Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1013 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 1013 conjugacy classes of subgroups — too many to draw. Only its 40 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 | C24×C3×A4 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22⋊(C22⋊(C22⋊C9)) | C3 | 1 | 2 | yes | maximal, complemented |
| 3 | C22×C22⋊(C22⋊C9) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #88907 | — | 4 | 1 | no | maximal |
| 5 | #88880 | — | 4 | 5 | no | maximal |
| 6 | C23×C3×A4 | — | 3 | 5 | no | — |
| 7 | C2×C22⋊(C22⋊C9) | — | 3 | 1 | no | — |
| 8 | C26×C3 | C32 | 1 | 1 | yes | characteristic |
| 9 | C24×A4 | — | 3 | 1 | no | — |
| 10 | C24×C32 | — | 4 | 1 | no | — |
| 11 | C22×C3×A4 | — | 3 | 10 | no | — |
| 12 | C22×C3×A4 | A4 | 1 | 5 | yes | — |
| 13 | C22⋊(C22⋊C9) | — | 4 | 2 | no | — |
| 14 | C22⋊(C22⋊C9) | — | 4 | 10 | no | — |
| 15 | C22⋊(C22⋊C9) | — | 12 | 10 | no | — |
| 16 | C22⋊(C22⋊C9) | A4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C22×C22⋊C9 | — | 4 | 5 | no | — |
| 18 | #1094 | — | 16 | 5 | no | — |
| 19 | C9⋊A4 | — | 16 | 1 | no | — |
| 20 | C25×C3 | — | 3 | 5 | no | — |
| 21 | C25×C3 | — | 3 | 1 | no | — |
| 22 | C25×C3 | — | 9 | 5 | no | — |
| 23 | C23×A4 | — | 3 | 15 | no | — |
| 24 | C23×C32 | — | 12 | 5 | no | — |
| 25 | C2×C3×A4 | — | 3 | 5 | no | — |
| 26 | C2×C22⋊C9 | — | 12 | 5 | no | — |
| 27 | C26 | 3−1+2 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 28 | C24×C3 | — | 3 | 10 | no | — |
| 29 | C24×C3 | — | 3 | 10 | no | — |
| 30 | C24×C3 | — | 9 | 5 | no | — |
| 31 | C24×C3 | — | 9 | 5 | no | — |
| 32 | C24×C3 | — | 9 | 5 | no | — |
| 33 | C24×C3 | — | 9 | 30 | no | — |
| 34 | C24×C3 | — | 9 | 20 | no | — |
| 35 | C24×C3 | — | 12 | 1 | no | — |
| 36 | C24×C3 | C3×A4 | 1 | 1 | yes | characteristic |
| 37 | C24×C3 | C3×A4 | 1 | 5 | yes | — |
| 38 | C22×A4 | — | 3 | 30 | no | — |
| 39 | C22×A4 | — | 3 | 5 | no | — |
| 40 | C22×C32 | — | 4 | 5 | no | — |
| 41 | C22×C32 | — | 12 | 10 | no | — |
| 42 | C3×A4 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 43 | C22×C9 | — | 16 | 1 | no | — |
| 44 | C22⋊C9 | — | 4 | 5 | no | — |
| 45 | C22⋊C9 | — | 16 | 10 | no | — |
| 46 | C22⋊C9 | — | 16 | 2 | no | — |
| 47 | C22⋊C9 | — | 48 | 10 | no | — |
| 48 | C25 | — | 3 | 5 | no | — |
| 49 | C25 | — | 3 | 1 | no | — |
| 50 | C25 | — | 9 | 5 | no | — |
| 51 | 3−1+2 | — | 64 | 1 | no | Sylow |
| 52 | C23×C3 | — | 3 | 5 | no | — |
| 53 | C23×C3 | — | 3 | 5 | no | — |
| 54 | C23×C3 | — | 3 | 5 | no | — |
| 55 | C23×C3 | — | 9 | 5 | no | — |
| 56 | C23×C3 | — | 9 | 10 | no | — |
| 57 | C23×C3 | — | 9 | 40 | no | — |
| 58 | C23×C3 | — | 9 | 30 | no | — |
| 59 | C23×C3 | — | 9 | 30 | no | — |
| 60 | C23×C3 | — | 9 | 30 | no | — |
| 61 | C23×C3 | — | 9 | 5 | no | — |
