Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
268 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 37 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 | #11723647 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11723094 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#63542 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10590016 | — | 3 | 1 | no | maximal |
| 5 | C4⋊#1113 | — | 2 | 1 | no | — |
| 6 | C4⋊(C3⋊#157) | — | 2 | 1 | no | — |
| 7 | C3×#3745 | — | 2 | 1 | no | — |
| 8 | C3×C8×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 9 | #88810 | C4 | 1 | 2 | yes | complemented |
| 10 | C32×C3⋊C16 | — | 2 | 1 | no | — |
| 11 | #63797 | — | 3 | 1 | no | — |
| 12 | D3×D24 | — | 6 | 1 | no | — |
| 13 | #63769 | — | 3 | 1 | no | — |
| 14 | C3×D3×C16 | — | 6 | 1 | no | — |
| 15 | #63542 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×#1113 | — | 4 | 1 | no | — |
| 17 | C2×C3⋊#157 | — | 4 | 1 | no | — |
| 18 | C3×C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 19 | C3×C4×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 20 | #6295 | — | 4 | 1 | no | — |
| 21 | C33×C8 | D4 | 1 | 1 | yes | characteristic |
| 22 | C3×C3⋊(C3⋊C8) | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | #4509 | — | 9 | 1 | no | maximal |
| 24 | #3802 | — | 6 | 1 | no | — |
| 25 | D3×D12 | — | 6 | 2 | no | — |
| 26 | C4⋊#157 | — | 6 | 1 | no | — |
| 27 | #3745 | — | 2 | 1 | no | — |
| 28 | C8×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | C3×D24 | — | 6 | 1 | no | — |
| 30 | C3×D24 | — | 6 | 1 | no | — |
| 31 | C3×D24 | — | 12 | 1 | no | — |
| 32 | C3×D3×C8 | — | 6 | 1 | no | — |
| 33 | C3×D3×C8 | — | 6 | 1 | no | — |
| 34 | #3714 | — | 6 | 2 | no | — |
| 35 | #3713 | — | 3 | 2 | no | — |
| 36 | C32×C16 | — | 6 | 1 | no | — |
| 37 | C3×C3⋊C16 | — | 2 | 1 | no | — |
| 38 | C3×C3⋊C16 | — | 2 | 1 | no | — |
| 39 | C3×C3⋊C16 | — | 4 | 1 | no | — |
| 40 | C2×C3×C3⋊D3 | — | 4 | 2 | no | — |
| 41 | C2×C3×C3⋊D3 | D8 | 1 | 1 | yes | characteristic |
| 42 | #1113 | — | 8 | 1 | no | — |
| 43 | #1113 | — | 8 | 1 | no | — |
| 44 | C3⋊#157 | — | 8 | 1 | no | — |
| 45 | C33×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 46 | C3×C3⋊Dic3 | SD16 | 1 | 1 | yes | characteristic |
| 47 | D3×D8 | — | 18 | 1 | no | — |
| 48 | C2×D24 | — | 9 | 1 | no | — |
| 49 | C2×C3×C16 | — | 9 | 1 | no | — |
| 50 | #818 | — | 9 | 1 | no | — |
| 51 | D3×C16 | — | 6 | 1 | no | — |
| 52 | C2×D32 | — | 12 | 2 | no | — |
| 53 | C2×D32 | — | 12 | 1 | no | — |
| 54 | C2×#157 | — | 12 | 1 | no | — |
| 55 | C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 56 | C4×C3⋊D3 | D12 | 1 | 1 | yes | characteristic |
| 57 | C3×D12 | — | 6 | 2 | no | — |
| 58 | C3×D12 | — | 6 | 2 | no | — |
| 59 | C3×D12 | — | 12 | 2 | no | — |
| 60 | C3×C4×D3 | — | 6 | 1 | no | — |
| 61 | C3×C4×D3 | — | 6 | 1 | no | — |
| 62 | #626 | — | 12 | 2 | no | — |
| 63 | #625 | — | 12 | 1 | no | — |
| 64 | C32×C8 | — | 2 | 1 | no | — |
| 65 | C32×C8 | — | 2 | 1 | no | — |
| 66 | C32×C8 | — | 4 | 1 | no | — |
| 67 | C32×C8 | — | 4 | 1 | no | — |
| 68 | C32×C8 | #67 | 1 | 1 | yes | characteristic |
| 69 | C3⋊(C3⋊C8) | C4×D3 | 1 | 1 | yes | characteristic |
