Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1771 conjugacy classes of subgroups; only the 301 normal subgroups are shown.
This group has 1771 conjugacy classes of subgroups — too many to draw. Only its 301 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23×C3 | C23×C32 |
| LCS | G ▷ C23×C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D4×H(3,𝔽3) | C2 | semidirect |
| 2. C2×C22⋊C4×H(3,𝔽3) | C2 | semidirect |
| 3. H(3,𝔽3)×#95 | C2 | semidirect |
| 4. C32×#392 | C3 | semidirect |
| 11. C22×C32×D4 | C2×C3 | semidirect |
| 12. C2×C32×C22⋊C4 | C2×C3 | semidirect |
| 13. C32×#95 | C2×C3 | semidirect |
| 18. C23×H(3,𝔽3) | D4 | semidirect |
| 23. C2×C4×H(3,𝔽3) | D4 | semidirect |
| 52. C23×C32 | C3×D4 | semidirect |
| 57. C2×C32×C4 | C3×D4 | semidirect |
| 58. #392 | H(3,𝔽3) | direct (N × H) |
| 85. C22×D4 | C2×H(3,𝔽3) | semidirect |
| 86. C2×C22⋊C4 | C2×H(3,𝔽3) | semidirect |
| 87. #95 | C2×H(3,𝔽3) | semidirect |
| 88. H(3,𝔽3) | #392 | direct (N × H) |
| 139. C32 | C3×#392 | semidirect |
| 144. C23 | D4×H(3,𝔽3) | semidirect |
| 149. C2×C4 | D4×H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×H(3,𝔽3) | 2 |
| C2×C22⋊C4×H(3,𝔽3) | 2 |
| H(3,𝔽3)×#95 | 2 |
| C32×#392 | 3 |