Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C16 | D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C8×D4 |
| prime radical | C3×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C16 ◁ C2×C3×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#504 | C2 | semidirect |
| 3. C3×#364 | C2 | semidirect |
| 4. C3×#350 | C2 | semidirect |
| 5. C3×C4×C16 | C2 | semidirect |
| 6. #2170 | C3 | direct (N × H) |
| 17. #504 | C2×C3 | semidirect |
| 19. #364 | C2×C3 | semidirect |
| 20. #350 | C2×C3 | semidirect |
| 21. C4×C16 | C2×C3 | semidirect |
| 97. C3 | #2170 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#504 | 2 |
| C3×#443 | 2 |
| C3×#364 | 2 |
| C3×#350 | 2 |
| C3×C4×C16 | 2 |
| #2170 | 3 |