Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
840 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 135 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 | #11722661 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11721840 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11721783 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×C9×#100 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11720558 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10592446 | — | 3 | 1 | no | maximal |
| 7 | #10588427 | — | 3 | 3 | no | maximal |
| 8 | C2×C3×D4×C9 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6233 | — | 2 | 2 | no | — |
| 10 | C2×#6233 | — | 2 | 2 | no | — |
| 11 | #88742 | — | 2 | 2 | no | — |
| 12 | D4×C3⋊D9 | — | 2 | 2 | no | — |
| 13 | #88736 | C22 | 1 | 2 | yes | — |
| 14 | C2×C4⋊(C3⋊D9) | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C3×C9×M16 | C22 | 1 | 2 | yes | complemented |
| 16 | C4×C3⋊Dic9 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | #88525 | — | 2 | 2 | no | — |
| 18 | #88524 | — | 2 | 2 | no | — |
| 19 | #64361 | — | 3 | 1 | no | — |
| 20 | #64149 | — | 3 | 1 | no | — |
| 21 | #64117 | — | 3 | 2 | no | — |
| 22 | #63710 | — | 3 | 3 | no | — |
| 23 | C32×#100 | D3 | 1 | 1 | yes | characteristic |
| 24 | #63652 | — | 3 | 2 | no | — |
| 25 | #63502 | — | 3 | 3 | no | — |
| 26 | #63470 | — | 3 | 6 | no | — |
| 27 | C9×#100 | D3 | 1 | 3 | yes | complemented |
| 28 | #63384 | — | 3 | 6 | no | — |
| 29 | C23×C3×C9 | D4 | 1 | 2 | yes | — |
| 30 | C22×C3⋊D9 | — | 2 | 2 | no | — |
| 31 | C22×C3⋊D9 | — | 2 | 1 | no | — |
| 32 | C3×D4×C9 | — | 2 | 2 | no | — |
| 33 | C2×C3×C4×C9 | C23 | 1 | 1 | yes | characteristic |
| 34 | #6233 | — | 2 | 4 | no | — |
| 35 | #6233 | — | 2 | 2 | no | — |
| 36 | #6233 | — | 4 | 4 | no | — |
| 37 | C2×C3⋊Dic9 | — | 2 | 2 | no | — |
| 38 | C2×C3⋊Dic9 | — | 2 | 1 | no | — |
| 39 | C4⋊(C3⋊D9) | — | 2 | 2 | no | — |
| 40 | C4⋊(C3⋊D9) | D4 | 1 | 2 | yes | complemented |
| 41 | C4×C3⋊D9 | — | 2 | 2 | no | — |
| 42 | #6226 | D4 | 1 | 2 | yes | complemented |
| 43 | C3×C8×C9 | — | 2 | 2 | no | — |
| 44 | #4747 | — | 9 | 3 | no | — |
| 45 | #4747 | — | 9 | 1 | no | — |
| 46 | C2×C32×D4 | C2×D3 | 1 | 1 | yes | characteristic |
| 47 | C2×#638 | — | 6 | 2 | no | — |
| 48 | C2×#638 | — | 6 | 2 | no | — |
| 49 | #3830 | — | 6 | 2 | no | — |
| 50 | D4×C3⋊D3 | — | 6 | 2 | no | — |
| 51 | #3828 | — | 3 | 2 | no | — |
| 52 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 53 | C32×M16 | C2×D3 | 1 | 2 | yes | — |
