Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
644 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 209 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, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C4⋊Q8 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | C2×C4⋊Q8 | D3 |
| hypocenter | C3 | #2857 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×C4⋊Q8 | C2 | semidirect |
| 4. #5229 | C2 | semidirect |
| 6. #4972 | C2 | semidirect |
| 7. #4935 | C2 | semidirect |
| 24. C2×C4⋊Q8 | D3 | semidirect |
| 46. C2×C4×D3 | Q8 | semidirect |
| 129. C2×D3 | C4⋊Q8 | semidirect |
| 132. C3×C4 | C22⋊Q8 | semidirect |
| 183. C4 | #946 | semidirect |
| 184. C3 | #2857 | 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×#946 | 2 |
| C2×C42×D3 | 2 |
| #5229 | 2 |
| #4987 | 2 |
| #4972 | 2 |
| #4935 | 2 |
| #2857 | 3 |