Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
458 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 50 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 | Q8×(C22⋊C9)⋊C2 | C2 | 1 | 2 | yes | maximal |
| 2 | Q16×C22⋊C9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10589518 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C8×(C22⋊C9)⋊C2 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #10589500 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | Q16×#67 | — | 3 | 1 | no | maximal |
| 7 | Q8×C22⋊C9 | C22 | 1 | 2 | yes | — |
| 8 | C4×(C22⋊C9)⋊C2 | — | 2 | 2 | no | — |
| 9 | C4×(C22⋊C9)⋊C2 | C22 | 1 | 1 | yes | characteristic |
| 10 | (C22⋊C9)⋊Q8 | — | 2 | 2 | no | — |
| 11 | (C22⋊C9)⋊Q8 | C22 | 1 | 2 | yes | — |
| 12 | Q16×D9 | — | 4 | 1 | no | maximal |
| 13 | C8×C22⋊C9 | C22 | 1 | 1 | yes | characteristic |
| 14 | (C22⋊C9)⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×C3×Q16 | D3 | 1 | 1 | yes | characteristic |
| 16 | Q8×#67 | — | 3 | 2 | no | — |
| 17 | C2×D3×Q16 | — | 3 | 1 | no | — |
| 18 | #5228 | — | 3 | 2 | no | — |
| 19 | #5188 | — | 3 | 1 | no | — |
| 20 | #5186 | — | 3 | 2 | no | — |
| 21 | #5183 | — | 3 | 1 | no | — |
| 22 | #5182 | — | 3 | 2 | no | — |
| 23 | Dic3×Q16 | — | 3 | 1 | no | — |
| 24 | #5116 | — | 3 | 1 | no | — |
| 25 | C8×#67 | — | 3 | 1 | no | — |
| 26 | C2×(C22⋊C9)⋊C2 | D4 | 1 | 1 | yes | characteristic |
| 27 | Q8×D9 | — | 4 | 2 | no | — |
| 28 | C4×C22⋊C9 | — | 2 | 2 | no | — |
| 29 | C4×C22⋊C9 | C23 | 1 | 1 | yes | characteristic |
| 30 | (C22⋊C9)⋊C4 | — | 2 | 2 | no | — |
| 31 | (C22⋊C9)⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 32 | C9×Q16 | — | 4 | 1 | no | — |
| 33 | #3674 | — | 4 | 2 | no | — |
| 34 | C8×D9 | — | 4 | 1 | no | — |
| 35 | Dic36 | — | 4 | 1 | no | — |
| 36 | D4×Q16 | — | 9 | 1 | no | Sylow |
| 37 | C22×C3×Q8 | C2×D3 | 1 | 2 | yes | — |
| 38 | C2×D3×Q8 | — | 3 | 2 | no | — |
| 39 | C2×C3×Q16 | — | 3 | 1 | no | — |
| 40 | C2×C3×Q16 | — | 3 | 2 | no | — |
| 41 | C2×C3×Q16 | — | 3 | 1 | no | — |
| 42 | C22×C3×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | #946 | — | 3 | 2 | no | — |
| 44 | #946 | — | 6 | 2 | no | — |
| 45 | Q8×Dic3 | — | 3 | 2 | no | — |
| 46 | #944 | — | 3 | 2 | no | — |
| 47 | #944 | — | 6 | 2 | no | — |
| 48 | C2×#216 | — | 3 | 2 | no | — |
| 49 | C2×#216 | — | 3 | 2 | no | — |
