Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
4544 conjugacy classes of subgroups; only the 805 normal subgroups are shown.
This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 805 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C26×C3 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C26 | D3 |
| hypocenter | C3 | C22×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C26 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C26×C3 | C2 | semidirect |
| 2. C23×#67 | C2 | semidirect |
| 3. C2×#953 | C2 | direct (N × H) |
| 4. C22×#217 | C2 | semidirect |
| 8. C25×C3 | C22 | semidirect |
| 9. C24×D3 | C22 | semidirect |
| 11. C22×#67 | C22 | semidirect |
| 12. C23×Dic3 | C22 | semidirect |
| 13. #953 | C22 | direct (N × H) |
| 14. C2×#217 | C22 | semidirect |
| 15. C26 | D3 | semidirect |
| 22. C24×C3 | D4 | semidirect |
| 25. C24×C3 | C23 | semidirect |
| 27. C23×D3 | C23 | semidirect |
| 29. C2×#67 | C23 | semidirect |
| 31. C22×Dic3 | C23 | semidirect |
| 32. #217 | C23 | semidirect |
| 36. C25 | C2×D3 | semidirect |
| 47. C23×C3 | C2×D4 | semidirect |
| 52. C22×D3 | C24 | semidirect |
| 55. C2×Dic3 | C24 | semidirect |
| 61. C24 | #67 | semidirect |
| 64. C24 | C22×D3 | semidirect |
| 73. C22×C3 | #120 | semidirect |
| 75. C22×C3 | C22×D4 | semidirect |
| 88. C23 | C2×#67 | semidirect |
| 95. C2×C3 | C2×#120 | semidirect |
| 103. C22 | #953 | direct (N × H) |
| 105. C22 | C22×#67 | semidirect |
| 108. C3 | C22×#120 | semidirect |
| 111. C2 | C2×#953 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C26×C3 | 2 |
| C23×#67 | 2 |
| C2×#953 | 2 |
| C22×#217 | 2 |
| C22×#120 | 3 |