Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
418 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22×C3 | C23×C32 |
| prime radical | C2×D4×H(3,𝔽3) | C2 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×H(3,𝔽3) | C2 | semidirect |
| 3. C22⋊C4×H(3,𝔽3) | C2 | semidirect |
| 4. C42×H(3,𝔽3) | C2 | semidirect |
| 5. C32×#124 | C3 | semidirect |
| 12. C2×C32×Q8 | C2×C3 | semidirect |
| 14. C32×C22⋊C4 | C2×C3 | semidirect |
| 15. C32×C42 | C2×C3 | semidirect |
| 47. #124 | H(3,𝔽3) | direct (N × H) |
| 48. H(3,𝔽3) | #124 | direct (N × H) |
| 64. C2×Q8 | C2×H(3,𝔽3) | semidirect |
| 66. C22⋊C4 | C2×H(3,𝔽3) | semidirect |
| 67. C42 | C2×H(3,𝔽3) | semidirect |
| 80. C32 | C3×#124 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×H(3,𝔽3) | 2 |
| C2×D4×H(3,𝔽3) | 2 |
| C22⋊C4×H(3,𝔽3) | 2 |
| C42×H(3,𝔽3) | 2 |
| C32×#124 | 3 |