Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
282 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 119 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 | C3⋊AGL(1,𝔽9) |
| commutator, hypocenter | C33 | C22×C16 |
| Frattini | C2 | C22×C3⋊AGL(1,𝔽9) |
| Fitting, socle, prime radical | C23×C33 | C8 |
| cosocle | C3×#622 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊#3771 | C2 | direct (N × H) |
| 7. C3⋊#3771 | C22 | direct (N × H) |
| 21. C22×C33 | C16 | semidirect |
| 37. C2×C33 | C2×C16 | semidirect |
| 48. C22×C32 | C3⋊C16 | semidirect |
| 56. C33 | C22×C16 | semidirect |
| 70. C2×C32 | C2×C3⋊C16 | semidirect |
| 80. C22×C3 | #3771 | semidirect |
| 88. C32 | C22×C3⋊C16 | semidirect |
| 98. C2×C3 | C2×#3771 | semidirect |
| 100. C22 | C3⋊#3771 | direct (N × H) |
| 106. C3 | C22×#3771 | semidirect |
| 107. C2 | C2×C3⋊#3771 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#622 | 2 |
| C2×C3⋊#3771 | 2 |
| C22×#3771 | 3 |
| C22×C3⋊C16 | 9 |