| 54 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 55 | #3745 | — | 6 | 2 | no | — |
| 56 | #3744 | — | 6 | 2 | no | — |
| 57 | C2×D4×C9 | C2×D3 | 1 | 3 | yes | — |
| 58 | C2×#611 | — | 6 | 6 | no | — |
| 59 | C2×#611 | — | 6 | 6 | no | — |
| 60 | #3699 | — | 6 | 6 | no | — |
| 61 | D4×D9 | — | 6 | 6 | no | — |
| 62 | #3697 | — | 3 | 6 | no | — |
| 63 | C2×D36 | — | 3 | 3 | no | — |
| 64 | C9×M16 | C2×D3 | 1 | 6 | yes | complemented |
| 65 | C4×Dic9 | — | 3 | 3 | no | — |
| 66 | D72 | — | 6 | 6 | no | — |
| 67 | #3664 | — | 6 | 6 | no | — |
| 68 | C22×C3×C9 | — | 2 | 2 | no | — |
| 69 | C22×C3×C9 | — | 4 | 2 | no | — |
| 70 | C22×C3×C9 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C2×C3⋊D9 | — | 2 | 2 | no | — |
| 72 | C2×C3⋊D9 | — | 4 | 2 | no | — |
| 73 | C2×C3⋊D9 | — | 4 | 1 | no | — |
| 74 | C2×C3⋊D9 | — | 8 | 2 | no | — |
| 75 | C3×C4×C9 | C2×D4 | 1 | 2 | yes | — |
| 76 | C3⋊Dic9 | — | 2 | 2 | no | — |
| 77 | C3⋊Dic9 | — | 2 | 2 | no | — |
| 78 | #1004 | — | 9 | 3 | no | — |
| 79 | #1004 | — | 9 | 1 | no | — |
| 80 | #940 | — | 9 | 3 | no | — |
| 81 | #940 | — | 9 | 1 | no | — |
| 82 | #908 | — | 9 | 6 | no | — |
| 83 | #908 | — | 9 | 2 | no | — |
| 84 | C3×#100 | D9 | 1 | 3 | yes | complemented |
| 85 | C3×#100 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 86 | #825 | — | 9 | 6 | no | — |
| 87 | #825 | — | 9 | 2 | no | — |
| 88 | C23×C32 | D12 | 1 | 2 | yes | — |
| 89 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 90 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 91 | C32×D4 | — | 2 | 2 | no | — |
| 92 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 93 | #638 | — | 6 | 4 | no | — |
| 94 | #638 | — | 6 | 2 | no | — |
| 95 | #638 | — | 12 | 4 | no | — |
| 96 | C2×C3⋊Dic3 | — | 6 | 2 | no | — |
| 97 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 98 | C4⋊(C3⋊D3) | — | 3 | 2 | no | — |
| 99 | C4⋊(C3⋊D3) | — | 6 | 2 | no | — |
| 100 | C4×C3⋊D3 | — | 6 | 2 | no | — |
| 101 | #634 | — | 3 | 2 | no | — |
| 102 | C23×C9 | D12 | 1 | 6 | yes | — |
| 103 | C22×D9 | — | 6 | 6 | no | — |
| 104 | C22×D9 | — | 6 | 3 | no | — |
| 105 | C32×C8 | — | 2 | 2 | no | — |
| 106 | D4×C9 | — | 2 | 6 | no | — |
| 107 | C2×C4×C9 | C22×D3 | 1 | 3 | yes | — |
| 108 | #611 | — | 6 | 12 | no | — |
| 109 | #611 | — | 6 | 6 | no | — |
| 110 | #611 | — | 12 | 12 | no | — |
| 111 | C2×Dic9 | — | 6 | 6 | no | — |
| 112 | C2×Dic9 | — | 6 | 3 | no | — |
| 113 | D36 | — | 3 | 6 | no | — |
| 114 | D36 | — | 6 | 6 | no | — |
| 115 | C4×D9 | — | 6 | 6 | no | — |
| 116 | Dic18 | — | 3 | 6 | no | — |
| 117 | C8×C9 | — | 2 | 6 | no | — |
| 118 | #453 | — | 27 | 1 | no | Sylow |
