Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
782 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 782 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 | 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. C23×C3×Q8 | C2 | semidirect |
| 3. C3×#395 | C2 | semidirect |
| 4. C3×#393 | C2 | semidirect |
| 5. C3×#381 | C2 | semidirect |
| 6. #2614 | C3 | direct (N × H) |
| 16. C23×Q8 | C2×C3 | semidirect |
| 18. #395 | C2×C3 | semidirect |
| 19. #393 | C2×C3 | semidirect |
| 20. #381 | C2×C3 | semidirect |
| 104. C3 | #2614 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×Q8 | 2 |
| C2×C3×#123 | 2 |
| C3×#395 | 2 |
| C3×#393 | 2 |
| C3×#381 | 2 |
| #2614 | 3 |