Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
400 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 200 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 | D8 |
| commutator, Frattini, cosocle | C8 | C24×C3 |
| prime radical | C22×C3×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×Q16 | C2 | semidirect |
| 3. C2×C3×SD32 | C2 | direct (N × H) |
| 4. C22×C3×C16 | C2 | semidirect |
| 5. C22×SD32 | C3 | direct (N × H) |
| 9. C2×C3×Q16 | C22 | semidirect |
| 13. C3×SD32 | C22 | direct (N × H) |
| 14. C2×C3×C16 | C22 | semidirect |
| 15. C22×Q16 | C2×C3 | semidirect |
| 17. C2×SD32 | C2×C3 | direct (N × H) |
| 18. C22×C16 | C2×C3 | semidirect |
| 27. C3×Q16 | C23 | semidirect |
| 32. C3×C16 | C23 | semidirect |
| 36. C2×Q16 | C22×C3 | semidirect |
| 40. SD32 | C22×C3 | direct (N × H) |
| 41. C2×C16 | C22×C3 | semidirect |
| 62. Q16 | C23×C3 | semidirect |
| 67. C16 | C23×C3 | semidirect |
| 70. C22×C3 | SD32 | direct (N × H) |
| 88. C2×C3 | C2×SD32 | direct (N × H) |
| 92. C22 | C3×SD32 | direct (N × H) |
| 97. C3 | C22×SD32 | direct (N × H) |
| 99. C2 | C2×C3×SD32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×Q16 | 2 |
| C22×C3×D8 | 2 |
| C2×C3×SD32 | 2 |
| C22×C3×C16 | 2 |
| C22×SD32 | 3 |