Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
456 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | C3⋊#157 |
| commutator, hypocenter | C33 | C22×C16 |
| Frattini | C4 | C22×C3⋊#157 |
| Fitting | C22×C33×C4 | C4 |
| socle, prime radical | C23×C33 | C8 |
| cosocle | C3×C3⋊(C3⋊C8) | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊#3708 | C2 | direct (N × H) |
| 7. C3⋊#3708 | C22 | direct (N × H) |
| 25. C22×C33 | C16 | semidirect |
| 46. C2×C33 | C2×C16 | semidirect |
| 60. C22×C32 | C3⋊C16 | semidirect |
| 70. C33 | C22×C16 | semidirect |
| 88. C2×C32 | C2×C3⋊C16 | semidirect |
| 98. C22×C3 | #3708 | semidirect |
| 108. C32 | C22×C3⋊C16 | semidirect |
| 118. C2×C3 | C2×#3708 | semidirect |
| 120. C22 | C3⋊#3708 | direct (N × H) |
| 126. C3 | C22×#3708 | semidirect |
| 127. C2 | C2×C3⋊#3708 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C3⋊(C3⋊C8) | 2 |
| C2×C3⋊#3708 | 2 |
| C22×#3708 | 3 |
| C22×C3⋊C16 | 9 |