Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
164 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C23 |
| commutator | C22 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C22×C8 |
| prime radical | C2×C3×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×M32 | C2 | semidirect |
| 2. C22×C3×C16 | C2 | semidirect |
| 4. C3×C4⋊C16 | C2 | semidirect |
| 5. C3×C22⋊C16 | C2 | semidirect |
| 6. C3×C22⋊C16 | C2 | semidirect |
| 7. C3×#346 | C2 | semidirect |
| 8. #2168 | C3 | direct (N × H) |
| 22. C2×M32 | C2×C3 | semidirect |
| 23. C22×C16 | C2×C3 | semidirect |
| 25. C4⋊C16 | C2×C3 | semidirect |
| 26. C22⋊C16 | C2×C3 | semidirect |
| 27. C22⋊C16 | C2×C3 | semidirect |
| 28. #346 | C2×C3 | semidirect |
| 47. C3×C16 | D4 | semidirect |
| 96. C16 | C3×D4 | semidirect |
| 102. C22×C3 | M32 | semidirect |
| 137. C22 | C3×M32 | semidirect |
| 145. C3 | #2168 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×M32 | 2 |
| C22×C3×C16 | 2 |
| C3×C8×D4 | 2 |
| C3×C4⋊C16 | 2 |
| C3×C22⋊C16 | 2 |
| C3×#346 | 2 |
| #2168 | 3 |