Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
514 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 89 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 | C9×#1007 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11721857 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11720885 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×#63394 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11720300 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10591646 | — | 3 | 1 | no | maximal |
| 7 | #10588926 | — | 3 | 1 | no | maximal |
| 8 | C2×C3×Q8×C9 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C4⋊(C3⋊D9) | C22 | 1 | 1 | yes | characteristic |
| 10 | C9×#239 | — | 2 | 2 | no | — |
| 11 | D3×Q8×C9 | — | 2 | 2 | no | — |
| 12 | C9×#235 | — | 2 | 2 | no | — |
| 13 | C9×#235 | C22 | 1 | 2 | yes | — |
| 14 | #88536 | — | 2 | 2 | no | — |
| 15 | C4×C3⋊Dic9 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×#3667 | C22 | 1 | 2 | yes | — |
| 17 | #88426 | — | 2 | 2 | no | — |
| 18 | #88422 | — | 2 | 2 | no | — |
| 19 | C3×#1007 | D3 | 1 | 1 | yes | characteristic |
| 20 | #64156 | — | 3 | 1 | no | — |
| 21 | #63825 | — | 3 | 2 | no | — |
| 22 | C9×2−1+4 | — | 3 | 1 | no | — |
| 23 | C3×#836 | — | 3 | 1 | no | — |
| 24 | #63568 | — | 3 | 2 | no | — |
| 25 | #63512 | — | 3 | 2 | no | — |
| 26 | #63509 | — | 3 | 1 | no | — |
| 27 | #63509 | — | 6 | 1 | no | — |
| 28 | #63396 | — | 3 | 2 | no | — |
| 29 | #63394 | D3 | 1 | 1 | yes | characteristic, complemented |
| 30 | C22×C3⋊D9 | — | 2 | 1 | no | — |
| 31 | C3×Q8×C9 | — | 2 | 2 | no | — |
| 32 | C2×C3×C4×C9 | D4 | 1 | 2 | yes | — |
| 33 | C2×C3×C4×C9 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C3⋊Dic9 | — | 2 | 1 | no | — |
| 35 | C4⋊(C3⋊D9) | — | 2 | 2 | no | — |
| 36 | C9×#67 | — | 2 | 2 | no | — |
| 37 | C9×D12 | — | 2 | 2 | no | — |
| 38 | C9×D12 | D4 | 1 | 2 | yes | — |
| 39 | C4×D3×C9 | — | 2 | 4 | no | — |
| 40 | C4×D3×C9 | — | 2 | 2 | no | — |
| 41 | C9×Dic6 | — | 2 | 2 | no | — |
| 42 | C9×Dic6 | D4 | 1 | 2 | yes | — |
| 43 | C3×C9⋊C8 | — | 2 | 2 | no | — |
| 44 | #5245 | — | 9 | 1 | no | — |
| 45 | #4753 | — | 9 | 1 | no | — |
| 46 | C2×C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 47 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 48 | C3×#239 | — | 2 | 2 | no | — |
| 49 | C3×D3×Q8 | — | 2 | 2 | no | — |
| 50 | C3×#235 | — | 2 | 2 | no | — |
| 51 | C3×#235 | C2×D3 | 1 | 2 | yes | — |
| 52 | #3752 | — | 6 | 2 | no | — |
| 53 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 54 | C3×#208 | — | 3 | 2 | no | — |
| 55 | #3717 | — | 6 | 2 | no | — |
| 56 | #3714 | — | 6 | 2 | no | — |
