Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1684 conjugacy classes of subgroups; only the 251 normal subgroups are shown.
This group has 1684 conjugacy classes of subgroups — too many to draw. Only its 251 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#1113 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C2×C33×Q8 | C22 |
| hypercenter | C2×Q8 | #1113 |
| hypocenter | C33 | #550 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#3832 | C2 | semidirect |
| 2. C3×Q8×C3⋊Dic3 | C2 | semidirect |
| 3. #11723697 | C2 | semidirect |
| 4. #11723696 | C2 | semidirect |
| 5. C4×#6295 | C2 | semidirect |
| 18. C2×#3832 | D3 | semidirect |
| 22. Q8×C3⋊Dic3 | D3 | semidirect |
| 33. C33×Q8 | D4 | semidirect |
| 83. C3×C3⋊Dic3 | #41 | semidirect |
| 105. C32×Q8 | D12 | semidirect |
| 106. C32×Q8 | #67 | semidirect |
| 195. C3⋊Dic3 | #239 | semidirect |
| 208. C33 | #550 | semidirect |
| 216. C3×Q8 | #625 | semidirect |
| 217. C3×Q8 | #626 | semidirect |
| 309. C32 | #5577 | semidirect |
| 310. C32 | #5834 | semidirect |
| 312. Q8 | #6295 | semidirect |
| 351. C3 | #10595410 | semidirect |
| 352. C3 | #10595412 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#3832 | 2 |
| C3×Q8×C3⋊Dic3 | 2 |
| #11723697 | 2 |
| #11723696 | 2 |
| C4×#6295 | 2 |
| #10595412 | 3 |
| #10595410 | 3 |