Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1112 conjugacy classes of subgroups; only the 581 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 581 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×C4⋊Q8 | C2 |
| socle, prime radical | C23×C3 | C24 |
| hypercenter | C2×C4⋊Q8 | D3 |
| hypocenter | C3 | C2×Q82 |
| 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 |
|---|---|---|
| 3. #5722 | C2 | direct (N × H) |
| 37. C2×Dic6 | Q8 | semidirect |
| 75. Dic6 | C2×Q8 | semidirect |
| 116. C2×C3 | Q82 | semidirect |
| 127. C3 | C2×Q82 | semidirect |
| 128. C2 | #5722 | direct (N × H) |
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×#944 | 2 |
| #5722 | 2 |
| C2×#888 | 2 |
| C2×#887 | 2 |
| C2×C4×Dic6 | 2 |
| C2×Q82 | 3 |