| 62 | C23×C3 | — | 12 | 15 | no | — |
| 63 | C2×A4 | — | 3 | 15 | no | — |
| 64 | C2×C32 | — | 12 | 5 | no | — |
| 65 | C2×C9 | — | 48 | 1 | no | — |
| 66 | C24 | — | 3 | 10 | no | — |
| 67 | C24 | — | 3 | 10 | no | — |
| 68 | C24 | — | 9 | 5 | no | — |
| 69 | C24 | — | 9 | 5 | no | — |
| 70 | C24 | — | 9 | 5 | no | — |
| 71 | C24 | — | 9 | 30 | no | — |
| 72 | C24 | — | 9 | 20 | no | — |
| 73 | C24 | C9⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 74 | C24 | #1094 | 1 | 5 | yes | complemented |
| 75 | C22×C3 | — | 3 | 10 | no | — |
| 76 | C22×C3 | — | 3 | 10 | no | — |
| 77 | C22×C3 | — | 9 | 5 | no | — |
| 78 | C22×C3 | — | 9 | 5 | no | — |
| 79 | C22×C3 | — | 9 | 5 | no | — |
| 80 | C22×C3 | — | 9 | 30 | no | — |
| 81 | C22×C3 | — | 9 | 20 | no | — |
| 82 | C22×C3 | — | 12 | 30 | no | — |
| 83 | C22×C3 | — | 12 | 5 | no | — |
| 84 | C22×C3 | A42 | 1 | 5 | yes | — |
| 85 | C22×C3 | C3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 86 | A4 | — | 3 | 1 | no | — |
| 87 | C32 | — | 4 | 1 | no | — |
| 88 | C9 | — | 16 | 1 | no | — |
| 89 | C9 | — | 64 | 2 | no | — |
| 90 | C23 | — | 3 | 5 | no | — |
| 91 | C23 | — | 3 | 5 | no | — |
| 92 | C23 | — | 3 | 5 | no | — |
| 93 | C23 | — | 9 | 5 | no | — |
| 94 | C23 | — | 9 | 10 | no | — |
| 95 | C23 | — | 9 | 40 | no | — |
| 96 | C23 | — | 9 | 30 | no | — |
| 97 | C23 | — | 9 | 30 | no | — |
| 98 | C23 | — | 9 | 30 | no | — |
| 99 | C23 | — | 9 | 5 | no | — |
| 100 | C2×C3 | — | 3 | 5 | no | — |
| 101 | C2×C3 | — | 3 | 1 | no | — |
| 102 | C2×C3 | — | 9 | 5 | no | — |
| 103 | C2×C3 | — | 12 | 15 | no | — |
| 104 | C22 | — | 3 | 10 | no | — |
| 105 | C22 | — | 3 | 10 | no | — |
| 106 | C22 | — | 9 | 5 | no | — |
| 107 | C22 | — | 9 | 5 | no | — |
| 108 | C22 | — | 9 | 5 | no | — |
| 109 | C22 | — | 9 | 30 | no | — |
| 110 | C22 | — | 9 | 20 | no | — |
| 111 | C22 | #88880 | 1 | 5 | yes | complemented |
| 112 | C22 | #88907 | 1 | 1 | yes | characteristic, complemented |
| 113 | C3 | — | 12 | 1 | no | — |
| 114 | C3 | A4×C22⋊A4 | 1 | 1 | yes | characteristic |
| 115 | C2 | — | 3 | 5 | no | — |
| 116 | C2 | — | 3 | 1 | no | — |
| 117 | C2 | — | 9 | 5 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | A4×C22⋊A4 |
| commutator, Fitting, socle, cosocle | C26×C3 | C32 |
| prime radical | C24×C3×A4 | C3 |
| hypocenter | C26 | 3−1+2 |
| LCS | G ▷ C26×C3 ▷ C26 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22⋊(C22⋊(C22⋊C9)) | C3 | semidirect |
| 3. C22×C22⋊(C22⋊C9) | C3 | semidirect |
| 16. C22⋊(C22⋊C9) | A4 | semidirect |
| 27. C26 | 3−1+2 | semidirect |
| 73. C24 | C9⋊A4 | semidirect |
| 74. C24 | #1094 | semidirect |
| 111. C22 | #88880 | semidirect |
| 112. C22 | #88907 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×A4 | 3 |
| C22⋊(C22⋊(C22⋊C9)) | 3 |
| C22×C22⋊(C22⋊C9) | 3 |
| #88907 | 4 |
| #88880 | 4 |