Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1072 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 230 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 |
| prime radical | C3×#3549 | C2 |
| 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 |
|---|---|---|
| 2. C3×#3152 | C2 | semidirect |
| 3. C3×#3141 | C2 | semidirect |
| 4. C3×#1659 | C2 | semidirect |
| 5. C3×#1525 | C2 | semidirect |
| 6. C3×#1525 | C2 | semidirect |
| 7. #19074 | C3 | direct (N × H) |
| 18. C2×C3×C4⋊Q8 | C22 | semidirect |
| 19. C3×#497 | C22 | semidirect |
| 22. C3×#489 | C22 | semidirect |
| 26. C3×C4×M16 | C22 | semidirect |
| 27. C3×C4×M16 | C22 | semidirect |
| 28. C3×#329 | C22 | semidirect |
| 29. C3×#329 | C22 | semidirect |
| 31. #3152 | C2×C3 | semidirect |
| 32. #3141 | C2×C3 | semidirect |
| 33. #1659 | C2×C3 | semidirect |
| 34. #1525 | C2×C3 | semidirect |
| 35. #1525 | C2×C3 | semidirect |
| 90. C2×C4⋊Q8 | C22×C3 | semidirect |
| 91. #497 | C22×C3 | semidirect |
| 94. #489 | C22×C3 | semidirect |
| 98. C4×M16 | C22×C3 | semidirect |
| 99. C4×M16 | C22×C3 | semidirect |
| 100. #329 | C22×C3 | semidirect |
| 101. #329 | C22×C3 | semidirect |
| 335. C3 | #19074 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3549 | 2 |
| C3×#3152 | 2 |
| C3×#3141 | 2 |
| C3×#1659 | 2 |
| C3×#1525 | 2 |
| #19074 | 3 |