Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
186 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C22×C3 | C4×D4 |
| prime radical | C23×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. #1922 | C3 | direct (N × H) |
| 16. C3×#107 | C4 | semidirect |
| 17. C3×#107 | C4 | semidirect |
| 18. C3×#106 | C4 | semidirect |
| 19. C3×#106 | C4 | semidirect |
| 64. #107 | C3×C4 | semidirect |
| 65. #107 | C3×C4 | semidirect |
| 66. #106 | C3×C4 | semidirect |
| 67. #106 | C3×C4 | semidirect |
| 171. C3 | #1922 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#427 | 2 |
| C3×#424 | 2 |
| C3×#405 | 2 |
| C3×#341 | 2 |
| C3×#339 | 2 |
| C3×#336 | 2 |
| #1922 | 3 |