Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
473 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 91 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×#99 |
| commutator | C2×C3×C4 | C2×C32×C4 |
| Frattini, cosocle | C2×C3×D4 | C22×C32 |
| prime radical | H(3,𝔽3)×#127 | C2 |
| LCS | G ▷ C2×C3×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×H(3,𝔽3) ◁ C2×D4×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. H(3,𝔽3)×#99 | C2 | semidirect |
| 3. C32×#353 | C3 | semidirect |
| 5. C2×D4×H(3,𝔽3) | C4 | semidirect |
| 8. C42×H(3,𝔽3) | C4 | semidirect |
| 10. C32×#99 | C2×C3 | semidirect |
| 22. C2×C32×D4 | C3×C4 | semidirect |
| 25. C32×C42 | C3×C4 | semidirect |
| 46. #353 | H(3,𝔽3) | direct (N × H) |
| 70. #99 | C2×H(3,𝔽3) | semidirect |
| 71. H(3,𝔽3) | #353 | direct (N × H) |
| 93. C2×D4 | C4×H(3,𝔽3) | semidirect |
| 96. C42 | C4×H(3,𝔽3) | semidirect |
| 113. C32 | C3×#353 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×#127 | 2 |
| H(3,𝔽3)×#99 | 2 |
| C32×#353 | 3 |