Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
354 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×A4 |
| commutator, socle | C23 | C22×C3×C4 |
| Frattini | C4 | C23×A4 |
| Fitting | C22×#131 | C3 |
| prime radical | A4×#41 | C2 |
| cosocle | C22×C4 | C23×C3 |
| hypercenter | #131 | A4 |
| hypocenter | C22 | C3×#131 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C8 ◁ #131 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×M16 | C2 | semidirect |
| 3. C2×C8×A4 | C2 | semidirect |
| 4. C22×#131 | C3 | semidirect |
| 16. C22×M16 | C2×C3 | semidirect |
| 17. C23×C8 | C2×C3 | semidirect |
| 37. #131 | A4 | direct (N × H) |
| 76. M16 | C2×A4 | semidirect |
| 85. C2×C8 | C2×A4 | semidirect |
| 87. A4 | #131 | direct (N × H) |
| 124. C22 | C3×#131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×#41 | 2 |
| A4×M16 | 2 |
| C2×C8×A4 | 2 |
| C22×#131 | 3 |
| C3×#131 | 4 |