Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
770 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 770 conjugacy classes of subgroups — too many to draw. Only its 238 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 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#2645 | C2 | semidirect |
| 3. C3×#1927 | C2 | semidirect |
| 4. C3×#1919 | C2 | semidirect |
| 7. C3×#1840 | C2 | semidirect |
| 8. #33092 | C3 | direct (N × H) |
| 26. #2645 | C2×C3 | semidirect |
| 28. #1927 | C2×C3 | semidirect |
| 29. #1919 | C2×C3 | semidirect |
| 32. #1840 | C2×C3 | semidirect |
| 228. C3 | #33092 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2645 | 2 |
| C3×#2275 | 2 |
| C3×#1927 | 2 |
| C3×#1919 | 2 |
| C3×#1877 | 2 |
| C3×#1864 | 2 |
| C3×#1840 | 2 |
| #33092 | 3 |