Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
150 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 27 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 | #11720652 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11720650 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #131⋊C27 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10588822 | — | 3 | 1 | no | maximal |
| 5 | #41⋊C27 | C22 | 1 | 1 | yes | characteristic |
| 6 | #88391 | — | 2 | 1 | no | — |
| 7 | #88390 | — | 2 | 1 | no | — |
| 8 | #88360 | — | 4 | 1 | no | maximal |
| 9 | C9×#131 | D3 | 1 | 1 | yes | characteristic |
| 10 | #63512 | — | 3 | 1 | no | — |
| 11 | #63511 | — | 3 | 1 | no | — |
| 12 | C2×#3664 | — | 3 | 1 | no | — |
| 13 | #63383 | — | 3 | 1 | no | — |
| 14 | #63382 | — | 3 | 1 | no | — |
| 15 | #63372 | — | 3 | 1 | no | — |
| 16 | D108 | — | 4 | 1 | no | — |
| 17 | Dic54 | — | 4 | 1 | no | — |
| 18 | Q8⋊C27 | D4 | 1 | 1 | yes | characteristic |
| 19 | C8×C27 | — | 4 | 1 | no | — |
| 20 | #5141 | — | 9 | 1 | no | — |
| 21 | C9×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 22 | C2×D36 | — | 3 | 1 | no | — |
| 23 | C2×Dic18 | — | 3 | 1 | no | — |
| 24 | C9×M16 | — | 3 | 1 | no | — |
| 25 | C2×C8×C9 | — | 3 | 1 | no | — |
| 26 | #3675 | — | 6 | 1 | no | — |
| 27 | #3674 | — | 6 | 1 | no | — |
| 28 | #3673 | — | 6 | 1 | no | — |
| 29 | #3672 | — | 6 | 1 | no | — |
| 30 | #3667 | — | 3 | 1 | no | — |
| 31 | #3667 | — | 3 | 1 | no | — |
| 32 | #3664 | — | 6 | 1 | no | — |
| 33 | #3664 | — | 6 | 1 | no | — |
| 34 | C2×D27 | — | 8 | 1 | no | — |
| 35 | C4×C27 | — | 4 | 1 | no | — |
| 36 | Dic27 | — | 8 | 1 | no | — |
| 37 | C3×#131 | D9 | 1 | 1 | yes | characteristic |
| 38 | #951 | — | 9 | 1 | no | — |
| 39 | #949 | — | 9 | 1 | no | — |
| 40 | C2×#204 | — | 9 | 1 | no | — |
| 41 | #824 | — | 9 | 1 | no | — |
| 42 | #823 | — | 9 | 1 | no | — |
| 43 | #819 | — | 9 | 1 | no | — |
| 44 | C22×D9 | — | 6 | 1 | no | — |
| 45 | Q8×C9 | D12 | 1 | 1 | yes | characteristic |
| 46 | D4×C9 | — | 3 | 1 | no | — |
| 47 | C2×C4×C9 | — | 3 | 1 | no | — |
| 48 | C2×Dic9 | — | 6 | 1 | no | — |
| 49 | D36 | — | 6 | 1 | no | — |
| 50 | D36 | — | 6 | 1 | no | — |
| 51 | Dic18 | — | 6 | 1 | no | — |
| 52 | Dic18 | — | 6 | 1 | no | — |
| 53 | C8×C9 | — | 3 | 1 | no | — |
| 54 | C8×C9 | S4 | 1 | 1 | yes | characteristic |
| 55 | C9⋊C8 | — | 6 | 1 | no | — |
| 56 | C9⋊C8 | — | 6 | 1 | no | — |
| 57 | #471 | — | 27 | 1 | no | Sylow |
| 58 | C2×C27 | — | 4 | 1 | no | — |
| 59 | D27 | — | 16 | 1 | no | — |
| 60 | C3×#41 | C2×D9 | 1 | 1 | yes | characteristic |
| 61 | C2×D12 | — | 9 | 1 | no | — |
| 62 | C2×Dic6 | — | 9 | 1 | no | — |
| 63 | C3×M16 | — | 3 | 1 | no | — |
| 64 | C2×C3×C8 | — | 3 | 1 | no | — |
| 65 | #216 | — | 18 | 1 | no | — |
| 66 | #215 | — | 18 | 1 | no | — |
| 67 | #214 | — | 18 | 1 | no | — |
| 68 | #213 | — | 18 | 1 | no | — |
| 69 | #208 | — | 9 | 1 | no | — |
| 70 | #208 | — | 9 | 1 | no | — |
| 71 | #204 | — | 18 | 1 | no | — |
| 72 | #204 | — | 18 | 1 | no | — |
| 73 | C22×C9 | — | 3 | 1 | no | — |
| 74 | C2×D9 | — | 12 | 1 | no | — |
| 75 | C2×D9 | — | 24 | 1 | no | — |
| 76 | C4×C9 | — | 3 | 1 | no | — |
