Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
280 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×SL(2,𝔽3) | C2 | 1 | 7 | yes | maximal, direct product factor |
| 2 | C23×Q8 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C24×C3 | — | 4 | 1 | no | maximal |
| 4 | C2×SL(2,𝔽3) | C22 | 1 | 7 | yes | direct product factor |
| 5 | C22×Q8 | — | 3 | 7 | no | — |
| 6 | C22×Q8 | C2×C3 | 1 | 7 | yes | complemented |
| 7 | C23×C4 | — | 3 | 1 | no | — |
| 8 | C23×C3 | — | 4 | 8 | no | — |
| 9 | C23×C3 | — | 4 | 7 | no | — |
| 10 | SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic, direct product factor |
| 11 | C24 | A4 | 1 | 1 | yes | characteristic |
| 12 | C2×Q8 | — | 3 | 14 | no | — |
| 13 | C2×Q8 | — | 3 | 21 | no | — |
| 14 | C2×Q8 | C22×C3 | 1 | 7 | yes | complemented |
| 15 | C22×C4 | — | 3 | 7 | no | — |
| 16 | C22×C4 | — | 3 | 7 | no | — |
| 17 | C22×C3 | — | 4 | 28 | no | — |
| 18 | C22×C3 | — | 4 | 7 | no | — |
| 19 | C23 | SL(2,𝔽3) | 1 | 8 | yes | direct product factor |
| 20 | C23 | C2×A4 | 1 | 7 | yes | — |
| 21 | Q8 | — | 3 | 7 | no | — |
| 22 | Q8 | — | 3 | 14 | no | — |
| 23 | Q8 | C23×C3 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×C4 | — | 3 | 21 | no | — |
| 25 | C2×C4 | — | 3 | 7 | no | — |
| 26 | C2×C3 | — | 4 | 14 | no | — |
| 27 | C2×C3 | — | 4 | 1 | no | — |
| 28 | C22 | C2×SL(2,𝔽3) | 1 | 28 | yes | direct product factor |
| 29 | C22 | C22×A4 | 1 | 7 | yes | — |
| 30 | C4 | — | 3 | 7 | no | — |
| 31 | C4 | — | 3 | 1 | no | — |
| 32 | C3 | — | 4 | 1 | no | Sylow |
| 33 | C2 | C22×SL(2,𝔽3) | 1 | 14 | yes | direct product factor |
| 34 | C2 | C23×A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | A4 |
| commutator, cosocle, hypocenter | Q8 | C23×C3 |
| Frattini | C2 | C23×A4 |
| Fitting | C23×Q8 | C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×SL(2,𝔽3) | C2 | direct (N × H) |
| 2. C23×Q8 | C3 | semidirect |
| 4. C2×SL(2,𝔽3) | C22 | direct (N × H) |
| 6. C22×Q8 | C2×C3 | semidirect |
| 10. SL(2,𝔽3) | C23 | direct (N × H) |
| 14. C2×Q8 | C22×C3 | semidirect |
| 19. C23 | SL(2,𝔽3) | direct (N × H) |
| 23. Q8 | C23×C3 | semidirect |
| 28. C22 | C2×SL(2,𝔽3) | direct (N × H) |
| 33. C2 | C22×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 |
|---|---|
| C22×SL(2,𝔽3) | 2 |
| C23×Q8 | 3 |
| C24×C3 | 4 |