| 57 | C9×#41 | — | 3 | 2 | no | — |
| 58 | C9×#41 | — | 6 | 2 | no | — |
| 59 | C9×#41 | — | 6 | 2 | no | — |
| 60 | C2×Q8×C9 | — | 2 | 1 | no | — |
| 61 | C2×Q8×C9 | — | 6 | 2 | no | — |
| 62 | C2×Q8×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 63 | C2×D36 | — | 3 | 1 | no | — |
| 64 | C2×D36 | — | 6 | 1 | no | — |
| 65 | #3675 | — | 6 | 2 | no | — |
| 66 | #3673 | — | 6 | 2 | no | — |
| 67 | (C2×D9)⋊C4 | — | 6 | 4 | no | — |
| 68 | (C2×D9)⋊C4 | — | 6 | 2 | no | — |
| 69 | C4×Dic9 | — | 3 | 1 | no | — |
| 70 | C4×Dic9 | — | 6 | 1 | no | — |
| 71 | #3667 | C2×D3 | 1 | 2 | yes | — |
| 72 | C22×C3×C9 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C2×C3⋊D9 | — | 4 | 2 | no | — |
| 74 | C2×C3⋊D9 | — | 4 | 1 | no | — |
| 75 | C2×D3×C9 | — | 2 | 2 | no | — |
| 76 | C3×C4×C9 | — | 2 | 2 | no | — |
| 77 | C3×C4×C9 | C2×D4 | 1 | 2 | yes | — |
| 78 | C3⋊Dic9 | — | 4 | 1 | no | — |
| 79 | C9×Dic3 | — | 2 | 2 | no | — |
| 80 | C3×2−1+4 | — | 3 | 1 | no | — |
| 81 | #1007 | D9 | 1 | 1 | yes | characteristic, complemented |
| 82 | #949 | — | 9 | 2 | no | — |
| 83 | #947 | — | 9 | 1 | no | — |
| 84 | #947 | — | 9 | 1 | no | — |
| 85 | #947 | — | 18 | 1 | no | — |
| 86 | #908 | — | 9 | 2 | no | — |
| 87 | C3×#101 | — | 9 | 1 | no | — |
| 88 | #837 | — | 9 | 2 | no | — |
| 89 | #836 | — | 3 | 1 | no | — |
| 90 | #825 | — | 9 | 2 | no | — |
| 91 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 92 | C32×Q8 | — | 2 | 2 | no | — |
| 93 | C2×C32×C4 | #67 | 1 | 2 | yes | — |
| 94 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 95 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 96 | C4⋊(C3⋊D3) | — | 6 | 2 | no | — |
| 97 | C3×#67 | — | 2 | 2 | no | — |
| 98 | C3×D12 | — | 2 | 2 | no | — |
| 99 | C3×D12 | #67 | 1 | 2 | yes | — |
| 100 | C3×C4×D3 | — | 2 | 4 | no | — |
| 101 | C3×C4×D3 | — | 2 | 2 | no | — |
| 102 | C3×Dic6 | — | 2 | 2 | no | — |
| 103 | C3×Dic6 | #67 | 1 | 2 | yes | — |
| 104 | C22×D9 | — | 6 | 2 | no | — |
| 105 | C22×D9 | — | 6 | 1 | no | — |
| 106 | C3×C3⋊C8 | — | 6 | 2 | no | — |
| 107 | Q8×C9 | — | 2 | 2 | no | — |
| 108 | Q8×C9 | — | 3 | 2 | no | — |
| 109 | Q8×C9 | — | 4 | 2 | no | — |
| 110 | Q8×C9 | — | 6 | 2 | no | — |
| 111 | D4×C9 | — | 3 | 2 | no | — |
| 112 | D4×C9 | — | 6 | 2 | no | — |
| 113 | D4×C9 | — | 6 | 2 | no | — |
| 114 | C2×C4×C9 | — | 2 | 2 | no | — |
| 115 | C2×C4×C9 | — | 2 | 1 | no | — |
| 116 | C2×C4×C9 | — | 6 | 4 | no | — |
| 117 | C2×C4×C9 | — | 6 | 2 | no | — |
| 118 | C2×C4×C9 | D12 | 1 | 2 | yes | — |
| 119 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 120 | C2×Dic9 | — | 6 | 2 | no | — |
| 121 | C2×Dic9 | — | 6 | 1 | no | — |
