Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
614 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #626 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C22×D32 |
| Fitting | C22×C32×C4 | C22 |
| socle | C22×C32 | C2×D4 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | C2×#135 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3828 | C2 | semidirect |
| 2. C2×C3×#235 | C2 | semidirect |
| 3. C22×C3×C3⋊C8 | C2 | semidirect |
| 4. C2×#3719 | C2 | semidirect |
| 5. C2×#3717 | C2 | semidirect |
| 6. #63829 | C2 | direct (N × H) |
| 7. C2×#3716 | C2 | semidirect |
| 8. C2×#3714 | C2 | semidirect |
| 14. #3828 | C22 | semidirect |
| 20. C3×#235 | C22 | semidirect |
| 26. C2×C3×C3⋊C8 | C22 | semidirect |
| 27. #3719 | C22 | semidirect |
| 28. #3717 | C22 | semidirect |
| 29. #3716 | C22 | semidirect |
| 30. #3714 | C22 | semidirect |
| 35. C2×#235 | D3 | semidirect |
| 42. C22×C3⋊C8 | D3 | semidirect |
| 92. #235 | C2×D3 | semidirect |
| 114. C2×C3⋊C8 | C2×D3 | semidirect |
| 199. C2×C32 | #135 | semidirect |
| 256. C32 | C2×#135 | semidirect |
| 286. C2×C3 | #904 | semidirect |
| 287. C2×C3 | #950 | semidirect |
| 307. C3 | C2×#904 | semidirect |
| 308. C3 | C2×#950 | semidirect |
| 312. C2 | #63829 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3828 | 2 |
| C2×C3×#235 | 2 |
| C22×C3×C3⋊C8 | 2 |
| C2×#3719 | 2 |
| C2×#3717 | 2 |
| #63829 | 2 |
| C2×#3716 | 2 |
| C2×#3714 | 2 |
| C2×#950 | 3 |
| C2×#904 | 3 |