Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
385 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 385 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 | C2×D4×H(3,𝔽3) | C22 |
| 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 |
|---|---|---|
| 3. C32×#354 | C3 | semidirect |
| 4. C2×Q8×H(3,𝔽3) | C4 | semidirect |
| 8. C42×H(3,𝔽3) | C4 | semidirect |
| 20. C2×C32×Q8 | C3×C4 | semidirect |
| 24. C32×C42 | C3×C4 | semidirect |
| 43. #354 | H(3,𝔽3) | direct (N × H) |
| 66. H(3,𝔽3) | #354 | direct (N × H) |
| 84. C2×Q8 | C4×H(3,𝔽3) | semidirect |
| 88. C42 | C4×H(3,𝔽3) | semidirect |
| 103. C32 | C3×#354 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×#124 | 2 |
| H(3,𝔽3)×#99 | 2 |
| C32×#354 | 3 |