Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
226 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #442 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| prime radical | C2×C3×D16 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#322 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#1594 | C2 | semidirect |
| 3. C3×#1368 | C2 | semidirect |
| 4. C3×#1333 | C2 | semidirect |
| 5. C3×#1313 | C2 | semidirect |
| 6. #12089 | C3 | direct (N × H) |
| 18. #1594 | C2×C3 | semidirect |
| 19. #1368 | C2×C3 | semidirect |
| 20. #1333 | C2×C3 | semidirect |
| 21. #1313 | C2×C3 | semidirect |
| 59. C3×C16 | M16 | semidirect |
| 100. C16 | C3×M16 | semidirect |
| 133. C3 | #12089 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2177 | 2 |
| C3×#1594 | 2 |
| C3×#1368 | 2 |
| C3×#1333 | 2 |
| C3×#1313 | 2 |
| #12089 | 3 |