| 119 | C2×C3×C9 | — | 2 | 1 | no | — |
| 120 | C2×C3×C9 | — | 4 | 2 | no | — |
| 121 | C2×C3×C9 | #120 | 1 | 1 | yes | characteristic |
| 122 | C3⋊D9 | — | 4 | 2 | no | — |
| 123 | C3⋊D9 | — | 8 | 1 | no | — |
| 124 | C2×C3×D4 | C2×D9 | 1 | 3 | yes | — |
| 125 | C2×C3×D4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 126 | C2×#67 | — | 18 | 6 | no | — |
| 127 | C2×#67 | — | 18 | 2 | no | — |
| 128 | C2×#67 | — | 18 | 6 | no | — |
| 129 | C2×#67 | — | 18 | 2 | no | — |
| 130 | #237 | — | 18 | 2 | no | — |
| 131 | #237 | — | 18 | 6 | no | — |
| 132 | D3×D4 | — | 18 | 2 | no | — |
| 133 | D3×D4 | — | 18 | 6 | no | — |
| 134 | #235 | — | 9 | 6 | no | — |
| 135 | #235 | — | 9 | 2 | no | — |
| 136 | C2×D12 | — | 9 | 3 | no | — |
| 137 | C2×D12 | — | 9 | 1 | no | — |
| 138 | C3×M16 | C2×D9 | 1 | 6 | yes | complemented |
| 139 | C3×M16 | C2×C3⋊D3 | 1 | 2 | yes | — |
| 140 | C4×Dic3 | — | 9 | 3 | no | — |
| 141 | C4×Dic3 | — | 9 | 1 | no | — |
| 142 | D24 | — | 18 | 2 | no | — |
| 143 | D24 | — | 18 | 6 | no | — |
| 144 | #204 | — | 18 | 2 | no | — |
| 145 | #204 | — | 18 | 6 | no | — |
| 146 | C22×C32 | — | 2 | 2 | no | — |
| 147 | C22×C32 | — | 4 | 2 | no | — |
| 148 | C22×C32 | C2×D12 | 1 | 1 | yes | characteristic |
| 149 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 150 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 151 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 152 | C2×C3⋊D3 | — | 24 | 2 | no | — |
| 153 | C32×C4 | D3×D4 | 1 | 2 | yes | — |
| 154 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 155 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 156 | C22×C9 | — | 2 | 6 | no | — |
| 157 | C22×C9 | — | 4 | 6 | no | — |
| 158 | C22×C9 | C2×D12 | 1 | 3 | yes | — |
| 159 | C2×D9 | — | 6 | 6 | no | — |
| 160 | C2×D9 | — | 12 | 6 | no | — |
| 161 | C2×D9 | — | 12 | 3 | no | — |
| 162 | C2×D9 | — | 24 | 6 | no | — |
| 163 | C4×C9 | D3×D4 | 1 | 6 | yes | — |
| 164 | Dic9 | — | 6 | 6 | no | — |
| 165 | Dic9 | — | 6 | 6 | no | — |
| 166 | H(5,𝔽2) | — | 27 | 1 | no | — |
| 167 | AGL(1,ℤ/8) | — | 27 | 2 | no | — |
| 168 | #127 | — | 27 | 1 | no | — |
| 169 | #104 | — | 27 | 2 | no | — |
| 170 | #100 | C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 171 | C3×C9 | #453 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 172 | C23×C3 | D36 | 1 | 6 | yes | — |
| 173 | C23×C3 | C4⋊(C3⋊D3) | 1 | 2 | yes | — |
| 174 | C22×D3 | — | 18 | 2 | no | — |
| 175 | C22×D3 | — | 18 | 6 | no | — |
| 176 | C22×D3 | — | 18 | 3 | no | — |
| 177 | C22×D3 | — | 18 | 1 | no | — |
| 178 | C3×D4 | — | 2 | 6 | no | — |
