Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
186 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×#41 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #3748 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×C3⋊(C3⋊C8) | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #948 | — | 3 | 4 | no | maximal |
| 5 | C32×Q8 | C4 | 1 | 1 | yes | characteristic |
| 6 | C32×D4 | C4 | 1 | 3 | yes | — |
| 7 | C2×C32×C4 | C22 | 1 | 3 | yes | — |
| 8 | C3⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 9 | C3⋊(C3⋊C8) | C22 | 1 | 3 | yes | — |
| 10 | C3×#41 | D3 | 1 | 4 | yes | — |
| 11 | #208 | — | 3 | 12 | no | — |
| 12 | C2×C3⋊C8 | — | 3 | 12 | no | — |
| 13 | C22×C32 | C2×C4 | 1 | 3 | yes | — |
| 14 | C32×C4 | C2×C4 | 1 | 3 | yes | — |
| 15 | C32×C4 | C23 | 1 | 1 | yes | characteristic |
| 16 | #131 | — | 9 | 1 | no | Sylow |
| 17 | C3×Q8 | Dic3 | 1 | 4 | yes | — |
| 18 | C3×D4 | Dic3 | 1 | 12 | yes | — |
| 19 | C2×C3×C4 | C2×D3 | 1 | 12 | yes | — |
| 20 | C3⋊C8 | — | 3 | 4 | no | — |
| 21 | C3⋊C8 | — | 3 | 12 | no | — |
| 22 | C2×C32 | — | 2 | 3 | no | — |
| 23 | C2×C32 | C22×C4 | 1 | 1 | yes | characteristic |
| 24 | #41 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 25 | M16 | — | 9 | 3 | no | — |
| 26 | C2×C8 | — | 9 | 3 | no | — |
| 27 | C22×C3 | C2×Dic3 | 1 | 12 | yes | — |
| 28 | C3×C4 | C2×Dic3 | 1 | 12 | yes | — |
| 29 | C3×C4 | C22×D3 | 1 | 4 | yes | — |
| 30 | C32 | #131 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 31 | Q8 | C3⋊Dic3 | 1 | 1 | yes | characteristic |
| 32 | D4 | C3⋊Dic3 | 1 | 3 | yes | — |
| 33 | C2×C4 | C2×C3⋊D3 | 1 | 3 | yes | — |
| 34 | C8 | — | 9 | 1 | no | — |
| 35 | C8 | — | 9 | 3 | no | — |
| 36 | C2×C3 | — | 2 | 12 | no | — |
| 37 | C2×C3 | C22×Dic3 | 1 | 4 | yes | — |
| 38 | C22 | C2×C3⋊Dic3 | 1 | 3 | yes | — |
| 39 | C4 | C2×C3⋊Dic3 | 1 | 3 | yes | — |
| 40 | C4 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 41 | C3 | #948 | 1 | 4 | yes | complemented |
| 42 | C2 | — | 2 | 3 | no | — |
| 43 | C2 | C22×C3⋊Dic3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×C3⋊D3 |
| commutator, socle | C2×C32 | C22×C4 |
| Fitting, prime radical | C32×#41 | C2 |
| cosocle | C32×C4 | C23 |
| hypercenter | #41 | C3⋊D3 |
| hypocenter | C32 | #131 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #3748 | C2 | semidirect |
| 3. C2×C3⋊(C3⋊C8) | C2 | semidirect |
| 30. C32 | #131 | semidirect |
| 41. C3 | #948 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#41 | 2 |
| #3748 | 2 |
| C2×C3⋊(C3⋊C8) | 2 |
| #948 | 3 |