| 50 | C4×#67 | — | 3 | 1 | no | — |
| 51 | C4×#67 | — | 6 | 2 | no | — |
| 52 | #924 | — | 3 | 2 | no | — |
| 53 | #924 | — | 6 | 2 | no | — |
| 54 | D3×Q16 | — | 6 | 2 | no | — |
| 55 | D3×Q16 | — | 6 | 1 | no | — |
| 56 | D3×Q16 | — | 6 | 1 | no | — |
| 57 | C2×Dic12 | — | 3 | 1 | no | — |
| 58 | C2×D3×C8 | — | 3 | 1 | no | — |
| 59 | #835 | — | 3 | 2 | no | — |
| 60 | #835 | — | 3 | 2 | no | — |
| 61 | #830 | — | 3 | 1 | no | — |
| 62 | (C2×D3)⋊C8 | — | 3 | 1 | no | — |
| 63 | #818 | — | 3 | 1 | no | — |
| 64 | #816 | — | 3 | 2 | no | — |
| 65 | Dic3⋊C8 | — | 3 | 1 | no | — |
| 66 | C8×Dic3 | — | 3 | 1 | no | — |
| 67 | C2×C22⋊C9 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | (C22⋊C9)⋊C2 | Q16 | 1 | 2 | yes | direct product factor |
| 69 | Q8×C9 | — | 4 | 2 | no | — |
| 70 | C4×D9 | — | 4 | 1 | no | — |
| 71 | C4×D9 | — | 8 | 2 | no | — |
| 72 | Dic18 | — | 4 | 2 | no | — |
| 73 | Dic18 | — | 8 | 2 | no | — |
| 74 | C8×C9 | — | 4 | 1 | no | — |
| 75 | C9⋊C8 | — | 4 | 1 | no | — |
| 76 | C22×Q16 | — | 9 | 1 | no | — |
| 77 | C22×Q16 | D9 | 1 | 1 | yes | characteristic, complemented |
| 78 | D4×Q8 | — | 9 | 2 | no | — |
| 79 | #494 | — | 9 | 1 | no | — |
| 80 | #467 | — | 9 | 1 | no | — |
| 81 | #467 | — | 9 | 1 | no | — |
| 82 | #462 | — | 9 | 2 | no | — |
| 83 | #451 | — | 9 | 2 | no | — |
| 84 | #451 | — | 9 | 2 | no | — |
| 85 | C4×Q16 | — | 9 | 1 | no | — |
| 86 | C8×D4 | — | 9 | 1 | no | — |
| 87 | C2×C3×Q8 | — | 3 | 2 | no | — |
| 88 | C2×C3×Q8 | — | 3 | 2 | no | — |
| 89 | C2×C3×Q8 | — | 6 | 2 | no | — |
| 90 | C22×C3×C4 | — | 2 | 2 | no | — |
| 91 | C22×C3×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 92 | C2×#67 | — | 3 | 1 | no | — |
| 93 | D3×Q8 | — | 6 | 2 | no | — |
| 94 | D3×Q8 | — | 6 | 2 | no | — |
| 95 | D3×Q8 | — | 12 | 2 | no | — |
| 96 | C2×C4×D3 | — | 3 | 1 | no | — |
| 97 | C2×C4×D3 | — | 6 | 2 | no | — |
| 98 | C2×Dic6 | — | 3 | 2 | no | — |
| 99 | C2×Dic6 | — | 6 | 2 | no | — |
| 100 | C3×Q16 | — | 3 | 2 | no | — |
| 101 | C3×Q16 | — | 3 | 1 | no | — |
| 102 | C3×Q16 | — | 6 | 1 | no | — |
| 103 | C3×Q16 | S4 | 1 | 1 | yes | characteristic |
| 104 | C2×C3×C8 | — | 3 | 1 | no | — |
| 105 | C2×C3×C8 | — | 3 | 1 | no | — |
| 106 | #217 | — | 3 | 1 | no | — |
| 107 | #217 | — | 6 | 2 | no | — |
| 108 | #216 | — | 6 | 2 | no | — |
| 109 | #216 | — | 6 | 2 | no | — |
