Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
148 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C27 | C23 |
| commutator | C22 | C23×C27 |
| Frattini, cosocle | C22×C9 | C23×C3 |
| socle | C22×C3 | C23×C9 |
| prime radical | C23×C3 | C22×C9 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C27 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×C27 | C2 | semidirect |
| 3. C4⋊C4×C27 | C2 | semidirect |
| 4. C4⋊C4×C27 | C2 | semidirect |
| 21. C22×C27 | Q8 | semidirect |
| 52. C22⋊Q8 | C27 | direct (N × H) |
| 53. C27 | C22⋊Q8 | direct (N × H) |
| 65. C2×Q8 | C2×C27 | semidirect |
| 67. C4⋊C4 | C2×C27 | semidirect |
| 68. C4⋊C4 | C2×C27 | semidirect |
| 92. C22 | Q8×C27 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×C27 | 2 |
| C22×C4×C27 | 2 |
| C4⋊C4×C27 | 2 |
| C22⋊C4×C27 | 2 |
| C9×C22⋊Q8 | 3 |