Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
101 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22⋊(C22⋊C9) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #953 | — | 3 | 1 | no | maximal |
| 3 | (C22⋊C9)⋊C2 | — | 4 | 3 | no | maximal |
| 4 | C24×C3 | D3 | 1 | 1 | yes | characteristic |
| 5 | C2×#67 | — | 3 | 3 | no | — |
| 6 | #217 | — | 3 | 3 | no | — |
| 7 | C22⋊C9 | — | 4 | 3 | no | — |
| 8 | C22⋊C9 | — | 8 | 1 | no | — |
| 9 | #120 | — | 9 | 1 | no | Sylow |
| 10 | C23×C3 | — | 3 | 3 | no | — |
| 11 | C23×C3 | — | 6 | 1 | no | — |
| 12 | C22×D3 | — | 3 | 1 | no | — |
| 13 | #67 | — | 6 | 3 | no | — |
| 14 | #67 | — | 6 | 3 | no | — |
| 15 | C2×Dic3 | — | 3 | 3 | no | — |
| 16 | D9 | — | 16 | 1 | no | — |
| 17 | C24 | D9 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×D4 | — | 9 | 3 | no | — |
| 19 | C22⋊C4 | — | 9 | 3 | no | — |
| 20 | C22×C3 | — | 2 | 1 | no | — |
| 21 | C22×C3 | — | 3 | 1 | no | — |
| 22 | C22×C3 | — | 3 | 3 | no | — |
| 23 | C22×C3 | — | 6 | 3 | no | — |
| 24 | C22×C3 | S4 | 1 | 3 | yes | — |
| 25 | C2×D3 | — | 6 | 3 | no | — |
| 26 | Dic3 | — | 6 | 3 | no | — |
| 27 | C9 | — | 16 | 1 | no | Sylow |
| 28 | C23 | — | 3 | 3 | no | — |
| 29 | C23 | — | 6 | 1 | no | — |
| 30 | C23 | — | 9 | 1 | no | — |
| 31 | D4 | — | 18 | 3 | no | — |
| 32 | D4 | — | 18 | 3 | no | — |
| 33 | C2×C4 | — | 9 | 3 | no | — |
| 34 | C2×C3 | — | 3 | 3 | no | — |
| 35 | C2×C3 | — | 6 | 1 | no | — |
| 36 | D3 | — | 12 | 1 | no | — |
| 37 | C22 | — | 2 | 1 | no | — |
| 38 | C22 | — | 3 | 1 | no | — |
| 39 | C22 | — | 3 | 3 | no | — |
| 40 | C22 | — | 6 | 3 | no | — |
| 41 | C22 | — | 18 | 3 | no | — |
| 42 | C22 | (C22⋊C9)⋊C2 | 1 | 3 | yes | complemented |
| 43 | C4 | — | 18 | 3 | no | — |
| 44 | C3 | #1020 | 1 | 1 | yes | characteristic |
| 45 | C2 | — | 3 | 3 | no | — |
| 46 | C2 | — | 6 | 1 | no | — |
| 47 | C2 | — | 36 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C22⋊C9) | C2 |
| Frattini | C3 | #1020 |
| Fitting, socle | C24×C3 | D3 |
| LCS | G ▷ C22⋊(C22⋊C9) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22⋊(C22⋊C9) | C2 | semidirect |
| 17. C24 | D9 | semidirect |
| 42. C22 | (C22⋊C9)⋊C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22⋊(C22⋊C9) | 2 |
| #953 | 3 |
| (C22⋊C9)⋊C2 | 4 |