Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
139433 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C22⋊(C22⋊A4) | C2 | 1 | 3 | yes | maximal, direct product factor |
| 2 | C28 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C22⋊(C22⋊A4) | C22 | 1 | 1 | yes | characteristic, direct product factor |
| 4 | C22×C22⋊A4 | — | 4 | 21 | no | maximal |
| 5 | C27 | — | 3 | 63 | no | — |
| 6 | C27 | — | 3 | 21 | no | — |
| 7 | C27 | C2×C3 | 1 | 3 | yes | complemented |
| 8 | C2×C22⋊A4 | — | 4 | 63 | no | — |
| 9 | C26 | — | 3 | 63 | no | — |
| 10 | C26 | — | 3 | 1890 | no | — |
| 11 | C26 | — | 3 | 1260 | no | — |
| 12 | C26 | — | 3 | 42 | no | — |
| 13 | C26 | — | 3 | 210 | no | — |
| 14 | C26 | — | 3 | 63 | no | — |
| 15 | C26 | — | 3 | 63 | no | — |
| 16 | C26 | A4 | 1 | 21 | yes | complemented |
| 17 | C26 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 18 | C22⋊A4 | — | 4 | 21 | no | — |
| 19 | C22×A4 | — | 16 | 21 | no | — |
| 20 | C25 | — | 3 | 315 | no | — |
| 21 | C25 | — | 3 | 1890 | no | — |
| 22 | C25 | — | 3 | 1890 | no | — |
| 23 | C25 | — | 3 | 7560 | no | — |
| 24 | C25 | — | 3 | 15120 | no | — |
| 25 | C25 | — | 3 | 2520 | no | — |
| 26 | C25 | — | 3 | 360 | no | — |
| 27 | C25 | — | 3 | 1890 | no | — |
| 28 | C25 | — | 3 | 630 | no | — |
| 29 | C25 | — | 3 | 63 | no | — |
| 30 | C25 | — | 3 | 21 | no | — |
| 31 | C25 | — | 3 | 105 | no | — |
| 32 | C25 | C2×A4 | 1 | 63 | yes | complemented |
| 33 | C2×A4 | — | 16 | 63 | no | — |
| 34 | C24 | — | 3 | 315 | no | — |
| 35 | C24 | — | 3 | 1890 | no | — |
| 36 | C24 | — | 3 | 1260 | no | — |
| 37 | C24 | — | 3 | 7560 | no | — |
| 38 | C24 | — | 3 | 22680 | no | — |
| 39 | C24 | — | 3 | 20160 | no | — |
| 40 | C24 | — | 3 | 7560 | no | — |
| 41 | C24 | — | 3 | 1080 | no | — |
| 42 | C24 | — | 3 | 210 | no | — |
| 43 | C24 | — | 3 | 1260 | no | — |
| 44 | C24 | — | 3 | 1890 | no | — |
| 45 | C24 | — | 3 | 210 | no | — |
| 46 | C24 | — | 3 | 210 | no | — |
| 47 | C24 | — | 3 | 315 | no | — |
| 48 | C24 | — | 3 | 315 | no | — |
| 49 | C24 | C22×A4 | 1 | 21 | yes | complemented |
| 50 | C24 | C22⋊A4 | 1 | 21 | yes | complemented |
| 51 | C22×C3 | — | 64 | 1 | no | — |
| 52 | A4 | — | 16 | 21 | no | — |
| 53 | C23 | — | 3 | 63 | no | — |
| 54 | C23 | — | 3 | 1890 | no | — |
| 55 | C23 | — | 3 | 1890 | no | — |
| 56 | C23 | — | 3 | 2520 | no | — |
| 57 | C23 | — | 3 | 15120 | no | — |
| 58 | C23 | — | 3 | 7560 | no | — |
| 59 | C23 | — | 3 | 360 | no | — |
| 60 | C23 | — | 3 | 1890 | no | — |
| 61 | C23 | — | 3 | 630 | no | — |
| 62 | C23 | — | 3 | 315 | no | — |
| 63 | C23 | — | 3 | 105 | no | — |
| 64 | C23 | — | 3 | 21 | no | — |
| 65 | C23 | C2×C22⋊A4 | 1 | 63 | yes | complemented |
| 66 | C2×C3 | — | 64 | 3 | no | — |
| 67 | C22 | — | 3 | 63 | no | — |
| 68 | C22 | — | 3 | 42 | no | — |
| 69 | C22 | — | 3 | 1260 | no | — |
| 70 | C22 | — | 3 | 1890 | no | — |
| 71 | C22 | — | 3 | 210 | no | — |
| 72 | C22 | — | 3 | 63 | no | — |
| 73 | C22 | — | 3 | 63 | no | — |
| 74 | C22 | C22×C22⋊A4 | 1 | 21 | yes | complemented |
| 75 | C22 | C22⋊(C22⋊A4) | 1 | 1 | yes | characteristic, direct product factor |
| 76 | C3 | — | 64 | 1 | no | Sylow |
| 77 | C2 | — | 3 | 63 | no | — |
| 78 | C2 | — | 3 | 21 | no | — |
| 79 | C2 | C2×C22⋊(C22⋊A4) | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C22⋊(C22⋊A4) |
| commutator, cosocle, hypocenter | C26 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C28 | C3 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C22⋊(C22⋊A4) | C2 | direct (N × H) |
| 2. C28 | C3 | semidirect |
| 3. C22⋊(C22⋊A4) | C22 | direct (N × H) |
| 7. C27 | C2×C3 | semidirect |
| 16. C26 | A4 | semidirect |
| 17. C26 | C22×C3 | semidirect |
| 32. C25 | C2×A4 | semidirect |
| 49. C24 | C22×A4 | semidirect |
| 50. C24 | C22⋊A4 | semidirect |
| 65. C23 | C2×C22⋊A4 | semidirect |
| 74. C22 | C22×C22⋊A4 | semidirect |
| 75. C22 | C22⋊(C22⋊A4) | direct (N × H) |
| 79. C2 | C2×C22⋊(C22⋊A4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊(C22⋊A4) | 2 |
| C28 | 3 |
| C22×C22⋊A4 | 4 |