Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
354 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 100 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×#4207 |
| commutator, cosocle | C2×C3×C9 | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×C3⋊D3 |
| Fitting | Q8×#737 | C2 |
| hypercenter | Q8 | #4207 |
| hypocenter | C3×C9 | C3×#41 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×#737 | C2 | semidirect |
| 2. C4⋊#4207 | C2 | semidirect |
| 3. C4×#4207 | C2 | semidirect |
| 4. #88746 | C3 | semidirect |
| 13. Q8×3−1+2 | D3 | semidirect |
| 16. C3×Q8×C9 | D3 | semidirect |
| 17. C3×Q8×C9 | C2×C3 | semidirect |
| 18. C4⋊(C3⋊D9) | C2×C3 | semidirect |
| 19. C4×C3⋊D9 | C2×C3 | semidirect |
| 47. #737 | #41 | semidirect |
| 94. 3−1+2 | #239 | semidirect |
| 97. C3×C9 | #239 | semidirect |
| 98. C3×C9 | C3×#41 | semidirect |
| 138. Q8 | #4207 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#737 | 2 |
| C4⋊#4207 | 2 |
| C4×#4207 | 2 |
| #88746 | 3 |
| #88724 | 3 |
| #88722 | 3 |
| C3×#3701 | 3 |