| 70 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 71 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 72 | #357 | — | 27 | 1 | no | Sylow |
| 73 | C2×C33 | #102 | 1 | 1 | yes | characteristic |
| 74 | C3×C3⋊D3 | — | 8 | 2 | no | — |
| 75 | C3×C3⋊D3 | D16 | 1 | 1 | yes | characteristic |
| 76 | C3×C3⋊D3 | SD32 | 1 | 1 | yes | characteristic |
| 77 | D3×D4 | — | 18 | 2 | no | — |
| 78 | C2×D12 | — | 18 | 1 | no | — |
| 79 | C3×D8 | — | 18 | 1 | no | — |
| 80 | C2×C3×C8 | — | 9 | 1 | no | — |
| 81 | #213 | — | 18 | 2 | no | — |
| 82 | C4⋊Dic3 | — | 18 | 1 | no | — |
| 83 | D24 | — | 6 | 1 | no | — |
| 84 | D24 | — | 6 | 1 | no | — |
| 85 | D24 | — | 9 | 2 | no | — |
| 86 | D24 | — | 12 | 1 | no | — |
| 87 | D3×C8 | — | 6 | 1 | no | — |
| 88 | D3×C8 | — | 6 | 1 | no | — |
| 89 | C3×C16 | — | 6 | 1 | no | — |
| 90 | C3×C16 | — | 6 | 1 | no | — |
| 91 | C3×C16 | — | 12 | 1 | no | — |
| 92 | C3⋊C16 | — | 2 | 1 | no | — |
| 93 | C2×C3⋊D3 | — | 4 | 2 | no | — |
| 94 | C2×C3⋊D3 | D24 | 1 | 1 | yes | characteristic |
| 95 | C2×C3×D3 | — | 6 | 1 | no | — |
| 96 | C2×C3×D3 | — | 6 | 1 | no | — |
| 97 | C2×C3×D3 | — | 12 | 2 | no | — |
| 98 | C2×C3×D3 | — | 12 | 2 | no | — |
| 99 | C2×C3×D3 | — | 24 | 2 | no | — |
| 100 | D32 | — | 24 | 2 | no | — |
| 101 | D32 | — | 24 | 2 | no | — |
| 102 | D32 | — | 24 | 1 | no | — |
| 103 | D32 | — | 24 | 1 | no | — |
| 104 | #157 | — | 24 | 1 | no | — |
| 105 | C32×C4 | — | 2 | 1 | no | — |
| 106 | C32×C4 | — | 2 | 1 | no | — |
| 107 | C32×C4 | — | 4 | 1 | no | — |
| 108 | C32×C4 | — | 4 | 1 | no | — |
| 109 | C32×C4 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 110 | C3⋊Dic3 | #204 | 1 | 1 | yes | characteristic |
| 111 | C3×Dic3 | — | 6 | 1 | no | — |
| 112 | C3×Dic3 | — | 6 | 1 | no | — |
| 113 | C2×D8 | — | 27 | 1 | no | — |
| 114 | C2×C16 | — | 9 | 1 | no | — |
| 115 | #107 | — | 27 | 1 | no | — |
| 116 | C33 | #357 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 117 | C22×D3 | — | 36 | 2 | no | — |
| 118 | C22×D3 | — | 36 | 1 | no | — |
| 119 | C3×D4 | — | 18 | 2 | no | — |
| 120 | C2×C3×C4 | — | 9 | 1 | no | — |
| 121 | #67 | — | 36 | 2 | no | — |
| 122 | C2×Dic3 | — | 36 | 1 | no | — |
| 123 | D12 | — | 6 | 2 | no | — |
| 124 | D12 | — | 6 | 2 | no | — |
| 125 | D12 | — | 12 | 2 | no | — |
| 126 | D12 | — | 36 | 1 | no | — |
| 127 | C4×D3 | — | 6 | 1 | no | — |
| 128 | C4×D3 | — | 6 | 1 | no | — |
| 129 | C3×C8 | — | 2 | 1 | no | — |
| 130 | C3×C8 | — | 2 | 1 | no | — |
| 131 | C3×C8 | — | 4 | 1 | no | — |
| 132 | C3×C8 | — | 4 | 1 | no | — |
| 133 | C3×C8 | — | 9 | 1 | no | — |
| 134 | C3×C8 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 135 | C3⋊C8 | — | 6 | 1 | no | — |
| 136 | C3⋊C8 | — | 6 | 1 | no | — |
| 137 | C2×C32 | — | 2 | 1 | no | — |
| 138 | C2×C32 | — | 2 | 1 | no | — |
| 139 | C2×C32 | — | 4 | 1 | no | — |
| 140 | C2×C32 | — | 4 | 1 | no | — |
| 141 | C2×C32 | #821 | 1 | 1 | yes | characteristic |
| 142 | C3⋊D3 | — | 8 | 2 | no | — |
| 143 | C3⋊D3 | D48 | 1 | 1 | yes | characteristic |
| 144 | C3⋊D3 | #800 | 1 | 1 | yes | characteristic |
| 145 | C3×D3 | — | 6 | 1 | no | — |
| 146 | C3×D3 | — | 6 | 1 | no | — |
| 147 | C3×D3 | — | 6 | 1 | no | — |
