Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
696 conjugacy classes of subgroups; only the 520 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 520 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator | C2 | C22×C3×C42 |
| Frattini, cosocle | C23 | C24×C3 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×C4×C4⋊C4 | C2 | semidirect |
| 5. #2274 | C3 | direct (N × H) |
| 8. C3×#117 | C4 | semidirect |
| 9. C2×C3×C4⋊C4 | C4 | semidirect |
| 18. C4×C4⋊C4 | C2×C3 | semidirect |
| 29. C3×C4⋊C4 | C2×C4 | semidirect |
| 32. C3×C42 | C2×C4 | semidirect |
| 35. #117 | C3×C4 | semidirect |
| 36. C2×C4⋊C4 | C3×C4 | semidirect |
| 52. C2×C3×C4 | C42 | semidirect |
| 67. C4⋊C4 | C2×C3×C4 | semidirect |
| 70. C42 | C2×C3×C4 | semidirect |
| 91. C2×C4 | C3×C42 | semidirect |
| 113. C3 | #2274 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#117 | 2 |
| C22×C3×C4⋊C4 | 2 |
| C3×C4×C4⋊C4 | 2 |
| C3×C4×C22⋊C4 | 2 |
| #2274 | 3 |