Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
70 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #120⋊C3 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | H(3,𝔽4) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C22⋊A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 4 | C22×A4 | — | 4 | 1 | no | maximal |
| 5 | #124 | — | 3 | 3 | no | — |
| 6 | #120 | — | 3 | 1 | no | — |
| 7 | #120 | C2×C3 | 1 | 3 | yes | complemented |
| 8 | C2×A4 | — | 4 | 3 | no | — |
| 9 | C24 | A4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C24 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×Q8 | — | 3 | 1 | no | — |
| 12 | C2×D4 | — | 3 | 3 | no | — |
| 13 | C22⋊C4 | — | 3 | 3 | no | — |
| 14 | C22⋊C4 | — | 3 | 3 | no | — |
| 15 | C42 | — | 3 | 1 | no | — |
| 16 | C22×C3 | — | 16 | 1 | no | — |
| 17 | A4 | — | 4 | 1 | no | — |
| 18 | A4 | — | 16 | 1 | no | — |
| 19 | C23 | — | 3 | 1 | no | — |
| 20 | C23 | — | 12 | 1 | no | — |
| 21 | C23 | — | 12 | 1 | no | — |
| 22 | C23 | C2×A4 | 1 | 3 | yes | — |
| 23 | Q8 | — | 12 | 1 | no | — |
| 24 | D4 | — | 12 | 3 | no | — |
| 25 | C2×C4 | — | 3 | 3 | no | — |
| 26 | C2×C3 | — | 16 | 3 | no | — |
| 27 | C22 | — | 4 | 1 | no | — |
| 28 | C22 | — | 4 | 1 | no | — |
| 29 | C22 | — | 6 | 3 | no | — |
| 30 | C22 | — | 6 | 3 | no | — |
| 31 | C22 | — | 12 | 1 | no | — |
| 32 | C22 | — | 12 | 1 | no | — |
| 33 | C22 | C22×A4 | 1 | 1 | yes | characteristic |
| 34 | C4 | — | 6 | 3 | no | — |
| 35 | C3 | — | 16 | 1 | no | Sylow |
| 36 | C2 | — | 3 | 1 | no | — |
| 37 | C2 | — | 4 | 3 | no | — |
| 38 | C2 | — | 12 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24 | C22×C3 |
| Frattini, socle | C22 | C22×A4 |
| Fitting | H(3,𝔽4) | C3 |
| LCS | G ▷ C24 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #120⋊C3 | C2 | semidirect |
| 2. H(3,𝔽4) | C3 | semidirect |
| 3. C22⋊A4 | C22 | semidirect |
| 7. #120 | C2×C3 | semidirect |
| 9. C24 | A4 | semidirect |
| 10. C24 | C22×C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #120⋊C3 | 2 |
| H(3,𝔽4) | 3 |
| C22×A4 | 4 |