Finite Group DatabasebetaFinite Group DatabasebetaOrder 12288.
Only the 41 normal subgroups are shown; non-normal subgroups are omitted.
Only the group's 41 normal subgroups are shown; non-normal subgroups are omitted.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #72340920 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #72241349 | D3 | 1 | 1 | yes | characteristic, complemented |
| 3 | #8934366 | S4 | 1 | 3 | yes | complemented |
| 4 | #8932885 | S4 | 1 | 4 | yes | complemented |
| 5 | C22×#127 | #1020 | 1 | 1 | yes | characteristic |
| 6 | C22×#120 | #1020 | 1 | 6 | yes | — |
| 7 | C26 | #5934 | 1 | 6 | yes | complemented |
| 8 | C22×C42 | #5934 | 1 | 1 | yes | characteristic, complemented |
| 9 | C25 | #87588 | 1 | 1 | yes | characteristic |
| 10 | C24 | #11715850 | 1 | 6 | yes | — |
| 11 | C24 | #11715850 | 1 | 1 | yes | characteristic |
| 12 | C23 | #72209641 | 1 | 4 | yes | — |
| 13 | C23 | #72209641 | 1 | 3 | yes | — |
| 14 | C22 | #72227766 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #72340920 | C2 |
| Frattini | C25 | #87588 |
| Fitting | #72241349 | D3 |
| socle | C22 | #72227766 |
| LCS | G ▷ #72340920 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72340920 | C2 | semidirect |
| 2. #72241349 | D3 | semidirect |
| 3. #8934366 | S4 | semidirect |
| 4. #8932885 | S4 | semidirect |
| 7. C26 | #5934 | semidirect |
| 8. C22×C42 | #5934 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72340920 | 2 |
| #72241355 | 3 |
| #72240900 | 4 |
| #72217073 | 4 |