Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
715 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 203 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 |
| prime radical | C2×D8×H(3,𝔽3) | C2 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C42×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q16×H(3,𝔽3) | C2 | semidirect |
| 2. C2×SD16×H(3,𝔽3) | C2 | semidirect |
| 4. H(3,𝔽3)×#124 | C2 | semidirect |
| 5. C4×C8×H(3,𝔽3) | C2 | semidirect |
| 6. C32×#495 | C3 | semidirect |
| 16. C2×C32×Q16 | C2×C3 | semidirect |
| 17. C2×C32×SD16 | C2×C3 | semidirect |
| 19. C32×#124 | C2×C3 | semidirect |
| 20. C32×C4×C8 | C2×C3 | semidirect |
| 69. #495 | H(3,𝔽3) | direct (N × H) |
| 98. C2×Q16 | C2×H(3,𝔽3) | semidirect |
| 99. C2×SD16 | C2×H(3,𝔽3) | semidirect |
| 101. #124 | C2×H(3,𝔽3) | semidirect |
| 102. C4×C8 | C2×H(3,𝔽3) | semidirect |
| 103. H(3,𝔽3) | #495 | direct (N × H) |
| 150. C32 | C3×#495 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q16×H(3,𝔽3) | 2 |
| C2×SD16×H(3,𝔽3) | 2 |
| C2×D8×H(3,𝔽3) | 2 |
| H(3,𝔽3)×#124 | 2 |
| C4×C8×H(3,𝔽3) | 2 |
| C32×#495 | 3 |