Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
146 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3802 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C2×D32 |
| Fitting | C32×C8 | C22 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C8 | D32 |
| hypocenter | C32 | AGL(1,ℤ/8) |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3802 | C2 | semidirect |
| 2. #3796 | C2 | semidirect |
| 3. #3743 | C2 | semidirect |
| 4. C3×D24 | C2 | semidirect |
| 5. C3×#204 | C2 | semidirect |
| 6. #3715 | C2 | semidirect |
| 7. #3713 | C2 | semidirect |
| 14. C3×D12 | C22 | semidirect |
| 15. C3×Dic6 | C22 | semidirect |
| 19. C32×C8 | C22 | semidirect |
| 20. C3⋊(C3⋊C8) | C22 | semidirect |
| 31. D24 | D3 | semidirect |
| 32. #204 | D3 | semidirect |
| 58. D12 | C2×D3 | semidirect |
| 63. Dic6 | C2×D3 | semidirect |
| 65. C3×C8 | C2×D3 | semidirect |
| 66. C3×C8 | C2×D3 | semidirect |
| 101. C32 | AGL(1,ℤ/8) | semidirect |
| 112. C8 | D32 | semidirect |
| 134. C3 | #911 | semidirect |
| 135. C3 | #914 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3802 | 2 |
| #3796 | 2 |
| #3743 | 2 |
| C3×D24 | 2 |
| C3×#204 | 2 |
| #3715 | 2 |
| #3713 | 2 |
| #914 | 3 |
| #911 | 3 |