Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
438 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×#41 | C2 | 1 | 10 | yes | maximal, complemented |
| 2 | C2×C32×Q8 | C2 | 1 | 5 | yes | maximal, complemented |
| 3 | C3×2−1+4 | C3 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C32×Q8 | C22 | 1 | 10 | yes | — |
| 5 | C32×D4 | C22 | 1 | 10 | yes | — |
| 6 | C2×C32×C4 | C22 | 1 | 15 | yes | — |
| 7 | C3×#41 | C2×C3 | 1 | 40 | yes | complemented |
| 8 | C2×C3×Q8 | C2×C3 | 1 | 20 | yes | complemented |
| 9 | C22×C32 | C23 | 1 | 5 | yes | — |
| 10 | C32×C4 | C23 | 1 | 10 | yes | — |
| 11 | 2−1+4 | C32 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 12 | C3×Q8 | C22×C3 | 1 | 40 | yes | — |
| 13 | C3×D4 | C22×C3 | 1 | 40 | yes | — |
| 14 | C2×C3×C4 | C22×C3 | 1 | 60 | yes | — |
| 15 | C2×C32 | — | 2 | 5 | no | — |
| 16 | C2×C32 | C24 | 1 | 1 | yes | characteristic |
| 17 | #41 | C2×C32 | 1 | 10 | yes | complemented |
| 18 | C2×Q8 | C2×C32 | 1 | 5 | yes | complemented |
| 19 | C22×C3 | C23×C3 | 1 | 20 | yes | — |
| 20 | C3×C4 | C23×C3 | 1 | 40 | yes | — |
| 21 | C32 | 2−1+4 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 22 | Q8 | C22×C32 | 1 | 10 | yes | — |
| 23 | D4 | C22×C32 | 1 | 10 | yes | — |
| 24 | C2×C4 | C22×C32 | 1 | 15 | yes | — |
| 25 | C2×C3 | — | 2 | 20 | no | — |
| 26 | C2×C3 | C24×C3 | 1 | 4 | yes | — |
| 27 | C22 | C23×C32 | 1 | 5 | yes | — |
| 28 | C4 | C23×C32 | 1 | 10 | yes | — |
| 29 | C3 | C3×2−1+4 | 1 | 4 | yes | direct product factor |
| 30 | C2 | — | 2 | 5 | no | — |
| 31 | C2 | C24×C32 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C24 |
| commutator, Frattini, cosocle | C2 | C24×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#41 | C2 | semidirect |
| 2. C2×C32×Q8 | C2 | semidirect |
| 3. C3×2−1+4 | C3 | direct (N × H) |
| 7. C3×#41 | C2×C3 | semidirect |
| 8. C2×C3×Q8 | C2×C3 | semidirect |
| 11. 2−1+4 | C32 | direct (N × H) |
| 17. #41 | C2×C32 | semidirect |
| 18. C2×Q8 | C2×C32 | semidirect |
| 21. C32 | 2−1+4 | direct (N × H) |
| 29. C3 | C3×2−1+4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#41 | 2 |
| C2×C32×Q8 | 2 |
| C3×2−1+4 | 3 |