Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
668 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×SD16 | C2 | semidirect |
| 2. C22×C3×D8 | C2 | semidirect |
| 3. C2×C3×#127 | C2 | semidirect |
| 4. C2×C3×#124 | C2 | semidirect |
| 5. C3×#496 | C2 | direct (N × H) |
| 6. C2×C3×#97 | C2 | semidirect |
| 7. C2×#496 | C3 | direct (N × H) |
| 13. C2×C3×SD16 | C22 | semidirect |
| 15. C2×C3×D8 | C22 | semidirect |
| 18. C3×#127 | C22 | semidirect |
| 20. C3×#124 | C22 | semidirect |
| 23. C3×#97 | C22 | semidirect |
| 24. C22×SD16 | C2×C3 | semidirect |
| 25. C22×D8 | C2×C3 | semidirect |
| 26. C2×#127 | C2×C3 | semidirect |
| 27. C2×#124 | C2×C3 | semidirect |
| 28. #496 | C2×C3 | direct (N × H) |
| 29. C2×#97 | C2×C3 | semidirect |
| 49. C2×C3×C8 | D4 | semidirect |
| 59. C2×SD16 | C22×C3 | semidirect |
| 61. C2×D8 | C22×C3 | semidirect |
| 64. #127 | C22×C3 | semidirect |
| 66. #124 | C22×C3 | semidirect |
| 69. #97 | C22×C3 | semidirect |
| 92. C3×C8 | C2×D4 | semidirect |
| 112. C2×C8 | C3×D4 | semidirect |
| 152. C8 | C2×C3×D4 | semidirect |
| 155. C2×C3 | #496 | direct (N × H) |
| 171. C3 | C2×#496 | direct (N × H) |
| 174. C2 | C3×#496 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×SD16 | 2 |
| C22×C3×D8 | 2 |
| C2×C3×#127 | 2 |
| C2×C3×#124 | 2 |
| C3×#496 | 2 |
| C2×C3×#97 | 2 |
| C2×#496 | 3 |