Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
494 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 85 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 | C3×#63713 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11721854 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11720869 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C9×#836 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11720291 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10591649 | — | 3 | 1 | no | maximal |
| 7 | #10588432 | — | 3 | 1 | no | maximal |
| 8 | C2×C3×Q8×C9 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6226 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#3701 | — | 2 | 2 | no | — |
| 11 | C3×Q8×D9 | — | 2 | 2 | no | — |
| 12 | C3×#3697 | — | 2 | 2 | no | — |
| 13 | C3×#3697 | C22 | 1 | 2 | yes | — |
| 14 | #88534 | — | 2 | 2 | no | — |
| 15 | C4×C3⋊Dic9 | C22 | 1 | 1 | yes | characteristic |
| 16 | C9×#208 | C22 | 1 | 2 | yes | — |
| 17 | #88418 | — | 2 | 2 | no | — |
| 18 | #88415 | — | 2 | 2 | no | — |
| 19 | C3×#1007 | — | 3 | 1 | no | — |
| 20 | #64153 | — | 3 | 1 | no | — |
| 21 | #63831 | — | 3 | 2 | no | — |
| 22 | #63713 | D3 | 1 | 1 | yes | characteristic |
| 23 | C3×#836 | D3 | 1 | 1 | yes | characteristic |
| 24 | #63568 | — | 3 | 2 | no | — |
| 25 | #63506 | — | 3 | 1 | no | — |
| 26 | #63506 | — | 6 | 1 | no | — |
| 27 | #63471 | — | 3 | 2 | no | — |
| 28 | C9×#101 | — | 3 | 1 | no | — |
| 29 | #63384 | — | 3 | 2 | no | — |
| 30 | C3×Q8×C9 | — | 2 | 2 | no | — |
| 31 | C2×C3×C4×C9 | D4 | 1 | 2 | yes | — |
| 32 | C2×C3×C4×C9 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×C3⋊Dic9 | — | 2 | 1 | no | — |
| 34 | C2×C3⋊Dic9 | — | 2 | 1 | no | — |
| 35 | #6226 | — | 2 | 2 | no | — |
| 36 | C3×#611 | — | 2 | 2 | no | — |
| 37 | C3×D36 | — | 2 | 2 | no | — |
| 38 | C3×D36 | D4 | 1 | 2 | yes | — |
| 39 | C3×C4×D9 | — | 2 | 4 | no | — |
| 40 | C3×C4×D9 | — | 2 | 2 | no | — |
| 41 | C3×Dic18 | — | 2 | 2 | no | — |
| 42 | C3×Dic18 | D4 | 1 | 2 | yes | — |
| 43 | C9×C3⋊C8 | — | 2 | 2 | no | — |
| 44 | #5246 | — | 9 | 1 | no | — |
| 45 | #4752 | — | 9 | 1 | no | — |
| 46 | C2×C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 47 | C2×#634 | — | 3 | 1 | no | — |
| 48 | C3×#239 | — | 6 | 2 | no | — |
| 49 | C3×D3×Q8 | — | 6 | 2 | no | — |
| 50 | C3×#235 | — | 3 | 2 | no | — |
| 51 | C3×#235 | — | 6 | 2 | no | — |
| 52 | #3750 | — | 6 | 2 | no | — |
| 53 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 54 | C3×#208 | C2×D3 | 1 | 2 | yes | — |
