Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
320 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×A4 |
| commutator, socle, cosocle | C24 | C23×C3 |
| Fitting | C22×#126 | C3 |
| prime radical | C23×A4 | C22 |
| hypercenter | #126 | A4 |
| hypocenter | C22 | C3×#126 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×C4⋊C4 | C2 | semidirect |
| 3. C42×A4 | C2 | semidirect |
| 4. C22×#126 | C3 | semidirect |
| 13. C22×C4⋊C4 | C2×C3 | semidirect |
| 15. C22×C42 | C2×C3 | semidirect |
| 32. #126 | A4 | direct (N × H) |
| 63. C4⋊C4 | C2×A4 | semidirect |
| 71. C42 | C2×A4 | semidirect |
| 74. A4 | #126 | direct (N × H) |
| 113. C22 | C3×#126 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×C4⋊C4 | 2 |
| A4×C22⋊C4 | 2 |
| C42×A4 | 2 |
| C22×#126 | 3 |
| C3×#126 | 4 |