Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
656 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C3×#532 | C2 | semidirect |
| 3. C2×C3×#127 | C2 | semidirect |
| 4. C3×#496 | C2 | semidirect |
| 5. C3×#493 | C2 | semidirect |
| 6. C3×#492 | C2 | semidirect |
| 7. C3×C4×M16 | C2 | semidirect |
| 8. #3151 | C3 | direct (N × H) |
| 13. C2×C3×SD16 | C22 | semidirect |
| 14. C2×C3×D8 | C22 | semidirect |
| 16. C3×C4⋊Q8 | C22 | semidirect |
| 20. C3×#127 | C22 | semidirect |
| 21. C3×#124 | C22 | semidirect |
| 25. C3×#97 | C22 | semidirect |
| 26. C3×C4×C8 | C22 | semidirect |
| 27. C2×AGL(1,ℤ/8) | C2×C3 | semidirect |
| 28. #532 | C2×C3 | semidirect |
| 29. C2×#127 | C2×C3 | semidirect |
| 30. #496 | C2×C3 | semidirect |
| 31. #493 | C2×C3 | semidirect |
| 32. #492 | C2×C3 | semidirect |
| 33. C4×M16 | C2×C3 | semidirect |
| 51. C3×M16 | D4 | semidirect |
| 62. C2×SD16 | C22×C3 | semidirect |
| 63. C2×D8 | C22×C3 | semidirect |
| 65. C4⋊Q8 | C22×C3 | semidirect |
| 69. #127 | C22×C3 | semidirect |
| 70. #124 | C22×C3 | semidirect |
| 74. #97 | C22×C3 | semidirect |
| 75. C4×C8 | C22×C3 | semidirect |
| 97. C3×C8 | C2×D4 | semidirect |
| 115. M16 | C3×D4 | semidirect |
| 133. C3×C4 | AGL(1,ℤ/8) | semidirect |
| 156. C8 | C2×C3×D4 | semidirect |
| 173. C4 | C3×AGL(1,ℤ/8) | semidirect |
| 175. C3 | #3151 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×AGL(1,ℤ/8) | 2 |
| C3×#532 | 2 |
| C2×C3×#127 | 2 |
| C3×#496 | 2 |
| C3×#493 | 2 |
| C3×#492 | 2 |
| C3×C4×M16 | 2 |
| #3151 | 3 |