| 55 | #3719 | — | 6 | 2 | no | — |
| 56 | #3716 | — | 6 | 2 | no | — |
| 57 | C2×Q8×C9 | — | 2 | 1 | no | — |
| 58 | C2×Q8×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 59 | #3701 | — | 2 | 2 | no | — |
| 60 | Q8×D9 | — | 2 | 2 | no | — |
| 61 | #3697 | — | 2 | 2 | no | — |
| 62 | #3697 | C2×D3 | 1 | 2 | yes | — |
| 63 | C2×Dic18 | — | 3 | 1 | no | — |
| 64 | C2×Dic18 | — | 6 | 1 | no | — |
| 65 | C9×M16 | — | 3 | 2 | no | — |
| 66 | Dic9⋊C4 | — | 6 | 4 | no | — |
| 67 | Dic9⋊C4 | — | 6 | 2 | no | — |
| 68 | C4×Dic9 | — | 3 | 1 | no | — |
| 69 | C4×Dic9 | — | 6 | 1 | no | — |
| 70 | #3664 | — | 6 | 2 | no | — |
| 71 | Dic36 | — | 6 | 2 | no | — |
| 72 | C22×C3×C9 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C2×C3×D9 | — | 2 | 2 | no | — |
| 74 | C3×C4×C9 | — | 2 | 2 | no | — |
| 75 | C3×C4×C9 | C2×D4 | 1 | 2 | yes | — |
| 76 | C3⋊Dic9 | — | 2 | 2 | no | — |
| 77 | C3⋊Dic9 | — | 4 | 1 | no | — |
| 78 | C3×Dic9 | — | 2 | 2 | no | — |
| 79 | C3×2−1+4 | — | 9 | 1 | no | — |
| 80 | #1007 | — | 3 | 1 | no | — |
| 81 | #951 | — | 9 | 2 | no | — |
| 82 | #944 | — | 9 | 1 | no | — |
| 83 | #944 | — | 9 | 1 | no | — |
| 84 | #944 | — | 18 | 1 | no | — |
| 85 | #909 | — | 9 | 2 | no | — |
| 86 | C3×#101 | — | 3 | 1 | no | — |
| 87 | #837 | — | 9 | 2 | no | — |
| 88 | #836 | D9 | 1 | 1 | yes | characteristic, complemented |
| 89 | #825 | — | 9 | 2 | no | — |
| 90 | C32×Q8 | — | 2 | 2 | no | — |
| 91 | C2×C32×C4 | D12 | 1 | 2 | yes | — |
| 92 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 93 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 94 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 95 | #634 | — | 6 | 2 | no | — |
| 96 | C3×#67 | — | 6 | 2 | no | — |
| 97 | C3×D12 | — | 3 | 2 | no | — |
| 98 | C3×D12 | — | 6 | 2 | no | — |
| 99 | C3×C4×D3 | — | 6 | 4 | no | — |
| 100 | C3×C4×D3 | — | 6 | 2 | no | — |
| 101 | C3×Dic6 | — | 3 | 2 | no | — |
| 102 | C3×Dic6 | — | 6 | 2 | no | — |
| 103 | C3×C3⋊C8 | — | 2 | 2 | no | — |
| 104 | Q8×C9 | — | 2 | 2 | no | — |
| 105 | Q8×C9 | — | 4 | 2 | no | — |
| 106 | C2×C4×C9 | — | 2 | 2 | no | — |
| 107 | C2×C4×C9 | — | 2 | 1 | no | — |
| 108 | C2×C4×C9 | #67 | 1 | 2 | yes | — |
| 109 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 110 | #611 | — | 2 | 2 | no | — |
| 111 | C2×Dic9 | — | 6 | 2 | no | — |
| 112 | C2×Dic9 | — | 6 | 1 | no | — |
| 113 | C2×Dic9 | — | 6 | 2 | no | — |
| 114 | C2×Dic9 | — | 6 | 1 | no | — |
| 115 | D36 | — | 2 | 2 | no | — |
