Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
594 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×#121 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C4×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽3)×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C2×D8×H(3,𝔽3) | C2 | semidirect |
| 3. H(3,𝔽3)×#131 | C2 | semidirect |
| 4. H(3,𝔽3)×#108 | C2 | semidirect |
| 5. H(3,𝔽3)×#100 | C2 | semidirect |
| 6. C32×#472 | C3 | semidirect |
| 18. C32×AGL(1,ℤ/8) | C2×C3 | semidirect |
| 19. C2×C32×D8 | C2×C3 | semidirect |
| 20. C32×#131 | C2×C3 | semidirect |
| 21. C32×#108 | C2×C3 | semidirect |
| 22. C32×#100 | C2×C3 | semidirect |
| 77. #472 | H(3,𝔽3) | direct (N × H) |
| 111. AGL(1,ℤ/8) | C2×H(3,𝔽3) | semidirect |
| 112. C2×D8 | C2×H(3,𝔽3) | semidirect |
| 113. #131 | C2×H(3,𝔽3) | semidirect |
| 114. #108 | C2×H(3,𝔽3) | semidirect |
| 115. #100 | C2×H(3,𝔽3) | semidirect |
| 116. H(3,𝔽3) | #472 | direct (N × H) |
| 170. C32 | C3×#472 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×AGL(1,ℤ/8) | 2 |
| C2×D8×H(3,𝔽3) | 2 |
| H(3,𝔽3)×#131 | 2 |
| H(3,𝔽3)×#108 | 2 |
| H(3,𝔽3)×#100 | 2 |
| C32×#472 | 3 |