Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #882 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×D4⋊C8 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | D4⋊C8 | D3 |
| hypocenter | C3 | #1624 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ D4⋊C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5563 | C2 | semidirect |
| 2. #5015 | C2 | semidirect |
| 3. #4843 | C2 | semidirect |
| 4. #4693 | C2 | semidirect |
| 5. C3×D4⋊C8 | C2 | semidirect |
| 6. #4481 | C2 | semidirect |
| 7. #4459 | C2 | semidirect |
| 32. D4⋊C8 | D3 | semidirect |
| 196. C3 | #1624 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5563 | 2 |
| #5015 | 2 |
| #4843 | 2 |
| #4693 | 2 |
| C3×D4⋊C8 | 2 |
| #4481 | 2 |
| #4459 | 2 |
| #1624 | 3 |