Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
715 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 189 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 ◁ C22×C4×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. C2×M16×H(3,𝔽3) | C2 | semidirect |
| 3. H(3,𝔽3)×#121 | C2 | semidirect |
| 4. H(3,𝔽3)×#106 | C2 | semidirect |
| 5. H(3,𝔽3)×#102 | C2 | semidirect |
| 6. C32×#469 | C3 | semidirect |
| 16. C2×C32×D8 | C2×C3 | semidirect |
| 17. C2×C32×M16 | C2×C3 | semidirect |
| 18. C32×#121 | C2×C3 | semidirect |
| 19. C32×#106 | C2×C3 | semidirect |
| 20. C32×#102 | C2×C3 | semidirect |
| 32. C8×H(3,𝔽3) | D4 | semidirect |
| 75. C32×C8 | C3×D4 | semidirect |
| 76. #469 | H(3,𝔽3) | direct (N × H) |
| 109. C2×D8 | C2×H(3,𝔽3) | semidirect |
| 110. C2×M16 | C2×H(3,𝔽3) | semidirect |
| 111. #121 | C2×H(3,𝔽3) | semidirect |
| 112. #106 | C2×H(3,𝔽3) | semidirect |
| 113. #102 | C2×H(3,𝔽3) | semidirect |
| 114. H(3,𝔽3) | #469 | direct (N × H) |
| 172. C32 | C3×#469 | semidirect |
| 184. C8 | D4×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 |
| C2×M16×H(3,𝔽3) | 2 |
| H(3,𝔽3)×#121 | 2 |
| H(3,𝔽3)×#106 | 2 |
| H(3,𝔽3)×#102 | 2 |
| C32×#469 | 3 |