Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
568 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 287 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C2×C3×C4×Q8 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | Q8×C4⋊C4 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. Q8×Dic3 | C4 | semidirect |
| 47. Dic3⋊C4 | Q8 | direct (N × H) |
| 66. C3×Q8 | C4⋊C4 | semidirect |
| 120. Dic3 | C4×Q8 | semidirect |
| 122. Q8 | Dic3⋊C4 | direct (N × H) |
| 165. C3 | Q8×C4⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×Q8 | 2 |
| C2×Q8×Dic3 | 2 |
| #4975 | 2 |
| #4972 | 2 |
| C4×Dic3⋊C4 | 2 |
| #4928 | 2 |
| Q8×C4⋊C4 | 3 |