| 148 | C3×D3 | — | 6 | 1 | no | — |
| 149 | C3×D3 | — | 24 | 2 | no | — |
| 150 | C3×D3 | — | 24 | 2 | no | — |
| 151 | C3×D3 | — | 48 | 2 | no | — |
| 152 | C2×D4 | — | 54 | 1 | no | — |
| 153 | D8 | — | 18 | 1 | no | — |
| 154 | D8 | — | 27 | 2 | no | — |
| 155 | C2×C8 | — | 9 | 1 | no | — |
| 156 | C4⋊C4 | — | 54 | 1 | no | — |
| 157 | C16 | — | 6 | 1 | no | — |
| 158 | C22×C3 | — | 9 | 1 | no | — |
| 159 | C22×C3 | — | 36 | 2 | no | — |
| 160 | C2×D3 | — | 6 | 1 | no | — |
| 161 | C2×D3 | — | 6 | 1 | no | — |
| 162 | C2×D3 | — | 12 | 2 | no | — |
| 163 | C2×D3 | — | 12 | 2 | no | — |
| 164 | C2×D3 | — | 24 | 2 | no | — |
| 165 | C2×D3 | — | 72 | 2 | no | — |
| 166 | C2×D3 | — | 72 | 2 | no | — |
| 167 | C2×D3 | — | 72 | 1 | no | — |
| 168 | C2×D3 | — | 72 | 1 | no | — |
| 169 | C3×C4 | — | 2 | 1 | no | — |
| 170 | C3×C4 | — | 2 | 1 | no | — |
| 171 | C3×C4 | — | 4 | 1 | no | — |
| 172 | C3×C4 | — | 4 | 1 | no | — |
| 173 | C3×C4 | — | 9 | 1 | no | — |
| 174 | C3×C4 | D32⋊C4 | 1 | 1 | yes | characteristic |
| 175 | Dic3 | — | 6 | 1 | no | — |
| 176 | Dic3 | — | 6 | 1 | no | — |
| 177 | Dic3 | — | 72 | 1 | no | — |
| 178 | C32 | — | 2 | 1 | no | — |
| 179 | C32 | — | 2 | 1 | no | — |
| 180 | C32 | — | 4 | 1 | no | — |
| 181 | C32 | — | 4 | 1 | no | — |
| 182 | C32 | #4509 | 1 | 1 | yes | characteristic, complemented |
| 183 | C23 | — | 108 | 1 | no | — |
| 184 | D4 | — | 18 | 2 | no | — |
| 185 | D4 | — | 108 | 1 | no | — |
| 186 | C2×C4 | — | 9 | 1 | no | — |
| 187 | C2×C4 | — | 108 | 1 | no | — |
| 188 | C8 | — | 9 | 1 | no | — |
| 189 | C8 | D32⋊D3 | 1 | 1 | yes | characteristic |
| 190 | C2×C3 | — | 2 | 1 | no | — |
| 191 | C2×C3 | — | 2 | 1 | no | — |
| 192 | C2×C3 | — | 4 | 1 | no | — |
| 193 | C2×C3 | — | 4 | 1 | no | — |
| 194 | C2×C3 | — | 9 | 1 | no | — |
| 195 | C2×C3 | — | 9 | 1 | no | — |
| 196 | C2×C3 | — | 72 | 2 | no | — |
| 197 | C2×C3 | #63729 | 1 | 1 | yes | characteristic |
| 198 | D3 | — | 6 | 1 | no | — |
| 199 | D3 | — | 6 | 1 | no | — |
| 200 | D3 | — | 6 | 1 | no | — |
| 201 | D3 | — | 6 | 1 | no | — |
| 202 | D3 | — | 24 | 2 | no | — |
| 203 | D3 | — | 24 | 2 | no | — |
| 204 | D3 | — | 48 | 2 | no | — |
| 205 | C22 | — | 9 | 1 | no | — |
| 206 | C22 | — | 36 | 2 | no | — |
| 207 | C22 | — | 216 | 1 | no | — |
| 208 | C22 | — | 216 | 1 | no | — |
| 209 | C4 | — | 9 | 1 | no | — |
| 210 | C4 | — | 216 | 1 | no | — |
| 211 | C4 | #88932 | 1 | 1 | yes | characteristic |
| 212 | C3 | — | 2 | 1 | no | — |
| 213 | C3 | — | 2 | 1 | no | — |
| 214 | C3 | — | 4 | 1 | no | — |
| 215 | C3 | — | 4 | 1 | no | — |
| 216 | C3 | #10590016 | 1 | 1 | yes | characteristic, complemented |
| 217 | C2 | — | 9 | 1 | no | — |
| 218 | C2 | — | 9 | 1 | no | — |
| 219 | C2 | — | 72 | 2 | no | — |
| 220 | C2 | #11722757 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11723094 | C2 | semidirect |
| 3. C3×#63542 | C2 | semidirect |
| 9. #88810 | C4 | semidirect |
| 15. #63542 | D3 | semidirect |
| 116. C33 | #357 | semidirect |
| 182. C32 | #4509 | semidirect |
| 216. C3 | #10590016 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11723647 | 2 |
| #11723094 | 2 |
| C3×#63542 | 2 |
| #10590016 | 3 |
| #4509 | 9 |