Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1504 conjugacy classes of subgroups; only the 525 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 525 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | D3×D4 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C22×C3×Q16 | C2 |
| socle | C23×C3 | C2×D4 |
| hypercenter | C22×Q16 | D3 |
| hypocenter | C3 | C22×#135 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×Q16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#239 | C2 | semidirect |
| 2. C22×C3×Q16 | C2 | semidirect |
| 3. C22×#215 | C2 | semidirect |
| 4. C2×#919 | C2 | direct (N × H) |
| 5. C22×D24 | C2 | semidirect |
| 6. C22×D3×C8 | C2 | semidirect |
| 10. C2×#239 | C22 | semidirect |
| 15. C2×C3×Q16 | C22 | semidirect |
| 17. C2×#215 | C22 | semidirect |
| 19. #919 | C22 | direct (N × H) |
| 20. C2×D24 | C22 | semidirect |
| 21. C2×D3×C8 | C22 | semidirect |
| 24. C22×Q16 | D3 | semidirect |
| 36. #239 | C23 | semidirect |
| 43. C3×Q16 | C23 | semidirect |
| 45. #215 | C23 | semidirect |
| 47. D24 | C23 | semidirect |
| 48. D3×C8 | C23 | semidirect |
| 57. C2×Q16 | C2×D3 | semidirect |
| 97. Q16 | C22×D3 | semidirect |
| 103. C22×C3 | #135 | semidirect |
| 132. C2×C3 | C2×#135 | semidirect |
| 140. C22 | #919 | direct (N × H) |
| 146. C3 | C22×#135 | semidirect |
| 149. C2 | C2×#919 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#239 | 2 |
| C22×C3×Q16 | 2 |
| C22×#215 | 2 |
| C2×#919 | 2 |
| C22×D24 | 2 |
| C22×D3×C8 | 2 |
| C22×#135 | 3 |