Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
462 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C32×D8 |
| commutator, Frattini, cosocle | C3×C8 | C23×C32 |
| socle | C2×C3 | C2×C32×D8 |
| LCS | G ▷ C3×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C4×H(3,𝔽3) ◁ C2×C8×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽3)×#135 | C2 | semidirect |
| 2. H(3,𝔽3)×Q32 | C2 | semidirect |
| 3. H(3,𝔽3)×SD32 | C2 | semidirect |
| 4. H(3,𝔽3)×D16 | C2 | semidirect |
| 5. C2×C16×H(3,𝔽3) | C2 | semidirect |
| 6. C32×#508 | C3 | semidirect |
| 14. C32×#135 | C2×C3 | semidirect |
| 15. C32×Q32 | C2×C3 | semidirect |
| 16. C32×SD32 | C2×C3 | semidirect |
| 17. C32×D16 | C2×C3 | semidirect |
| 18. C2×C32×C16 | C2×C3 | semidirect |
| 57. #508 | H(3,𝔽3) | direct (N × H) |
| 80. #135 | C2×H(3,𝔽3) | semidirect |
| 81. Q32 | C2×H(3,𝔽3) | semidirect |
| 82. SD32 | C2×H(3,𝔽3) | semidirect |
| 83. D16 | C2×H(3,𝔽3) | semidirect |
| 84. C2×C16 | C2×H(3,𝔽3) | semidirect |
| 85. H(3,𝔽3) | #508 | direct (N × H) |
| 120. C32 | C3×#508 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×#135 | 2 |
| H(3,𝔽3)×Q32 | 2 |
| H(3,𝔽3)×SD32 | 2 |
| H(3,𝔽3)×D16 | 2 |
| C2×C16×H(3,𝔽3) | 2 |
| C32×#508 | 3 |