Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2278 conjugacy classes of subgroups; only the 515 normal subgroups are shown.
This group has 2278 conjugacy classes of subgroups — too many to draw. Only its 515 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator, socle, prime radical, cosocle | C24×C3 | C24 |
| Fitting | C3×C4×#95 | C2 |
| hypercenter | C4×#95 | D3 |
| hypocenter | C3 | #15190 |
| LCS | G ▷ C24×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C4×#95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 34. C3×#95 | Q8 | semidirect |
| 84. #95 | Dic6 | semidirect |
| 96. C2×C3×C4 | C4⋊Q8 | semidirect |
| 155. C2×C4 | #869 | semidirect |
| 175. C3 | #15190 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#869 | 2 |
| C2×#4929 | 2 |
| C3×C4×#95 | 2 |
| #69419 | 2 |
| #15190 | 3 |