Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
373 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | (C22⋊C27)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C27 | C23 |
| Frattini | C2×C9 | C22×S4 |
| Fitting | C22×Q8×C9 | D3 |
| socle, prime radical | C23×C3 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C27 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#88390 | C2 | direct (N × H) |
| 6. #88390 | C22 | direct (N × H) |
| 105. C22 | #88390 | direct (N × H) |
| 111. C2 | C2×#88390 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8⋊C27 | 2 |
| C2×#88390 | 2 |
| C22×#3674 | 3 |
| C22×Dic27 | 4 |