Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1494 conjugacy classes of subgroups; only the 310 normal subgroups are shown.
This group has 1494 conjugacy classes of subgroups — too many to draw. Only its 310 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×#131 | C2 | semidirect |
| 2. C2×C3×#466 | C2 | semidirect |
| 3. C3×#2615 | C2 | semidirect |
| 4. C3×#1895 | C2 | semidirect |
| 5. C3×#1808 | C2 | semidirect |
| 6. #32933 | C3 | direct (N × H) |
| 20. C22×#131 | C2×C3 | semidirect |
| 21. C2×#466 | C2×C3 | semidirect |
| 22. #2615 | C2×C3 | semidirect |
| 23. #1895 | C2×C3 | semidirect |
| 24. #1808 | C2×C3 | semidirect |
| 37. C22×C3×C8 | D4 | semidirect |
| 101. C22×C8 | C3×D4 | semidirect |
| 225. C3 | #32933 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#131 | 2 |
| C2×C3×#466 | 2 |
| C3×#2615 | 2 |
| C3×#1895 | 2 |
| C3×#1808 | 2 |
| #32933 | 3 |