Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
114 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C4×C9 | C2 | 1 | 3 | yes | maximal |
| 2 | C3×#95 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C23×C9 | C22 | 1 | 1 | yes | characteristic |
| 4 | C2×C4×C9 | — | 2 | 6 | no | — |
| 5 | C2×C4×C9 | C4 | 1 | 6 | yes | complemented |
| 6 | C22×C3×C4 | C2×C3 | 1 | 3 | yes | — |
| 7 | C22×C9 | C2×C4 | 1 | 3 | yes | — |
| 8 | C22×C9 | D4 | 1 | 3 | yes | — |
| 9 | C22×C9 | Q8 | 1 | 1 | yes | characteristic |
| 10 | C4×C9 | — | 2 | 6 | no | — |
| 11 | #95 | C9 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 12 | C23×C3 | C22×C3 | 1 | 1 | yes | characteristic |
| 13 | C2×C3×C4 | — | 2 | 6 | no | — |
| 14 | C2×C3×C4 | C3×C4 | 1 | 6 | yes | — |
| 15 | C2×C9 | C42 | 1 | 1 | yes | characteristic |
| 16 | C2×C9 | C22⋊C4 | 1 | 3 | yes | — |
| 17 | C2×C9 | C4⋊C4 | 1 | 3 | yes | — |
| 18 | C22×C4 | C2×C9 | 1 | 3 | yes | — |
| 19 | C22×C3 | C2×C3×C4 | 1 | 3 | yes | — |
| 20 | C22×C3 | C3×D4 | 1 | 3 | yes | — |
| 21 | C22×C3 | C3×Q8 | 1 | 1 | yes | characteristic |
| 22 | C3×C4 | — | 2 | 6 | no | — |
| 23 | C9 | #95 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 24 | C23 | C22×C9 | 1 | 1 | yes | characteristic |
| 25 | C2×C4 | — | 2 | 6 | no | — |
| 26 | C2×C4 | C4×C9 | 1 | 6 | yes | complemented |
| 27 | C2×C3 | C3×C42 | 1 | 1 | yes | characteristic |
| 28 | C2×C3 | C3×C22⋊C4 | 1 | 3 | yes | — |
| 29 | C2×C3 | C3×C4⋊C4 | 1 | 3 | yes | — |
| 30 | C22 | C2×C4×C9 | 1 | 3 | yes | — |
| 31 | C22 | D4×C9 | 1 | 3 | yes | — |
| 32 | C22 | Q8×C9 | 1 | 1 | yes | characteristic |
| 33 | C4 | — | 2 | 6 | no | — |
| 34 | C3 | C3×#95 | 1 | 1 | yes | characteristic |
| 35 | C2 | C42×C9 | 1 | 1 | yes | characteristic |
| 36 | C2 | C9×C22⋊C4 | 1 | 3 | yes | — |
| 37 | C2 | C9×C4⋊C4 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C22 |
| commutator | C2 | C42×C9 |
| Frattini, socle, prime radical, cosocle | C23×C3 | C22×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C2×C4×C9 | C4 | semidirect |
| 11. #95 | C9 | direct (N × H) |
| 23. C9 | #95 | direct (N × H) |
| 26. C2×C4 | C4×C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×C9 | 2 |
| C3×#95 | 3 |