Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1278 conjugacy classes of subgroups; only the 290 normal subgroups are shown.
This group has 1278 conjugacy classes of subgroups — too many to draw. Only its 290 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#3059 | C2 | semidirect |
| 4. C3×#1909 | C2 | semidirect |
| 5. C3×#1812 | C2 | semidirect |
| 6. #32949 | C3 | direct (N × H) |
| 8. C22×C3×Q16 | C22 | semidirect |
| 17. C3×#427 | C22 | semidirect |
| 18. C3×#419 | C22 | semidirect |
| 20. C3×H(3,ℤ/4) | C22 | semidirect |
| 22. #3059 | C2×C3 | semidirect |
| 24. #1909 | C2×C3 | semidirect |
| 25. #1812 | C2×C3 | semidirect |
| 56. C22×Q16 | C22×C3 | semidirect |
| 65. #427 | C22×C3 | semidirect |
| 66. #419 | C22×C3 | semidirect |
| 68. H(3,ℤ/4) | C22×C3 | semidirect |
| 229. C3 | #32949 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3573 | 2 |
| C3×#3059 | 2 |
| C3×#3028 | 2 |
| C3×#1909 | 2 |
| C3×#1812 | 2 |
| #32949 | 3 |