Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
280 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#4184 |
| commutator, cosocle | C2×#731 | C23 |
| Frattini | C2×C32 | C22×C3⋊D3 |
| Fitting | Q8×#731 | C2 |
| socle | C2×C3 | C22×#272 |
| hypercenter | Q8 | #4184 |
| hypocenter | #731 | #41 |
| LCS | G ▷ C2×#731 ▷ #731 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×#731 | C2 | semidirect |
| 2. C4⋊#4184 | C2 | semidirect |
| 3. C4×#4184 | C2 | semidirect |
| 10. Q8×3−1+2 | D3 | semidirect |
| 12. C3×Q8×C9 | D3 | semidirect |
| 34. #731 | #41 | semidirect |
| 68. 3−1+2 | #239 | semidirect |
| 70. C3×C9 | #239 | semidirect |
| 102. Q8 | #4184 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#731 | 2 |
| C4⋊#4184 | 2 |
| C4×#4184 | 2 |
| #88724 | 3 |
| #88722 | 3 |
| C3×#3701 | 3 |