Finite Group DatabasebetaFinite Group DatabasebetaOrder 6144.
At least 10725 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has at least 10725 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#72209716 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | #72210409 | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 3 | #72209716 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 4 | C29 | A4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×#62817 | A4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#8101 | A4 | 1 | 3 | yes | complemented |
| 7 | C28 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 8 | #62817 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 9 | #8101 | C2×A4 | 1 | 3 | yes | complemented |
| 10 | C27 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 11 | C26 | #997 | 1 | 3 | yes | — |
| 12 | C26 | #997 | 1 | 3 | yes | — |
| 13 | C26 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 14 | C25 | #5464 | 1 | 1 | yes | characteristic |
| 15 | C25 | #5465 | 1 | 1 | yes | characteristic |
| 16 | C25 | #5465 | 1 | 3 | yes | — |
| 17 | C25 | C2×#997 | 1 | 3 | yes | — |
| 18 | C24 | #2679⋊C3 | 1 | 3 | yes | — |
| 19 | C24 | #2679⋊C3 | 1 | 3 | yes | — |
| 20 | C24 | C2×#5464 | 1 | 1 | yes | characteristic |
| 21 | C24 | C2×#5465 | 1 | 1 | yes | characteristic |
| 22 | C23 | #11709313 | 1 | 1 | yes | characteristic |
| 23 | C23 | #11709313 | 1 | 3 | yes | — |
| 24 | C23 | C2×#2679⋊C3 | 1 | 3 | yes | — |
| 25 | C22 | C2×#11709313 | 1 | 1 | yes | characteristic |
| 26 | C2 | #72210409 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72210409 |
| commutator, cosocle, hypocenter | #72209716 | C2×C3 |
| Frattini | C26 | C2×C22⋊A4 |
| Fitting | C2×#72209716 | C3 |
| socle | C23 | #11709313 |
| LCS | G ▷ #72209716 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#72209716 | C3 | semidirect |
| 2. #72210409 | C2 | direct (N × H) |
| 3. #72209716 | C2×C3 | semidirect |
| 4. C29 | A4 | semidirect |
| 5. C2×#62817 | A4 | semidirect |
| 6. C2×#8101 | A4 | semidirect |
| 7. C28 | C2×A4 | semidirect |
| 8. #62817 | C2×A4 | semidirect |
| 9. #8101 | C2×A4 | semidirect |
| 26. C2 | #72210409 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#8101⋊C3 | 4 |
| C2×#11710731 | 4 |
| C23×C22⋊(C22⋊A4) | 4 |
| C2×#72209716 | 3 |
| #72210409 | 2 |