Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
394 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C3×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#137 | C2 | semidirect |
| 3. C2×C3×#124 | C2 | semidirect |
| 4. C2×C3×#104 | C2 | semidirect |
| 5. C3×#419 | C2 | semidirect |
| 6. C2×C3×#100 | C2 | semidirect |
| 7. C3×#339 | C2 | semidirect |
| 8. #2004 | C3 | direct (N × H) |
| 30. C2×#137 | C2×C3 | semidirect |
| 32. C2×#124 | C2×C3 | semidirect |
| 33. C2×#104 | C2×C3 | semidirect |
| 34. #419 | C2×C3 | semidirect |
| 35. C2×#100 | C2×C3 | semidirect |
| 36. #339 | C2×C3 | semidirect |
| 48. C2×C3×Q8 | D4 | semidirect |
| 67. C3×M16 | D4 | semidirect |
| 76. C3×C42 | D4 | semidirect |
| 147. C2×Q8 | C3×D4 | semidirect |
| 166. M16 | C3×D4 | semidirect |
| 175. C42 | C3×D4 | semidirect |
| 256. C3 | #2004 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#137 | 2 |
| C3×#535 | 2 |
| C2×C3×#124 | 2 |
| C2×C3×#104 | 2 |
| C3×#419 | 2 |
| C2×C3×#100 | 2 |
| C3×#339 | 2 |
| #2004 | 3 |