Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
908 conjugacy classes of subgroups; only the 328 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 328 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| prime radical | C25×C3 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C4⋊C4 | C2 | semidirect |
| 3. C3×#396 | C2 | direct (N × H) |
| 4. C2×C3×#95 | C2 | semidirect |
| 5. C2×#396 | C3 | direct (N × H) |
| 10. C2×C3×C4⋊C4 | C22 | semidirect |
| 13. C3×#95 | C22 | semidirect |
| 14. C22×C4⋊C4 | C2×C3 | semidirect |
| 16. #396 | C2×C3 | direct (N × H) |
| 17. C2×#95 | C2×C3 | semidirect |
| 37. C2×C4⋊C4 | C22×C3 | semidirect |
| 40. #95 | C22×C3 | semidirect |
| 105. C2×C3 | #396 | direct (N × H) |
| 121. C3 | C2×#396 | direct (N × H) |
| 123. C2 | C3×#396 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C4⋊C4 | 2 |
| C22×C3×C22⋊C4 | 2 |
| C3×#396 | 2 |
| C2×C3×#95 | 2 |
| C2×#396 | 3 |