Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #940 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×#492 | C2 |
| socle | C22×C3 | #127 |
| hypercenter | #492 | D3 |
| hypocenter | C3 | #1721 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #492 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#492 | C2 | semidirect |
| 2. #5088 | C2 | semidirect |
| 3. #5073 | C2 | semidirect |
| 4. #4695 | C2 | semidirect |
| 5. #4455 | C2 | semidirect |
| 18. #492 | D3 | semidirect |
| 54. C3×C8 | D8 | semidirect |
| 90. C8 | #213 | semidirect |
| 104. C3 | #1721 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#492 | 2 |
| #5088 | 2 |
| #5073 | 2 |
| #4695 | 2 |
| #4455 | 2 |
| #1721 | 3 |