Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
Only the 31 normal subgroups are shown; non-normal subgroups are omitted.
Only the group's 31 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 | #72344167 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #72344160 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #72340959 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #72340920 | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | #72241353 | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | #72241349 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | #72236189 | S4 | 1 | 1 | yes | characteristic, complemented |
| 8 | #72210483 | S4 | 1 | 2 | yes | complemented |
| 9 | #8934366 | C2×S4 | 1 | 1 | yes | characteristic |
| 10 | #8934366 | C2×S4 | 1 | 2 | yes | complemented |
| 11 | C2×#2867 | #1020 | 1 | 1 | yes | characteristic |
| 12 | C22×#127 | #5934 | 1 | 1 | yes | characteristic, complemented |
| 13 | C22×#127 | C2×#1020 | 1 | 1 | yes | characteristic |
| 14 | C22×#120 | #1020⋊C2 | 1 | 2 | yes | — |
| 15 | C2×C43 | #5934 | 1 | 1 | yes | characteristic, complemented |
| 16 | C26 | #73026 | 1 | 2 | yes | — |
| 17 | C22×C42 | C2×#5934 | 1 | 1 | yes | characteristic |
| 18 | C25 | #87588⋊C2 | 1 | 1 | yes | characteristic |
| 19 | C24 | #72209653 | 1 | 1 | yes | characteristic |
| 20 | C24 | #72209660 | 1 | 2 | yes | — |
| 21 | C23 | #72209832 | 1 | 2 | yes | — |
| 22 | C23 | #72216913 | 1 | 1 | yes | characteristic |
| 23 | C22 | #72344151 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #72340920 | C22 |
| Frattini | C22×#127 | C2×#1020 |
| Fitting | #72241353 | D3 |
| socle | C22 | #72344151 |
| LCS | G ▷ #72340920 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72344167 | C2 | semidirect |
| 2. #72344160 | C2 | semidirect |
| 3. #72340959 | C2 | semidirect |
| 4. #72340920 | C22 | semidirect |
| 5. #72241353 | D3 | semidirect |
| 6. #72241349 | C2×D3 | semidirect |
| 7. #72236189 | S4 | semidirect |
| 8. #72210483 | S4 | semidirect |
| 10. #8934366 | C2×S4 | semidirect |
| 12. C22×#127 | #5934 | semidirect |
| 15. C2×C43 | #5934 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72344175 | 4 |
| #72344167 | 2 |
| #72344160 | 2 |
| #72340959 | 2 |
| #72241374 | 3 |
| #72240903 | 4 |