| 77 | C4×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 78 | Dic9 | — | 12 | 1 | no | — |
| 79 | #137 | — | 27 | 1 | no | — |
| 80 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 81 | C2×SD16 | — | 27 | 1 | no | — |
| 82 | #131 | D27 | 1 | 1 | yes | characteristic, complemented |
| 83 | #108 | — | 27 | 1 | no | — |
| 84 | #101 | — | 27 | 1 | no | — |
| 85 | #100 | — | 27 | 1 | no | — |
| 86 | C27 | — | 4 | 1 | no | Sylow |
| 87 | C22×D3 | — | 18 | 1 | no | — |
| 88 | C3×Q8 | D36 | 1 | 1 | yes | characteristic |
| 89 | C3×D4 | — | 3 | 1 | no | — |
| 90 | C2×C3×C4 | — | 3 | 1 | no | — |
| 91 | C2×Dic3 | — | 18 | 1 | no | — |
| 92 | D12 | — | 18 | 1 | no | — |
| 93 | D12 | — | 18 | 1 | no | — |
| 94 | Dic6 | — | 18 | 1 | no | — |
| 95 | Dic6 | — | 18 | 1 | no | — |
| 96 | C3×C8 | — | 3 | 1 | no | — |
| 97 | C3×C8 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 98 | C3⋊C8 | — | 18 | 1 | no | — |
| 99 | C3⋊C8 | — | 18 | 1 | no | — |
| 100 | C2×C9 | — | 6 | 1 | no | — |
| 101 | C2×C9 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 102 | D9 | — | 24 | 1 | no | — |
| 103 | #41 | C2×D27 | 1 | 1 | yes | characteristic |
| 104 | C2×Q8 | — | 27 | 1 | no | — |
| 105 | C2×D4 | — | 27 | 1 | no | — |
| 106 | Q16 | — | 54 | 1 | no | — |
| 107 | SD16 | — | 54 | 1 | no | — |
| 108 | SD16 | — | 54 | 1 | no | — |
| 109 | SD16 | — | 54 | 1 | no | — |
| 110 | SD16 | — | 54 | 1 | no | — |
| 111 | D8 | — | 54 | 1 | no | — |
| 112 | M16 | — | 3 | 1 | no | — |
| 113 | M16 | — | 27 | 1 | no | — |
| 114 | M16 | — | 27 | 1 | no | — |
| 115 | C2×C8 | — | 3 | 1 | no | — |
| 116 | C22×C3 | — | 3 | 1 | no | — |
| 117 | C2×D3 | — | 36 | 1 | no | — |
| 118 | C2×D3 | — | 72 | 1 | no | — |
| 119 | C3×C4 | — | 3 | 1 | no | — |
| 120 | C3×C4 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 121 | Dic3 | — | 36 | 1 | no | — |
| 122 | C9 | #5406 | 1 | 1 | yes | characteristic |
| 123 | C23 | — | 54 | 1 | no | — |
| 124 | Q8 | — | 54 | 1 | no | — |
| 125 | Q8 | — | 54 | 1 | no | — |
| 126 | Q8 | D108 | 1 | 1 | yes | characteristic |
| 127 | D4 | — | 3 | 1 | no | — |
| 128 | D4 | — | 54 | 1 | no | — |
| 129 | D4 | — | 54 | 1 | no | — |
| 130 | C2×C4 | — | 3 | 1 | no | — |
| 131 | C2×C4 | — | 54 | 1 | no | — |
| 132 | C8 | — | 3 | 1 | no | — |
| 133 | C8 | — | 54 | 1 | no | — |
| 134 | C8 | — | 54 | 1 | no | — |
| 135 | C8 | (C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic |
| 136 | C2×C3 | — | 6 | 1 | no | — |
| 137 | C2×C3 | C4⋊((C22⋊C9)⋊C2) | 1 | 1 | yes | characteristic |
| 138 | D3 | — | 72 | 1 | no | — |
| 139 | C22 | — | 3 | 1 | no | — |
| 140 | C22 | — | 108 | 1 | no | — |
| 141 | C22 | — | 216 | 1 | no | — |
| 142 | C4 | — | 3 | 1 | no | — |
| 143 | C4 | — | 108 | 1 | no | — |
| 144 | C4 | C2×(C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic |
| 145 | C3 | #10589508 | 1 | 1 | yes | characteristic |
| 146 | C2 | — | 6 | 1 | no | — |
| 147 | C2 | — | 216 | 1 | no | — |
| 148 | C2 | C4⋊((C22⋊C27)⋊C2) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11720650 | C2 | semidirect |
| 3. #131⋊C27 | C2 | semidirect |
| 82. #131 | D27 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11720652 | 2 |
| #11720650 | 2 |
| #131⋊C27 | 2 |
| #10588822 | 3 |
| #88360 | 4 |