Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
910 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 910 conjugacy classes of subgroups — too many to draw. Only its 218 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 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3568 | C2 | semidirect |
| 2. C3×#3217 | C2 | semidirect |
| 3. C3×#1657 | C2 | semidirect |
| 4. C3×#1655 | C2 | semidirect |
| 5. C3×#1535 | C2 | semidirect |
| 6. #19094 | C3 | direct (N × H) |
| 7. C3×#563 | C22 | semidirect |
| 8. C3×H(3,𝔽4) | C22 | semidirect |
| 13. C2×C3×#127 | C22 | semidirect |
| 16. C3×#490 | C22 | semidirect |
| 21. C3×#432 | C22 | semidirect |
| 22. C3×#330 | C22 | semidirect |
| 23. C3×#329 | C22 | semidirect |
| 24. #3568 | C2×C3 | semidirect |
| 25. #3217 | C2×C3 | semidirect |
| 26. #1657 | C2×C3 | semidirect |
| 27. #1655 | C2×C3 | semidirect |
| 28. #1535 | C2×C3 | semidirect |
| 70. #563 | C22×C3 | semidirect |
| 71. H(3,𝔽4) | C22×C3 | semidirect |
| 76. C2×#127 | C22×C3 | semidirect |
| 79. #490 | C22×C3 | semidirect |
| 84. #432 | C22×C3 | semidirect |
| 85. #330 | C22×C3 | semidirect |
| 86. #329 | C22×C3 | semidirect |
| 278. C3 | #19094 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3568 | 2 |
| C3×#3217 | 2 |
| C3×#1657 | 2 |
| C3×#1655 | 2 |
| C3×#1535 | 2 |
| #19094 | 3 |