Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
460 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×C4⋊(C3⋊D3) |
| commutator, cosocle | C33×C4 | C23 |
| Frattini | C4 | C2×D3×C3⋊D3 |
| Fitting | C33×C8 | C22 |
| socle | C2×C33 | C2×D4 |
| hypercenter | C8 | D3×C3⋊D3 |
| hypocenter | C33 | #135 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #89022 | C2 | semidirect |
| 2. #88852 | C2 | semidirect |
| 3. C3×#3746 | C2 | semidirect |
| 4. C32×D3×C8 | C2 | semidirect |
| 5. #88797 | C2 | semidirect |
| 18. #3746 | D3 | semidirect |
| 24. C3×D3×C8 | D3 | semidirect |
| 62. D3×C8 | C3⋊D3 | semidirect |
| 76. C33 | #135 | semidirect |
| 119. C32 | #904 | semidirect |
| 120. C32 | #919 | semidirect |
| 142. C3 | #63808 | semidirect |
| 143. C3 | #64113 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #89022 | 2 |
| #88852 | 2 |
| C3×#3746 | 2 |
| C32×D3×C8 | 2 |
| #88797 | 2 |
| #64113 | 3 |
| #63808 | 3 |