Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
176 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #564⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 2 | C2×#564 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C2×C42⋊C3 | — | 4 | 5 | no | maximal |
| 4 | #564 | — | 3 | 5 | no | — |
| 5 | #564 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#125 | — | 3 | 5 | no | — |
| 7 | C42⋊C3 | — | 4 | 5 | no | — |
| 8 | #125 | — | 3 | 30 | no | — |
| 9 | #125 | — | 3 | 5 | no | — |
| 10 | #125 | — | 3 | 5 | no | — |
| 11 | C2×C4⋊C4 | — | 3 | 10 | no | — |
| 12 | C2×C42 | A4 | 1 | 5 | yes | — |
| 13 | C2×A4 | — | 16 | 1 | no | — |
| 14 | C22×C4 | — | 3 | 5 | no | — |
| 15 | C4⋊C4 | — | 3 | 10 | no | — |
| 16 | C4⋊C4 | — | 3 | 30 | no | — |
| 17 | C42 | — | 3 | 5 | no | — |
| 18 | C42 | C2×A4 | 1 | 5 | yes | — |
| 19 | A4 | — | 16 | 1 | no | — |
| 20 | C23 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 21 | C2×C4 | — | 3 | 5 | no | — |
| 22 | C2×C4 | — | 3 | 5 | no | — |
| 23 | C2×C4 | — | 6 | 5 | no | — |
| 24 | C2×C4 | — | 6 | 5 | no | — |
| 25 | C2×C3 | — | 64 | 1 | no | — |
| 26 | C22 | — | 3 | 1 | no | — |
| 27 | C22 | — | 3 | 1 | no | — |
| 28 | C22 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 29 | C4 | — | 6 | 5 | no | — |
| 30 | C4 | — | 6 | 5 | no | — |
| 31 | C3 | — | 64 | 1 | no | Sylow |
| 32 | C2 | — | 3 | 1 | no | — |
| 33 | C2 | — | 3 | 1 | no | — |
| 34 | C2 | #564⋊C3 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #564⋊C3 |
| commutator, cosocle, hypocenter | #564 | C2×C3 |
| Frattini | C22 | C2×C22⋊A4 |
| Fitting | C2×#564 | C3 |
| socle | C23 | C22⋊A4 |
| LCS | G ▷ #564 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #564⋊C3 | C2 | direct (N × H) |
| 2. C2×#564 | C3 | semidirect |
| 5. #564 | C2×C3 | semidirect |
| 34. C2 | #564⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #564⋊C3 | 2 |
| C2×#564 | 3 |
| C2×C42⋊C3 | 4 |