| 116 | D36 | #67 | 1 | 2 | yes | — |
| 117 | C4×D9 | — | 2 | 4 | no | — |
| 118 | C4×D9 | — | 2 | 2 | no | — |
| 119 | Dic18 | — | 2 | 2 | no | — |
| 120 | Dic18 | — | 6 | 2 | no | — |
| 121 | Dic18 | — | 12 | 2 | no | — |
| 122 | Dic18 | #67 | 1 | 2 | yes | — |
| 123 | C8×C9 | — | 6 | 2 | no | — |
| 124 | #456 | — | 27 | 1 | no | Sylow |
| 125 | C2×C3×C9 | — | 2 | 1 | no | — |
| 126 | C2×C3×C9 | #120 | 1 | 1 | yes | characteristic |
| 127 | C3×D9 | — | 4 | 2 | no | — |
| 128 | C3×#41 | — | 9 | 2 | no | — |
| 129 | C3×#41 | — | 18 | 2 | no | — |
| 130 | C3×#41 | — | 18 | 2 | no | — |
| 131 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 132 | C2×C3×Q8 | — | 18 | 2 | no | — |
| 133 | C2×C3×Q8 | C2×D9 | 1 | 1 | yes | characteristic |
| 134 | C2×C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 135 | #239 | — | 6 | 2 | no | — |
| 136 | D3×Q8 | — | 6 | 2 | no | — |
| 137 | #235 | — | 3 | 2 | no | — |
| 138 | #235 | — | 6 | 2 | no | — |
| 139 | C2×Dic6 | — | 9 | 1 | no | — |
| 140 | C2×Dic6 | — | 9 | 1 | no | — |
| 141 | C2×Dic6 | — | 18 | 1 | no | — |
| 142 | C3×M16 | — | 3 | 2 | no | — |
| 143 | #216 | — | 18 | 2 | no | — |
| 144 | #214 | — | 18 | 2 | no | — |
| 145 | Dic3⋊C4 | — | 18 | 4 | no | — |
| 146 | Dic3⋊C4 | — | 18 | 2 | no | — |
| 147 | Dic3⋊C4 | — | 18 | 2 | no | — |
| 148 | C4×Dic3 | — | 9 | 1 | no | — |
| 149 | C4×Dic3 | — | 9 | 1 | no | — |
| 150 | C4×Dic3 | — | 18 | 1 | no | — |
| 151 | #208 | C2×D9 | 1 | 2 | yes | — |
| 152 | Dic12 | — | 18 | 2 | no | — |
| 153 | #204 | — | 18 | 2 | no | — |
| 154 | C22×C32 | C2×D12 | 1 | 1 | yes | characteristic |
| 155 | C2×C3×D3 | — | 6 | 2 | no | — |
| 156 | C32×C4 | — | 2 | 2 | no | — |
| 157 | C32×C4 | D3×D4 | 1 | 2 | yes | — |
| 158 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 159 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 160 | C3×Dic3 | — | 6 | 2 | no | — |
| 161 | C22×C9 | — | 2 | 1 | no | — |
| 162 | C22×C9 | C2×#67 | 1 | 1 | yes | characteristic |
| 163 | C2×D9 | — | 2 | 2 | no | — |
| 164 | C4×C9 | — | 2 | 2 | no | — |
| 165 | C4×C9 | — | 2 | 2 | no | — |
| 166 | C4×C9 | — | 4 | 2 | no | — |
| 167 | C4×C9 | C2×#67 | 1 | 2 | yes | — |
| 168 | Dic9 | — | 2 | 2 | no | — |
| 169 | Dic9 | — | 6 | 4 | no | — |
| 170 | Dic9 | — | 6 | 2 | no | — |
| 171 | Dic9 | — | 12 | 2 | no | — |
| 172 | Dic9 | — | 12 | 1 | no | — |
| 173 | 2−1+4 | — | 9 | 1 | no | — |
| 174 | #137 | — | 27 | 2 | no | — |
| 175 | C4⋊Q8 | — | 27 | 1 | no | — |
| 176 | #104 | — | 27 | 2 | no | — |
