Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1624 conjugacy classes of subgroups; only the 725 normal subgroups are shown.
This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 725 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C23×C3×Q8 | C2 |
| socle, prime radical | C24×C3 | C23 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | C22×C4⋊Q8 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#944 | C2 | direct (N × H) |
| 15. #944 | C22 | direct (N × H) |
| 31. C22×Dic3 | Q8 | semidirect |
| 58. C2×Dic3 | C2×Q8 | semidirect |
| 77. C22×C3 | C4⋊Q8 | semidirect |
| 84. Dic3 | C22×Q8 | semidirect |
| 98. C2×C3 | C2×C4⋊Q8 | semidirect |
| 101. C22 | #944 | direct (N × H) |
| 109. C3 | C22×C4⋊Q8 | semidirect |
| 110. C2 | C2×#944 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×Q8 | 2 |
| C23×Dic6 | 2 |
| C2×#944 | 2 |
| C22×Dic3⋊C4 | 2 |
| C22×C4×Dic3 | 2 |
| C22×C4⋊Q8 | 3 |