| 122 | D36 | — | 6 | 2 | no | — |
| 123 | D36 | — | 12 | 2 | no | — |
| 124 | C9⋊C8 | — | 2 | 2 | no | — |
| 125 | #454 | — | 27 | 1 | no | Sylow |
| 126 | C2×C3×C9 | — | 2 | 1 | no | — |
| 127 | C2×C3×C9 | #120 | 1 | 1 | yes | characteristic |
| 128 | C3⋊D9 | — | 8 | 1 | no | — |
| 129 | D3×C9 | — | 4 | 2 | no | — |
| 130 | C3×#41 | — | 3 | 2 | no | — |
| 131 | C3×#41 | — | 6 | 2 | no | — |
| 132 | C3×#41 | — | 6 | 2 | no | — |
| 133 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 134 | C2×C3×Q8 | — | 6 | 2 | no | — |
| 135 | C2×C3×Q8 | C2×D9 | 1 | 1 | yes | characteristic |
| 136 | C2×C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 137 | #239 | — | 2 | 2 | no | — |
| 138 | D3×Q8 | — | 2 | 2 | no | — |
| 139 | #235 | — | 2 | 2 | no | — |
| 140 | #235 | C2×D9 | 1 | 2 | yes | — |
| 141 | C2×D12 | — | 9 | 1 | no | — |
| 142 | C2×D12 | — | 9 | 1 | no | — |
| 143 | C2×D12 | — | 18 | 1 | no | — |
| 144 | C3×M16 | — | 9 | 2 | no | — |
| 145 | #215 | — | 18 | 2 | no | — |
| 146 | #213 | — | 18 | 2 | no | — |
| 147 | (C2×D3)⋊C4 | — | 18 | 4 | no | — |
| 148 | (C2×D3)⋊C4 | — | 18 | 2 | no | — |
| 149 | (C2×D3)⋊C4 | — | 18 | 2 | no | — |
| 150 | C4×Dic3 | — | 9 | 1 | no | — |
| 151 | C4×Dic3 | — | 9 | 1 | no | — |
| 152 | C4×Dic3 | — | 18 | 1 | no | — |
| 153 | #208 | — | 3 | 2 | no | — |
| 154 | D24 | — | 18 | 2 | no | — |
| 155 | #204 | — | 18 | 2 | no | — |
| 156 | C22×C32 | C2×#67 | 1 | 1 | yes | characteristic |
| 157 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 158 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 159 | C2×C3×D3 | — | 2 | 2 | no | — |
| 160 | C32×C4 | — | 2 | 2 | no | — |
| 161 | C32×C4 | C2×#67 | 1 | 2 | yes | — |
| 162 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 163 | C3×Dic3 | — | 2 | 2 | no | — |
| 164 | C22×C9 | — | 2 | 1 | no | — |
| 165 | C22×C9 | — | 6 | 2 | no | — |
| 166 | C22×C9 | C2×D12 | 1 | 1 | yes | characteristic |
| 167 | C2×D9 | — | 12 | 2 | no | — |
| 168 | C2×D9 | — | 12 | 4 | no | — |
| 169 | C2×D9 | — | 12 | 2 | no | — |
| 170 | C2×D9 | — | 12 | 1 | no | — |
| 171 | C4×C9 | — | 2 | 2 | no | — |
| 172 | C4×C9 | — | 2 | 2 | no | — |
| 173 | C4×C9 | — | 4 | 2 | no | — |
| 174 | C4×C9 | — | 6 | 2 | no | — |
| 175 | C4×C9 | D3×D4 | 1 | 2 | yes | — |
| 176 | Dic9 | — | 12 | 2 | no | — |
| 177 | Dic9 | — | 12 | 1 | no | — |
| 178 | 2−1+4 | — | 3 | 1 | no | — |
| 179 | AGL(1,ℤ/8) | — | 27 | 2 | no | — |
| 180 | #124 | — | 27 | 1 | no | — |
| 181 | #104 | — | 27 | 2 | no | — |
| 182 | #101 | — | 9 | 1 | no | — |
| 183 | C3×C9 | #454 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 184 | C22×D3 | — | 18 | 2 | no | — |
