Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
360 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 184 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 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C3 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×Q16 | C2 | semidirect |
| 3. C3×#510 | C2 | direct (N × H) |
| 4. C2×C3×Q32 | C2 | semidirect |
| 5. C2×C3×SD32 | C2 | semidirect |
| 6. C2×C3×M32 | C2 | semidirect |
| 7. C2×#510 | C3 | direct (N × H) |
| 16. C2×C3×Q16 | C22 | semidirect |
| 20. C3×Q32 | C22 | semidirect |
| 21. C3×SD32 | C22 | semidirect |
| 22. C3×M32 | C22 | semidirect |
| 25. C22×Q16 | C2×C3 | semidirect |
| 26. #510 | C2×C3 | direct (N × H) |
| 27. C2×Q32 | C2×C3 | semidirect |
| 28. C2×SD32 | C2×C3 | semidirect |
| 29. C2×M32 | C2×C3 | semidirect |
| 63. C2×Q16 | C22×C3 | semidirect |
| 67. Q32 | C22×C3 | semidirect |
| 68. SD32 | C22×C3 | semidirect |
| 69. M32 | C22×C3 | semidirect |
| 150. C2×C3 | #510 | direct (N × H) |
| 162. C3 | C2×#510 | direct (N × H) |
| 165. C2 | C3×#510 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#135 | 2 |
| C22×C3×Q16 | 2 |
| C3×#510 | 2 |
| C2×C3×Q32 | 2 |
| C2×C3×SD32 | 2 |
| C2×C3×M32 | 2 |
| C2×#510 | 3 |