Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 234 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, socle | C2×C3 | C2×#99 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C2×D4 | C23×C3 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C3×#352 | C2 | semidirect |
| 5. #2123 | C3 | direct (N × H) |
| 7. C22×C3×Q8 | C4 | semidirect |
| 11. C3×#123 | C4 | semidirect |
| 16. C3×#117 | C4 | semidirect |
| 24. #352 | C2×C3 | semidirect |
| 53. C22×Q8 | C3×C4 | semidirect |
| 57. #123 | C3×C4 | semidirect |
| 62. #117 | C3×C4 | semidirect |
| 179. C3 | #2123 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#536 | 2 |
| C3×#413 | 2 |
| C3×#410 | 2 |
| C3×#352 | 2 |
| #2123 | 3 |