| 185 | C22×D3 | — | 18 | 1 | no | — |
| 186 | C22×D3 | — | 18 | 1 | no | — |
| 187 | C3×Q8 | — | 2 | 2 | no | — |
| 188 | C3×Q8 | — | 2 | 2 | no | — |
| 189 | C3×Q8 | — | 3 | 2 | no | — |
| 190 | C3×Q8 | — | 4 | 2 | no | — |
| 191 | C3×Q8 | — | 6 | 2 | no | — |
| 192 | C3×D4 | — | 3 | 2 | no | — |
| 193 | C3×D4 | — | 6 | 2 | no | — |
| 194 | C3×D4 | — | 6 | 2 | no | — |
| 195 | C2×C3×C4 | — | 2 | 2 | no | — |
| 196 | C2×C3×C4 | — | 2 | 1 | no | — |
| 197 | C2×C3×C4 | — | 6 | 4 | no | — |
| 198 | C2×C3×C4 | — | 6 | 2 | no | — |
| 199 | C2×C3×C4 | #611 | 1 | 2 | yes | — |
| 200 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 201 | C2×C3×C4 | #626 | 1 | 2 | yes | — |
| 202 | C2×C3×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 203 | #67 | — | 2 | 2 | no | — |
| 204 | C2×Dic3 | — | 18 | 2 | no | — |
| 205 | C2×Dic3 | — | 18 | 1 | no | — |
| 206 | C2×Dic3 | — | 18 | 1 | no | — |
| 207 | D12 | — | 2 | 2 | no | — |
| 208 | D12 | — | 18 | 2 | no | — |
| 209 | D12 | — | 18 | 2 | no | — |
| 210 | D12 | — | 36 | 2 | no | — |
| 211 | D12 | #611 | 1 | 2 | yes | — |
| 212 | C4×D3 | — | 2 | 4 | no | — |
| 213 | C4×D3 | — | 2 | 2 | no | — |
| 214 | Dic6 | — | 2 | 2 | no | — |
| 215 | Dic6 | #611 | 1 | 2 | yes | — |
| 216 | C3×C8 | — | 18 | 2 | no | — |
| 217 | C3⋊C8 | — | 6 | 2 | no | — |
| 218 | C2×C32 | — | 2 | 1 | no | — |
| 219 | C2×C32 | #953 | 1 | 1 | yes | characteristic |
| 220 | C3⋊D3 | — | 24 | 1 | no | — |
| 221 | C3×D3 | — | 4 | 2 | no | — |
| 222 | C2×C9 | — | 2 | 2 | no | — |
| 223 | C2×C9 | — | 2 | 1 | no | — |
| 224 | C2×C9 | — | 2 | 1 | no | — |
| 225 | C2×C9 | — | 12 | 2 | no | — |
| 226 | C2×C9 | #882 | 1 | 1 | yes | characteristic |
| 227 | D9 | — | 24 | 2 | no | — |
| 228 | D9 | — | 24 | 1 | no | — |
| 229 | #41 | — | 3 | 2 | no | — |
| 230 | #41 | — | 6 | 2 | no | — |
| 231 | #41 | — | 6 | 2 | no | — |
| 232 | C2×Q8 | — | 6 | 2 | no | — |
| 233 | C2×Q8 | D3×D9 | 1 | 1 | yes | characteristic |
| 234 | C2×D4 | — | 27 | 1 | no | — |
| 235 | SD16 | — | 54 | 2 | no | — |
| 236 | D8 | — | 54 | 2 | no | — |
| 237 | M16 | — | 9 | 2 | no | — |
| 238 | C22⋊C4 | — | 54 | 2 | no | — |
| 239 | C42 | — | 27 | 1 | no | — |
| 240 | C22×C3 | — | 2 | 1 | no | — |
| 241 | C22×C3 | — | 6 | 2 | no | — |
| 242 | C22×C3 | C2×#611 | 1 | 1 | yes | characteristic |
| 243 | C22×C3 | C2×#626 | 1 | 1 | yes | characteristic |
| 244 | C2×D3 | — | 2 | 2 | no | — |
| 245 | C2×D3 | — | 36 | 2 | no | — |
| 246 | C2×D3 | — | 36 | 2 | no | — |
| 247 | C2×D3 | — | 36 | 4 | no | — |
| 248 | C2×D3 | — | 36 | 2 | no | — |
| 249 | C2×D3 | — | 36 | 1 | no | — |
| 250 | C2×D3 | — | 36 | 1 | no | — |
| 251 | C3×C4 | — | 2 | 2 | no | — |
