21 | 1 | no | — |
| 12 | C16 | — | 21 | 1 | no | — |
| 13 | C2×C7 | — | 8 | 1 | no | — |
| 14 | C3×C4 | — | 28 | 1 | no | — |
| 15 | Dic3 | — | 28 | 1 | no | — |
| 16 | Dic3 | — | 28 | 1 | no | — |
| 17 | Q8 | — | 14 | 1 | no | — |
| 18 | Q8 | — | 42 | 1 | no | — |
| 19 | C8 | — | 21 | 1 | no | — |
| 20 | C7 | — | 8 | 1 | no | Sylow |
| 21 | C2×C3 | — | 28 | 1 | no | — |
| 22 | C4 | — | 21 | 1 | no | — |
| 23 | C4 | — | 28 | 1 | no | — |
| 24 | C3 | — | 28 | 1 | no | Sylow |
| 25 | C2 | SO(3,𝔽7) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, hypercenter | C2 | SO(3,𝔽7) |
| commutator, prime radical, cosocle, hypocenter | SL(2,𝔽7) | C2 |
| LCS | G ▷ SL(2,𝔽7) | |
| UCS | C1 ◁ C2 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽7) | 2 |
| ΓL(1,𝔽8)⋊C4 | 8 |
| Q32 | 21 |
| Dic6 | 28 |