Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
420 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 176 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 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#401 | C2 | semidirect |
| 2. C3×#399 | C2 | semidirect |
| 3. C3×#396 | C2 | semidirect |
| 5. C3×#389 | C2 | semidirect |
| 6. C3×#388 | C2 | semidirect |
| 7. C3×#388 | C2 | semidirect |
| 8. #2801 | C3 | direct (N × H) |
| 22. #401 | C2×C3 | semidirect |
| 23. #399 | C2×C3 | semidirect |
| 24. #396 | C2×C3 | semidirect |
| 26. #389 | C2×C3 | semidirect |
| 27. #388 | C2×C3 | semidirect |
| 28. #388 | C2×C3 | semidirect |
| 149. C3 | #2801 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#401 | 2 |
| C3×#399 | 2 |
| C3×#396 | 2 |
| C3×#394 | 2 |
| C3×#389 | 2 |
| C3×#388 | 2 |
| #2801 | 3 |