Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D27 |
| commutator, cosocle | C22×C27 | C23 |
| Frattini | C22×C9 | C22×D3 |
| Fitting | C4⋊C4×C27 | C2 |
| socle | C22×C3 | C22×D9 |
| hypercenter | C4⋊C4 | D27 |
| hypocenter | C27 | #121 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D108 | C2 | semidirect |
| 2. C2×D108 | C2 | semidirect |
| 3. C2×C4×D27 | C2 | semidirect |
| 4. C4⋊C4×C27 | C2 | semidirect |
| 5. (C2×D27)⋊C4 | C2 | semidirect |
| 26. C2×D27 | D4 | semidirect |
| 28. C4×C27 | D4 | semidirect |
| 60. C27 | #121 | semidirect |
| 77. C4⋊C4 | D27 | semidirect |
| 112. C4 | D108 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D108 | 2 |
| C2×C4×D27 | 2 |
| C4⋊C4×C27 | 2 |
| (C2×D27)⋊C4 | 2 |
| #63457 | 3 |