| 179 | C3×D4 | — | 2 | 2 | no | — |
| 180 | C2×C3×C4 | C22×D9 | 1 | 3 | yes | — |
| 181 | C2×C3×C4 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 182 | #67 | — | 18 | 4 | no | — |
| 183 | #67 | — | 18 | 12 | no | — |
| 184 | #67 | — | 18 | 6 | no | — |
| 185 | #67 | — | 18 | 2 | no | — |
| 186 | #67 | — | 36 | 4 | no | — |
| 187 | #67 | — | 36 | 12 | no | — |
| 188 | C2×Dic3 | — | 18 | 2 | no | — |
| 189 | C2×Dic3 | — | 18 | 6 | no | — |
| 190 | C2×Dic3 | — | 18 | 3 | no | — |
| 191 | C2×Dic3 | — | 18 | 1 | no | — |
| 192 | D12 | — | 9 | 6 | no | — |
| 193 | D12 | — | 9 | 2 | no | — |
| 194 | D12 | — | 18 | 6 | no | — |
| 195 | D12 | — | 18 | 2 | no | — |
| 196 | C4×D3 | — | 18 | 6 | no | — |
| 197 | C4×D3 | — | 18 | 2 | no | — |
| 198 | Dic6 | — | 9 | 6 | no | — |
| 199 | Dic6 | — | 9 | 2 | no | — |
| 200 | C3×C8 | — | 2 | 6 | no | — |
| 201 | C3×C8 | — | 2 | 2 | no | — |
| 202 | C2×C32 | — | 2 | 1 | no | — |
| 203 | C2×C32 | — | 4 | 2 | no | — |
| 204 | C2×C32 | #882 | 1 | 1 | yes | characteristic |
| 205 | C3⋊D3 | — | 12 | 2 | no | — |
| 206 | C3⋊D3 | — | 24 | 1 | no | — |
| 207 | C2×C9 | — | 2 | 3 | no | — |
| 208 | C2×C9 | — | 4 | 6 | no | — |
| 209 | C2×C9 | #882 | 1 | 3 | yes | — |
| 210 | D9 | — | 12 | 6 | no | — |
| 211 | D9 | — | 24 | 3 | no | — |
| 212 | #41 | — | 27 | 2 | no | — |
| 213 | #41 | — | 54 | 2 | no | — |
| 214 | C2×D4 | — | 27 | 1 | no | — |
| 215 | C2×D4 | — | 54 | 2 | no | — |
| 216 | C2×D4 | — | 54 | 2 | no | — |
| 217 | C2×D4 | — | 54 | 2 | no | — |
| 218 | C2×D4 | C2×C3⋊D9 | 1 | 1 | yes | characteristic |
| 219 | SD16 | — | 54 | 2 | no | — |
| 220 | D8 | — | 54 | 2 | no | — |
| 221 | M16 | C2×C3⋊D9 | 1 | 2 | yes | complemented |
| 222 | C42 | — | 27 | 1 | no | — |
| 223 | C22×C3 | — | 2 | 2 | no | — |
| 224 | C22×C3 | — | 2 | 6 | no | — |
| 225 | C22×C3 | — | 4 | 6 | no | — |
| 226 | C22×C3 | — | 4 | 2 | no | — |
| 227 | C22×C3 | C2×D36 | 1 | 3 | yes | — |
| 228 | C22×C3 | C2×C4⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 229 | C2×D3 | — | 18 | 2 | no | — |
| 230 | C2×D3 | — | 18 | 6 | no | — |
| 231 | C2×D3 | — | 36 | 2 | no | — |
| 232 | C2×D3 | — | 36 | 6 | no | — |
| 233 | C2×D3 | — | 36 | 1 | no | — |
| 234 | C2×D3 | — | 36 | 3 | no | — |
| 235 | C2×D3 | — | 72 | 2 | no | — |
| 236 | C2×D3 | — | 72 | 6 | no | — |
| 237 | C3×C4 | D4×D9 | 1 | 6 | yes | — |
| 238 | C3×C4 | D4×C3⋊D3 | 1 | 2 | yes | — |
| 239 | Dic3 | — | 18 | 2 | no | — |
| 240 | Dic3 | — | 18 | 6 | no | — |
| 241 | Dic3 | — | 18 | 2 | no | — |
| 242 | Dic3 | — | 18 | 6 | no | — |
| 243 | C32 | #4747 | 1 | 1 | yes | characteristic |
