Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
715 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 343 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C32×D4 |
| commutator | C3×C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C23×H(3,𝔽3) | D4 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C4×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. H(3,𝔽3)×#106 | C2 | direct (N × H) |
| 4. C2×C32×#106 | C3 | semidirect |
| 7. C2×C8×H(3,𝔽3) | C4 | semidirect |
| 13. C32×#106 | C2×C3 | semidirect |
| 21. C8×H(3,𝔽3) | C2×C4 | semidirect |
| 26. C2×C32×C8 | C3×C4 | semidirect |
| 50. C32×C8 | C2×C3×C4 | semidirect |
| 51. C2×#106 | H(3,𝔽3) | direct (N × H) |
| 52. C2×H(3,𝔽3) | #106 | direct (N × H) |
| 76. #106 | C2×H(3,𝔽3) | direct (N × H) |
| 77. H(3,𝔽3) | C2×#106 | direct (N × H) |
| 94. C2×C32 | C3×#106 | semidirect |
| 99. C2×C8 | C4×H(3,𝔽3) | semidirect |
| 117. C32 | C2×C3×#106 | semidirect |
| 125. C8 | C2×C4×H(3,𝔽3) | semidirect |
| 141. C2 | H(3,𝔽3)×#106 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C8×H(3,𝔽3) | 2 |
| C2×C4⋊C4×H(3,𝔽3) | 2 |
| H(3,𝔽3)×#106 | 2 |
| C2×C32×#106 | 3 |