Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
770 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 770 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C32×D4 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C42×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D8×H(3,𝔽3) | C2 | semidirect |
| 2. H(3,𝔽3)×#127 | C2 | semidirect |
| 3. C4×D4×H(3,𝔽3) | C2 | semidirect |
| 4. H(3,𝔽3)×C4⋊C8 | C2 | semidirect |
| 5. H(3,𝔽3)×#102 | C2 | semidirect |
| 6. C32×#459 | C3 | semidirect |
| 17. C2×C32×D8 | C2×C3 | semidirect |
| 18. C32×#127 | C2×C3 | semidirect |
| 19. C32×C4×D4 | C2×C3 | semidirect |
| 20. C32×C4⋊C8 | C2×C3 | semidirect |
| 21. C32×#102 | C2×C3 | semidirect |
| 29. D4×H(3,𝔽3) | D4 | semidirect |
| 57. C4×H(3,𝔽3) | D8 | semidirect |
| 77. C32×D4 | C3×D4 | semidirect |
| 84. #459 | H(3,𝔽3) | direct (N × H) |
| 119. C32×C4 | C3×D8 | semidirect |
| 122. C2×D8 | C2×H(3,𝔽3) | semidirect |
| 123. #127 | C2×H(3,𝔽3) | semidirect |
| 124. C4×D4 | C2×H(3,𝔽3) | semidirect |
| 125. C4⋊C8 | C2×H(3,𝔽3) | semidirect |
| 126. #102 | C2×H(3,𝔽3) | semidirect |
| 127. H(3,𝔽3) | #459 | direct (N × H) |
| 195. C32 | C3×#459 | semidirect |
| 203. D4 | D4×H(3,𝔽3) | semidirect |
| 229. C4 | D8×H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D8×H(3,𝔽3) | 2 |
| H(3,𝔽3)×#127 | 2 |
| C4×D4×H(3,𝔽3) | 2 |
| H(3,𝔽3)×C4⋊C8 | 2 |
| H(3,𝔽3)×#102 | 2 |
| C32×#459 | 3 |