Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
944 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 944 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| socle | C23×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×#41 | C2 | semidirect |
| 2. C2×C3×C4×D4 | C2 | semidirect |
| 3. C3×#395 | C2 | semidirect |
| 4. C3×#394 | C2 | semidirect |
| 5. C3×#393 | C2 | semidirect |
| 6. C3×#392 | C2 | semidirect |
| 7. C3×C4×C22⋊C4 | C2 | semidirect |
| 8. #2388 | C3 | direct (N × H) |
| 23. C22×#41 | C2×C3 | semidirect |
| 24. C2×C4×D4 | C2×C3 | semidirect |
| 25. #395 | C2×C3 | semidirect |
| 26. #394 | C2×C3 | semidirect |
| 27. #393 | C2×C3 | semidirect |
| 28. #392 | C2×C3 | semidirect |
| 29. C4×C22⋊C4 | C2×C3 | semidirect |
| 41. C22×C3×C4 | D4 | semidirect |
| 73. C23×C3 | #41 | semidirect |
| 88. C2×C3×C4 | #41 | semidirect |
| 100. C22×C4 | C3×D4 | semidirect |
| 134. C23 | C3×#41 | semidirect |
| 149. C2×C4 | C3×#41 | semidirect |
| 171. C3 | #2388 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#41 | 2 |
| C2×C3×C4×D4 | 2 |
| C3×#395 | 2 |
| C3×#394 | 2 |
| C3×#393 | 2 |
| C3×#392 | 2 |
| C3×C4×C22⋊C4 | 2 |
| #2388 | 3 |