| 177 | #101 | — | 3 | 1 | no | — |
| 178 | C3×C9 | #456 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 179 | C3×Q8 | — | 2 | 2 | no | — |
| 180 | C3×Q8 | — | 2 | 2 | no | — |
| 181 | C3×Q8 | — | 4 | 2 | no | — |
| 182 | C3×Q8 | — | 9 | 2 | no | — |
| 183 | C3×Q8 | — | 18 | 2 | no | — |
| 184 | C3×D4 | — | 9 | 2 | no | — |
| 185 | C3×D4 | — | 18 | 2 | no | — |
| 186 | C3×D4 | — | 18 | 2 | no | — |
| 187 | C2×C3×C4 | — | 2 | 2 | no | — |
| 188 | C2×C3×C4 | — | 2 | 1 | no | — |
| 189 | C2×C3×C4 | — | 18 | 4 | no | — |
| 190 | C2×C3×C4 | — | 18 | 2 | no | — |
| 191 | C2×C3×C4 | D36 | 1 | 2 | yes | — |
| 192 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 193 | C2×C3×C4 | #626 | 1 | 2 | yes | — |
| 194 | C2×C3×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 195 | #67 | — | 6 | 2 | no | — |
| 196 | C2×Dic3 | — | 18 | 2 | no | — |
| 197 | C2×Dic3 | — | 18 | 1 | no | — |
| 198 | C2×Dic3 | — | 18 | 1 | no | — |
| 199 | C2×Dic3 | — | 18 | 2 | no | — |
| 200 | C2×Dic3 | — | 18 | 1 | no | — |
| 201 | C2×Dic3 | — | 18 | 1 | no | — |
| 202 | D12 | — | 3 | 2 | no | — |
| 203 | D12 | — | 6 | 2 | no | — |
| 204 | C4×D3 | — | 6 | 4 | no | — |
| 205 | C4×D3 | — | 6 | 2 | no | — |
| 206 | Dic6 | — | 3 | 2 | no | — |
| 207 | Dic6 | — | 6 | 2 | no | — |
| 208 | Dic6 | — | 18 | 2 | no | — |
| 209 | Dic6 | — | 18 | 2 | no | — |
| 210 | Dic6 | — | 36 | 2 | no | — |
| 211 | C3×C8 | — | 6 | 2 | no | — |
| 212 | C3⋊C8 | — | 2 | 2 | no | — |
| 213 | C2×C32 | — | 2 | 1 | no | — |
| 214 | C2×C32 | #882 | 1 | 1 | yes | characteristic |
| 215 | C3×D3 | — | 12 | 2 | no | — |
| 216 | C2×C9 | — | 2 | 2 | no | — |
| 217 | C2×C9 | — | 2 | 1 | no | — |
| 218 | C2×C9 | — | 2 | 1 | no | — |
| 219 | C2×C9 | #953 | 1 | 1 | yes | characteristic |
| 220 | D9 | — | 4 | 2 | no | — |
| 221 | #41 | — | 9 | 2 | no | — |
| 222 | #41 | — | 18 | 2 | no | — |
| 223 | #41 | — | 18 | 2 | no | — |
| 224 | C2×Q8 | — | 18 | 2 | no | — |
| 225 | C2×Q8 | — | 27 | 1 | no | — |
| 226 | C2×Q8 | D3×D9 | 1 | 1 | yes | characteristic |
| 227 | Q16 | — | 54 | 2 | no | — |
| 228 | SD16 | — | 54 | 2 | no | — |
| 229 | M16 | — | 3 | 2 | no | — |
| 230 | C4⋊C4 | — | 54 | 2 | no | — |
| 231 | C42 | — | 27 | 1 | no | — |
| 232 | C22×C3 | — | 2 | 1 | no | — |
| 233 | C22×C3 | — | 18 | 2 | no | — |
| 234 | C22×C3 | C2×D36 | 1 | 1 | yes | characteristic |
| 235 | C22×C3 | C2×#626 | 1 | 1 | yes | characteristic |
| 236 | C2×D3 | — | 6 | 2 | no | — |
| 237 | C3×C4 | — | 2 | 2 | no | — |
