Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
372 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×C3⋊#157 |
| commutator, socle | C2×C33 | C22×C8 |
| Fitting, prime radical | C33×#41 | C4 |
| cosocle | C3×C3⋊(C3⋊C8) | C23 |
| hypercenter | #41 | C3⋊#157 |
| hypocenter | C33 | #504 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11723098 | C2 | semidirect |
| 3. C2×C3⋊#3708 | C2 | semidirect |
| 73. C33 | #504 | semidirect |
| 109. C32 | #5140 | semidirect |
| 126. C3 | #10593838 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#64157 | 2 |
| #11723098 | 2 |
| C2×C3⋊#3708 | 2 |
| #10593838 | 3 |
| #5140 | 9 |