Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#213 |
| commutator, cosocle | C3×C8 | C24 |
| Frattini | C8 | C23×D3 |
| Fitting | C3×#578 | C2 |
| socle | C2×C3 | C22×D8 |
| hypercenter | #578 | D3 |
| hypocenter | C3 | #3416 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #578 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#578 | C2 | semidirect |
| 2. #5752 | C2 | semidirect |
| 3. #5194 | C2 | semidirect |
| 4. #5193 | C2 | semidirect |
| 5. #5179 | C2 | semidirect |
| 6. C2×#828 | C2 | semidirect |
| 7. #5140 | C2 | semidirect |
| 12. C3×#135 | C22 | semidirect |
| 13. C2×C3×Q16 | C22 | semidirect |
| 18. #829 | C22 | semidirect |
| 19. #828 | C22 | semidirect |
| 20. #828 | C22 | semidirect |
| 21. #827 | C22 | semidirect |
| 22. #826 | C22 | semidirect |
| 23. #812 | C22 | semidirect |
| 24. C2×C3⋊C16 | C22 | semidirect |
| 25. #578 | D3 | semidirect |
| 58. #135 | C2×D3 | semidirect |
| 59. C2×Q16 | C2×D3 | semidirect |
| 144. C3 | #3416 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#578 | 2 |
| #5752 | 2 |
| #5194 | 2 |
| #5193 | 2 |
| #5179 | 2 |
| C2×#828 | 2 |
| #5140 | 2 |
| #3416 | 3 |