Finite Group DatabasebetaFinite Group DatabasebetaOrder 336.
36 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 2 | C2×C7×Q8 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C22×C3×C7 | — | 4 | 1 | no | maximal |
| 4 | C7×Q8 | — | 3 | 1 | no | — |
| 5 | C7×Q8 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×C4×C7 | — | 3 | 1 | no | — |
| 7 | C2×SL(2,𝔽3) | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 8 | C2×C3×C7 | — | 4 | 2 | no | — |
| 9 | C2×C3×C7 | — | 4 | 1 | no | — |
| 10 | C22×C7 | A4 | 1 | 1 | yes | characteristic |
| 11 | C4×C7 | — | 3 | 1 | no | — |
| 12 | C4×C7 | — | 3 | 1 | no | — |
| 13 | SL(2,𝔽3) | C2×C7 | 1 | 1 | yes | characteristic, direct product factor |
| 14 | C3×C7 | — | 4 | 1 | no | Hall |
| 15 | C2×Q8 | C3×C7 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | C2×C7 | SL(2,𝔽3) | 1 | 2 | yes | direct product factor |
| 17 | C2×C7 | C2×A4 | 1 | 1 | yes | characteristic |
| 18 | C22×C3 | — | 4 | 1 | no | — |
| 19 | Q8 | — | 3 | 1 | no | — |
| 20 | Q8 | C2×C3×C7 | 1 | 1 | yes | characteristic, complemented |
| 21 | C2×C4 | — | 3 | 1 | no | — |
| 22 | C7 | C2×SL(2,𝔽3) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 23 | C2×C3 | — | 4 | 2 | no | — |
| 24 | C2×C3 | — | 4 | 1 | no | — |
| 25 | C22 | C7×A4 | 1 | 1 | yes | characteristic |
| 26 | C4 | — | 3 | 1 | no | — |
| 27 | C4 | — | 3 | 1 | no | — |
| 28 | C3 | — | 4 | 1 | no | Sylow |
| 29 | C2 | C7×SL(2,𝔽3) | 1 | 2 | yes | direct product factor |
| 30 | C2 | C2×C7×A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C7 | A4 |
| commutator, cosocle, hypocenter | Q8 | C2×C3×C7 |
| Frattini | C2 | C2×C7×A4 |
| Fitting | C2×C7×Q8 | C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×SL(2,𝔽3) | C2 | direct (N × H) |
| 2. C2×C7×Q8 | C3 | semidirect |
| 5. C7×Q8 | C2×C3 | semidirect |
| 7. C2×SL(2,𝔽3) | C7 | direct (N × H) |
| 13. SL(2,𝔽3) | C2×C7 | direct (N × H) |
| 15. C2×Q8 | C3×C7 | semidirect |
| 16. C2×C7 | SL(2,𝔽3) | direct (N × H) |
| 20. Q8 | C2×C3×C7 | semidirect |
| 22. C7 | C2×SL(2,𝔽3) | direct (N × H) |
| 29. C2 | C7×SL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×SL(2,𝔽3) | 2 |
| C2×C7×Q8 | 3 |
| C22×C3×C7 | 4 |
| C2×SL(2,𝔽3) | 7 |