| 238 | C3×C4 | — | 2 | 2 | no | — |
| 239 | C3×C4 | — | 2 | 2 | no | — |
| 240 | C3×C4 | — | 4 | 2 | no | — |
| 241 | C3×C4 | — | 18 | 2 | no | — |
| 242 | C3×C4 | D4×D9 | 1 | 2 | yes | — |
| 243 | C3×C4 | D3×#67 | 1 | 2 | yes | — |
| 244 | Dic3 | — | 6 | 2 | no | — |
| 245 | Dic3 | — | 18 | 2 | no | — |
| 246 | Dic3 | — | 18 | 4 | no | — |
| 247 | Dic3 | — | 18 | 2 | no | — |
| 248 | Dic3 | — | 36 | 2 | no | — |
| 249 | Dic3 | — | 36 | 1 | no | — |
| 250 | Dic3 | — | 36 | 1 | no | — |
| 251 | C32 | #4752 | 1 | 1 | yes | characteristic |
| 252 | C9 | — | 2 | 1 | no | — |
| 253 | C9 | #5246 | 1 | 1 | yes | characteristic, complemented |
| 254 | Q8 | — | 2 | 2 | no | — |
| 255 | Q8 | — | 9 | 2 | no | — |
| 256 | Q8 | — | 18 | 2 | no | — |
| 257 | Q8 | — | 54 | 2 | no | — |
| 258 | D4 | — | 9 | 2 | no | — |
| 259 | D4 | — | 18 | 2 | no | — |
| 260 | D4 | — | 18 | 2 | no | — |
| 261 | C2×C4 | — | 18 | 4 | no | — |
| 262 | C2×C4 | — | 18 | 2 | no | — |
| 263 | C2×C4 | — | 54 | 1 | no | — |
| 264 | C2×C4 | — | 54 | 1 | no | — |
| 265 | C2×C4 | #6192 | 1 | 2 | yes | — |
| 266 | C2×C4 | C2×D3×D9 | 1 | 1 | yes | characteristic |
| 267 | C8 | — | 6 | 2 | no | — |
| 268 | C2×C3 | — | 2 | 2 | no | — |
| 269 | C2×C3 | — | 2 | 1 | no | — |
| 270 | C2×C3 | — | 2 | 1 | no | — |
| 271 | C2×C3 | — | 2 | 1 | no | — |
| 272 | C2×C3 | — | 36 | 2 | no | — |
| 273 | C2×C3 | #63444 | 1 | 1 | yes | characteristic |
| 274 | C2×C3 | #63977 | 1 | 1 | yes | characteristic |
| 275 | D3 | — | 12 | 2 | no | — |
| 276 | C22 | — | 18 | 2 | no | — |
| 277 | C22 | C2×#6192 | 1 | 1 | yes | characteristic |
| 278 | C4 | — | 2 | 2 | no | — |
| 279 | C4 | — | 18 | 2 | no | — |
| 280 | C4 | — | 54 | 2 | no | — |
| 281 | C4 | — | 108 | 1 | no | — |
| 282 | C4 | D9×#67 | 1 | 2 | yes | — |
| 283 | C3 | — | 2 | 1 | no | — |
| 284 | C3 | #10588432 | 1 | 1 | yes | characteristic, complemented |
| 285 | C3 | #10591649 | 1 | 1 | yes | characteristic |
| 286 | C2 | — | 2 | 1 | no | — |
| 287 | C2 | — | 36 | 2 | no | — |
| 288 | C2 | #11721276 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11721854 | C2 | semidirect |
| 3. #11720869 | C2 | semidirect |
| 4. C9×#836 | C2 | semidirect |
| 5. #11720291 | C2 | semidirect |
| 88. #836 | D9 | semidirect |
| 178. C3×C9 | #456 | semidirect |
| 253. C9 | #5246 | semidirect |
| 284. C3 | #10588432 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63713 | 2 |
| #11721854 | 2 |
| #11720869 | 2 |
| C9×#836 | 2 |
| #11720291 | 2 |
| #10591649 | 3 |
| #10588432 | 3 |