Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2006 conjugacy classes of subgroups; only the 286 normal subgroups are shown.
This group has 2006 conjugacy classes of subgroups — too many to draw. Only its 286 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×C22⋊C4 |
| commutator | C24 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C22×C3×#127 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×#100 | C2 | semidirect |
| 3. C3×#1728 | C2 | semidirect |
| 4. #10981 | C3 | direct (N × H) |
| 15. C22×#100 | C2×C3 | semidirect |
| 16. #1728 | C2×C3 | semidirect |
| 31. C2×C3×M16 | D4 | semidirect |
| 93. C2×M16 | C3×D4 | semidirect |
| 117. C2×C3×C4 | #100 | semidirect |
| 191. C2×C4 | C3×#100 | semidirect |
| 219. C3 | #10981 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#127 | 2 |
| C22×C3×#100 | 2 |
| C3×#1728 | 2 |
| #10981 | 3 |