Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
564 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 216 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, Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#135 | C2 | semidirect |
| 2. C3×#532 | C2 | semidirect |
| 3. C3×#497 | C2 | semidirect |
| 4. C3×#496 | C2 | semidirect |
| 5. C2×C3×#97 | C2 | semidirect |
| 6. #3150 | C3 | direct (N × H) |
| 20. C2×#135 | C2×C3 | semidirect |
| 21. #532 | C2×C3 | semidirect |
| 22. #497 | C2×C3 | semidirect |
| 23. #496 | C2×C3 | semidirect |
| 24. C2×#97 | C2×C3 | semidirect |
| 36. C2×C3×C8 | D4 | semidirect |
| 80. C2×C8 | C3×D4 | semidirect |
| 122. C3 | #3150 | 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 |
| C3×#532 | 2 |
| C3×#497 | 2 |
| C3×#496 | 2 |
| C2×C3×#97 | 2 |
| #3150 | 3 |