| 110 | #216 | — | 12 | 2 | no | — |
| 111 | (C2×D3)⋊C4 | — | 3 | 1 | no | — |
| 112 | (C2×D3)⋊C4 | — | 6 | 2 | no | — |
| 113 | C4⋊Dic3 | — | 3 | 2 | no | — |
| 114 | C4⋊Dic3 | — | 6 | 2 | no | — |
| 115 | Dic3⋊C4 | — | 3 | 1 | no | — |
| 116 | Dic3⋊C4 | — | 6 | 2 | no | — |
| 117 | Dic3⋊C4 | — | 6 | 2 | no | — |
| 118 | Dic3⋊C4 | — | 6 | 2 | no | — |
| 119 | Dic3⋊C4 | — | 6 | 2 | no | — |
| 120 | C4×Dic3 | — | 3 | 1 | no | — |
| 121 | C4×Dic3 | — | 6 | 2 | no | — |
| 122 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 123 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 124 | Dic12 | — | 6 | 1 | no | — |
| 125 | Dic12 | — | 6 | 1 | no | — |
| 126 | D3×C8 | — | 6 | 1 | no | — |
| 127 | D3×C8 | — | 6 | 1 | no | — |
| 128 | C2×D9 | — | 4 | 1 | no | — |
| 129 | C22⋊C9 | C2×Q16 | 1 | 1 | yes | characteristic, complemented |
| 130 | C4×C9 | — | 4 | 1 | no | — |
| 131 | C4×C9 | — | 8 | 2 | no | — |
| 132 | Dic9 | — | 4 | 1 | no | — |
| 133 | Dic9 | — | 8 | 2 | no | — |
| 134 | C22×Q8 | — | 9 | 2 | no | — |
| 135 | C22×Q8 | C2×D9 | 1 | 2 | yes | — |
| 136 | C2×Q16 | — | 3 | 1 | no | — |
| 137 | C2×Q16 | — | 3 | 2 | no | — |
| 138 | C2×Q16 | — | 3 | 1 | no | — |
| 139 | C2×Q16 | — | 9 | 1 | no | — |
| 140 | C2×Q16 | — | 9 | 2 | no | — |
| 141 | C2×Q16 | — | 9 | 2 | no | — |
| 142 | C2×Q16 | — | 18 | 2 | no | — |
| 143 | C2×Q16 | — | 18 | 1 | no | — |
| 144 | C2×Q16 | — | 18 | 1 | no | — |
| 145 | C22×C8 | — | 9 | 1 | no | — |
| 146 | C22×C8 | C2×D9 | 1 | 1 | yes | characteristic |
| 147 | C4⋊Q8 | — | 9 | 2 | no | — |
| 148 | C4⋊Q8 | — | 18 | 2 | no | — |
| 149 | C22⋊Q8 | — | 9 | 2 | no | — |
| 150 | C22⋊Q8 | — | 9 | 2 | no | — |
| 151 | C22⋊Q8 | — | 18 | 2 | no | — |
| 152 | C22⋊Q8 | — | 18 | 2 | no | — |
| 153 | C4×Q8 | — | 9 | 2 | no | — |
| 154 | C4×D4 | — | 9 | 1 | no | — |
| 155 | C4×D4 | — | 18 | 2 | no | — |
| 156 | #107 | — | 9 | 1 | no | — |
| 157 | C4⋊C8 | — | 9 | 1 | no | — |
| 158 | #103 | — | 9 | 2 | no | — |
| 159 | #103 | — | 9 | 2 | no | — |
| 160 | #103 | — | 9 | 2 | no | — |
| 161 | C22⋊C8 | — | 9 | 1 | no | — |
| 162 | C22⋊C8 | — | 9 | 1 | no | — |
| 163 | C4×C8 | — | 9 | 1 | no | — |
| 164 | C23×C3 | D3×D4 | 1 | 1 | yes | characteristic |
| 165 | C22×D3 | — | 3 | 1 | no | — |
| 166 | C3×Q8 | — | 3 | 2 | no | — |
| 167 | C3×Q8 | — | 6 | 2 | no | — |
| 168 | C3×Q8 | — | 6 | 2 | no | — |
