Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
620 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D27 |
| commutator, cosocle | C22×C27 | C24 |
| Frattini | C22×C9 | C23×D3 |
| Fitting | C2×C42×C27 | C2 |
| socle | C22×C3 | C23×D9 |
| hypercenter | C2×C42 | D27 |
| hypocenter | C27 | #532 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#88399 | C2 | semidirect |
| 2. C2×C42×C27 | C2 | semidirect |
| 3. #11720179 | C2 | semidirect |
| 4. #11720124 | C2 | semidirect |
| 5. #11720123 | C2 | semidirect |
| 6. C4×D108 | C2 | semidirect |
| 7. #11720118 | C2 | semidirect |
| 32. C2×C4×C27 | D4 | semidirect |
| 65. C4×C27 | #41 | semidirect |
| 122. C2×C42 | D27 | semidirect |
| 123. C27 | #532 | semidirect |
| 175. C2×C4 | D108 | semidirect |
| 193. C4 | #88399 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88399 | 2 |
| C2×C42×C27 | 2 |
| #11720179 | 2 |
| #11720124 | 2 |
| #11720123 | 2 |
| C4×D108 | 2 |
| #11720118 | 2 |
| #10589579 | 3 |