Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
492 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 344 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 | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C22×D4 |
| prime radical | C24×C3 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#428 | C2 | semidirect |
| 4. C3×#427 | C2 | semidirect |
| 5. #2911 | C3 | direct (N × H) |
| 7. C2×C3×M16 | C4 | semidirect |
| 17. C3×#107 | C22 | semidirect |
| 18. C3×#106 | C22 | semidirect |
| 21. #428 | C2×C3 | semidirect |
| 22. #427 | C2×C3 | semidirect |
| 31. C3×M16 | C2×C4 | semidirect |
| 32. C2×C3×C8 | C2×C4 | semidirect |
| 41. C2×M16 | C3×C4 | semidirect |
| 51. #107 | C22×C3 | semidirect |
| 52. #106 | C22×C3 | semidirect |
| 77. M16 | C2×C3×C4 | semidirect |
| 78. C2×C8 | C2×C3×C4 | semidirect |
| 128. C3 | #2911 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×M16 | 2 |
| C2×C3×#117 | 2 |
| C3×#428 | 2 |
| C3×#427 | 2 |
| #2911 | 3 |