Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4242 conjugacy classes of subgroups; only the 1047 normal subgroups are shown.
This group has 4242 conjugacy classes of subgroups — too many to draw. Only its 1047 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | #1113 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C23×#1113 |
| Fitting, socle, prime radical | C24×C33 | C22 |
| hypocenter | C33 | C23×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#6296 | C2 | direct (N × H) |
| 6. C2×#6296 | C22 | direct (N × H) |
| 11. C23×C33 | Q8 | semidirect |
| 14. #6296 | C23 | direct (N × H) |
| 23. C22×C33 | C2×Q8 | semidirect |
| 34. C23×C32 | Dic6 | semidirect |
| 41. C2×C33 | C22×Q8 | semidirect |
| 54. C22×C32 | C2×Dic6 | semidirect |
| 61. C33 | C23×Q8 | semidirect |
| 66. C23×C3 | #627 | semidirect |
| 76. C2×C32 | C22×Dic6 | semidirect |
| 85. C22×C3 | C2×#627 | semidirect |
| 92. C32 | C23×Dic6 | semidirect |
| 93. C23 | #6296 | direct (N × H) |
| 101. C2×C3 | C22×#627 | semidirect |
| 103. C22 | C2×#6296 | direct (N × H) |
| 108. C3 | C23×#627 | semidirect |
| 109. C2 | C22×#6296 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C3⋊Dic3 | 2 |
| C22×#6296 | 2 |
| C23×#627 | 3 |