Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
812 conjugacy classes of subgroups; only the 476 normal subgroups are shown.
This group has 812 conjugacy classes of subgroups — too many to draw. Only its 476 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C33 | D8 |
| commutator | C8 | C2×C33×C4 |
| Frattini, cosocle | C2×C8 | C22×C33 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ C2×C33×C4 ◁ C2×C33×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C32×#358 | C3 | direct (N × H) |
| 7. C33×Q16 | C4 | semidirect |
| 21. C3×#358 | C32 | direct (N × H) |
| 24. C32×Q16 | C3×C4 | semidirect |
| 43. #358 | C33 | direct (N × H) |
| 49. C3×Q16 | C32×C4 | semidirect |
| 61. C33 | #358 | direct (N × H) |
| 74. Q16 | C33×C4 | semidirect |
| 83. C32 | C3×#358 | direct (N × H) |
| 99. C3 | C32×#358 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C33×Q16 | 2 |
| C2×C33×C16 | 2 |
| C33×#107 | 2 |
| C32×#358 | 3 |