Finite Group DatabasebetaFinite Group DatabasebetaOrder 168.
15 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | S4 | — | 7 | 2 | no | maximal, Hall |
| 2 | ΓL(1,𝔽8) | — | 8 | 1 | no | maximal, Hall |
| 3 | A4 | — | 7 | 2 | no | — |
| 4 | D4 | — | 21 | 1 | no | Sylow |
| 5 | C7 | — | 8 | 1 | no | Sylow |
| 6 | D3 | — | 28 | 1 | no | — |
| 7 | C22 | — | 7 | 2 | no | — |
| 8 | C4 | — | 21 | 1 | no | — |
| 9 | C3 | — | 28 | 1 | no | Sylow |
| 10 | C2 | — | 21 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| S4 | 7 |
| ΓL(1,𝔽8) | 8 |