| 244 | C9 | #4747 | 1 | 3 | yes | complemented |
| 245 | C23 | — | 54 | 2 | no | — |
| 246 | C23 | — | 54 | 1 | no | — |
| 247 | C23 | C4⋊(C3⋊D9) | 1 | 2 | yes | — |
| 248 | Q8 | — | 27 | 2 | no | — |
| 249 | D4 | — | 2 | 2 | no | — |
| 250 | D4 | — | 27 | 2 | no | — |
| 251 | D4 | — | 54 | 4 | no | — |
| 252 | D4 | — | 54 | 2 | no | — |
| 253 | D4 | — | 54 | 2 | no | — |
| 254 | D4 | — | 108 | 4 | no | — |
| 255 | C2×C4 | — | 54 | 2 | no | — |
| 256 | C2×C4 | — | 54 | 1 | no | — |
| 257 | C2×C4 | — | 54 | 2 | no | — |
| 258 | C2×C4 | C22×C3⋊D9 | 1 | 1 | yes | characteristic |
| 259 | C8 | — | 2 | 2 | no | — |
| 260 | C2×C3 | — | 2 | 3 | no | — |
| 261 | C2×C3 | — | 2 | 1 | no | — |
| 262 | C2×C3 | — | 4 | 6 | no | — |
| 263 | C2×C3 | — | 4 | 2 | no | — |
| 264 | C2×C3 | #63444 | 1 | 3 | yes | — |
| 265 | C2×C3 | #64091 | 1 | 1 | yes | characteristic |
| 266 | D3 | — | 36 | 6 | no | — |
| 267 | D3 | — | 36 | 2 | no | — |
| 268 | D3 | — | 72 | 3 | no | — |
| 269 | D3 | — | 72 | 1 | no | — |
| 270 | C22 | — | 2 | 2 | no | — |
| 271 | C22 | — | 4 | 2 | no | — |
| 272 | C22 | — | 54 | 2 | no | — |
| 273 | C22 | — | 108 | 2 | no | — |
| 274 | C22 | — | 108 | 1 | no | — |
| 275 | C22 | — | 216 | 2 | no | — |
| 276 | C22 | C2×C4⋊(C3⋊D9) | 1 | 1 | yes | characteristic |
| 277 | C4 | — | 54 | 2 | no | — |
| 278 | C4 | — | 54 | 2 | no | — |
| 279 | C4 | D4×C3⋊D9 | 1 | 2 | yes | — |
| 280 | C3 | #10588427 | 1 | 3 | yes | complemented |
| 281 | C3 | #10592446 | 1 | 1 | yes | characteristic |
| 282 | C2 | — | 2 | 1 | no | — |
| 283 | C2 | — | 4 | 2 | no | — |
| 284 | C2 | — | 108 | 2 | no | — |
| 285 | C2 | — | 216 | 1 | no | — |
| 286 | C2 | #11721728 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722661 | C2 | semidirect |
| 2. #11721840 | C2 | semidirect |
| 3. #11721783 | C2 | semidirect |
| 4. C3×C9×#100 | C2 | semidirect |
| 5. #11720558 | C2 | semidirect |
| 14. C2×C4⋊(C3⋊D9) | C22 | semidirect |
| 15. C3×C9×M16 | C22 | semidirect |
| 16. C4×C3⋊Dic9 | C22 | semidirect |
| 27. C9×#100 | D3 | semidirect |
| 40. C4⋊(C3⋊D9) | D4 | semidirect |
| 42. #6226 | D4 | semidirect |
| 64. C9×M16 | C2×D3 | semidirect |
| 84. C3×#100 | D9 | semidirect |
| 138. C3×M16 | C2×D9 | semidirect |
| 170. #100 | C3⋊D9 | semidirect |
| 171. C3×C9 | #453 | semidirect |
| 221. M16 | C2×C3⋊D9 | semidirect |
| 244. C9 | #4747 | semidirect |
| 280. C3 | #10588427 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722661 | 2 |
| #11721840 | 2 |
| #11721783 | 2 |
| C3×C9×#100 | 2 |
| #11720558 | 2 |
| #10592446 | 3 |
| #10588427 | 3 |