Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
702 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 702 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | A4×D9 |
| commutator, cosocle, hypocenter | Q8×C9 | C23×C3 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting | C22×Q8×C9 | C2×C3 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C9×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×#88615 | C2 | direct (N × H) |
| 4. C22×#3701 | C3 | semidirect |
| 6. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 7. #88615 | C22 | direct (N × H) |
| 12. C2×#3701 | C2×C3 | semidirect |
| 18. C9×SL(2,𝔽3) | C23 | semidirect |
| 31. #3701 | C22×C3 | semidirect |
| 47. C22×SL(2,𝔽3) | D9 | semidirect |
| 84. C2×SL(2,𝔽3) | C2×D9 | semidirect |
| 90. C22×C9 | #41⋊C3 | semidirect |
| 120. SL(2,𝔽3) | C22×D9 | semidirect |
| 125. C2×C9 | C2×#41⋊C3 | semidirect |
| 154. C9 | C22×#41⋊C3 | semidirect |
| 177. C22 | #88615 | direct (N × H) |
| 186. C2 | C2×#88615 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C9×SL(2,𝔽3) | 2 |
| C2×#88615 | 2 |
| C22×C3⋊(#41⋊C3) | 3 |
| C22×#3701 | 3 |
| C22×C3×Dic9 | 4 |