| 252 | C3×C4 | — | 2 | 2 | no | — |
| 253 | C3×C4 | — | 2 | 2 | no | — |
| 254 | C3×C4 | — | 4 | 2 | no | — |
| 255 | C3×C4 | — | 6 | 2 | no | — |
| 256 | C3×C4 | C2×#611 | 1 | 2 | yes | — |
| 257 | C3×C4 | D3×#67 | 1 | 2 | yes | — |
| 258 | Dic3 | — | 2 | 2 | no | — |
| 259 | Dic3 | — | 36 | 1 | no | — |
| 260 | Dic3 | — | 36 | 2 | no | — |
| 261 | Dic3 | — | 36 | 1 | no | — |
| 262 | C32 | #5245 | 1 | 1 | yes | characteristic |
| 263 | C9 | — | 2 | 1 | no | — |
| 264 | C9 | #4753 | 1 | 1 | yes | characteristic, complemented |
| 265 | C23 | — | 54 | 1 | no | — |
| 266 | Q8 | — | 2 | 2 | no | — |
| 267 | Q8 | — | 3 | 2 | no | — |
| 268 | Q8 | — | 6 | 2 | no | — |
| 269 | D4 | — | 3 | 2 | no | — |
| 270 | D4 | — | 6 | 2 | no | — |
| 271 | D4 | — | 6 | 2 | no | — |
| 272 | D4 | — | 54 | 2 | no | — |
| 273 | C2×C4 | — | 6 | 4 | no | — |
| 274 | C2×C4 | — | 6 | 2 | no | — |
| 275 | C2×C4 | — | 54 | 1 | no | — |
| 276 | C2×C4 | #6195 | 1 | 2 | yes | — |
| 277 | C2×C4 | C2×D3×D9 | 1 | 1 | yes | characteristic |
| 278 | C8 | — | 18 | 2 | no | — |
| 279 | C2×C3 | — | 2 | 2 | no | — |
| 280 | C2×C3 | — | 2 | 1 | no | — |
| 281 | C2×C3 | — | 2 | 1 | no | — |
| 282 | C2×C3 | — | 2 | 1 | no | — |
| 283 | C2×C3 | — | 12 | 2 | no | — |
| 284 | C2×C3 | #63515 | 1 | 1 | yes | characteristic |
| 285 | C2×C3 | #63977 | 1 | 1 | yes | characteristic |
| 286 | D3 | — | 4 | 2 | no | — |
| 287 | D3 | — | 72 | 2 | no | — |
| 288 | D3 | — | 72 | 1 | no | — |
| 289 | D3 | — | 72 | 1 | no | — |
| 290 | C22 | — | 6 | 2 | no | — |
| 291 | C22 | — | 108 | 2 | no | — |
| 292 | C22 | — | 108 | 1 | no | — |
| 293 | C22 | C2×#6195 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 2 | no | — |
| 295 | C4 | — | 6 | 2 | no | — |
| 296 | C4 | — | 108 | 1 | no | — |
| 297 | C4 | D3×#611 | 1 | 2 | yes | — |
| 298 | C3 | — | 2 | 1 | no | — |
| 299 | C3 | #10588926 | 1 | 1 | yes | characteristic, complemented |
| 300 | C3 | #10591646 | 1 | 1 | yes | characteristic |
| 301 | C2 | — | 2 | 1 | no | — |
| 302 | C2 | — | 12 | 2 | no | — |
| 303 | C2 | — | 216 | 1 | no | — |
| 304 | C2 | #11721287 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#1007 | C2 | semidirect |
| 2. #11721857 | C2 | semidirect |
| 3. #11720885 | C2 | semidirect |
| 4. C3×#63394 | C2 | semidirect |
| 5. #11720300 | C2 | semidirect |
| 29. #63394 | D3 | semidirect |
| 81. #1007 | D9 | semidirect |
| 183. C3×C9 | #454 | semidirect |
| 264. C9 | #4753 | semidirect |
| 299. C3 | #10588926 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#1007 | 2 |
| #11721857 | 2 |
| #11720885 | 2 |
| C3×#63394 | 2 |
| #11720300 | 2 |
| #10591646 | 3 |
| #10588926 | 3 |