Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
324 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 150 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×C32 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C2×C32×D8 | C2 | semidirect |
| 3. C32×#131 | C2 | semidirect |
| 4. C32×#108 | C2 | semidirect |
| 5. C32×#100 | C2 | semidirect |
| 6. C3×#472 | C3 | direct (N × H) |
| 18. C3×AGL(1,ℤ/8) | C2×C3 | semidirect |
| 19. C2×C3×D8 | C2×C3 | semidirect |
| 20. C3×#131 | C2×C3 | semidirect |
| 21. C3×#108 | C2×C3 | semidirect |
| 22. C3×#100 | C2×C3 | semidirect |
| 35. #472 | C32 | direct (N × H) |
| 54. AGL(1,ℤ/8) | C2×C32 | semidirect |
| 55. C2×D8 | C2×C32 | semidirect |
| 56. #131 | C2×C32 | semidirect |
| 57. #108 | C2×C32 | semidirect |
| 58. #100 | C2×C32 | semidirect |
| 93. C32 | #472 | direct (N × H) |
| 117. C3 | C3×#472 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×AGL(1,ℤ/8) | 2 |
| C2×C32×D8 | 2 |
| C32×#131 | 2 |
| C32×#108 | 2 |
| C32×#100 | 2 |
| C3×#472 | 3 |