Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
280 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 184 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 | D4 |
| commutator | C4 | C2×C3×C42 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle, prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C4×#106 | C3 | direct (N × H) |
| 11. C3×#106 | C4 | direct (N × H) |
| 12. C3×C4×C8 | C4 | semidirect |
| 36. #106 | C3×C4 | direct (N × H) |
| 37. C4×C8 | C3×C4 | semidirect |
| 52. C3×C8 | C42 | semidirect |
| 72. C3×C4 | #106 | direct (N × H) |
| 89. C8 | C3×C42 | semidirect |
| 100. C4 | C3×#106 | direct (N × H) |
| 104. C3 | C4×#106 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#106 | 2 |
| C2×C3×C4×C8 | 2 |
| C3×C4×C4⋊C4 | 2 |
| C3×#340 | 2 |
| C4×#106 | 3 |