Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
440 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 119 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 | C32×C22⋊C4 |
| commutator | C2×C3×C4 | C2×C32×C4 |
| Frattini, cosocle | C22×C3×C4 | C22×C32 |
| prime radical | C23×H(3,𝔽3) | D4 |
| LCS | G ▷ C2×C3×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C4×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C32×#328 | C3 | semidirect |
| 5. C2×Q8×H(3,𝔽3) | C4 | semidirect |
| 10. C4⋊C4×H(3,𝔽3) | C4 | semidirect |
| 28. C2×C32×Q8 | C3×C4 | semidirect |
| 33. C32×C4⋊C4 | C3×C4 | semidirect |
| 62. #328 | H(3,𝔽3) | direct (N × H) |
| 95. H(3,𝔽3) | #328 | direct (N × H) |
| 123. C2×Q8 | C4×H(3,𝔽3) | semidirect |
| 128. C4⋊C4 | C4×H(3,𝔽3) | semidirect |
| 150. C32 | C3×#328 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×C22⋊Q8 | 2 |
| H(3,𝔽3)×C22⋊C8 | 2 |
| H(3,𝔽3)×#95 | 2 |
| C32×#328 | 3 |