| 169 | C3×Q8 | C2×S4 | 1 | 2 | yes | — |
| 170 | C2×C3×C4 | — | 3 | 1 | no | — |
| 171 | C2×C3×C4 | — | 3 | 1 | no | — |
| 172 | C2×C3×C4 | — | 6 | 2 | no | — |
| 173 | C2×C3×C4 | — | 6 | 2 | no | — |
| 174 | #67 | — | 3 | 2 | no | — |
| 175 | #67 | — | 3 | 2 | no | — |
| 176 | C2×Dic3 | — | 3 | 1 | no | — |
| 177 | C2×Dic3 | — | 3 | 1 | no | — |
| 178 | C2×Dic3 | — | 3 | 1 | no | — |
| 179 | C2×Dic3 | — | 6 | 2 | no | — |
| 180 | C2×Dic3 | — | 6 | 2 | no | — |
| 181 | C4×D3 | — | 6 | 1 | no | — |
| 182 | C4×D3 | — | 6 | 1 | no | — |
| 183 | C4×D3 | — | 12 | 2 | no | — |
| 184 | C4×D3 | — | 12 | 2 | no | — |
| 185 | Dic6 | — | 6 | 2 | no | — |
| 186 | Dic6 | — | 6 | 2 | no | — |
| 187 | Dic6 | — | 12 | 2 | no | — |
| 188 | Dic6 | — | 12 | 2 | no | — |
| 189 | C3×C8 | — | 3 | 1 | no | — |
| 190 | C3×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 191 | C3⋊C8 | — | 6 | 1 | no | — |
| 192 | C3⋊C8 | — | 6 | 1 | no | — |
| 193 | C2×C9 | — | 4 | 1 | no | — |
| 194 | D9 | — | 4 | 2 | no | — |
| 195 | C2×Q8 | — | 3 | 2 | no | — |
| 196 | C2×Q8 | — | 3 | 2 | no | — |
| 197 | C2×Q8 | — | 6 | 2 | no | — |
| 198 | C2×Q8 | — | 9 | 2 | no | — |
| 199 | C2×Q8 | — | 18 | 2 | no | — |
| 200 | C2×Q8 | — | 18 | 2 | no | — |
| 201 | C2×Q8 | — | 18 | 2 | no | — |
| 202 | C2×Q8 | — | 36 | 2 | no | — |
| 203 | C2×D4 | — | 9 | 1 | no | — |
| 204 | C22×C4 | — | 2 | 2 | no | — |
| 205 | C22×C4 | — | 9 | 1 | no | — |
| 206 | C22×C4 | — | 18 | 2 | no | — |
| 207 | C22×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 208 | Q16 | — | 3 | 2 | no | — |
| 209 | Q16 | — | 3 | 1 | no | — |
| 210 | Q16 | — | 6 | 1 | no | — |
| 211 | Q16 | — | 18 | 1 | no | — |
| 212 | Q16 | — | 18 | 2 | no | — |
| 213 | Q16 | — | 18 | 2 | no | — |
| 214 | Q16 | — | 18 | 1 | no | — |
| 215 | Q16 | — | 36 | 2 | no | — |
| 216 | Q16 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic, direct product factor |
| 217 | C2×C8 | — | 3 | 1 | no | — |
| 218 | C2×C8 | — | 3 | 1 | no | — |
| 219 | C2×C8 | — | 9 | 1 | no | — |
| 220 | C2×C8 | — | 9 | 1 | no | — |
| 221 | C2×C8 | — | 18 | 1 | no | — |
| 222 | C2×C8 | — | 18 | 1 | no | — |
| 223 | C4⋊C4 | — | 9 | 1 | no | — |
| 224 | C4⋊C4 | — | 9 | 2 | no | — |
| 225 | C4⋊C4 | — | 18 | 2 | no | — |
| 226 | C4⋊C4 | — | 18 | 2 | no | — |
| 227 | C4⋊C4 | — | 18 | 2 | no | — |
| 228 | C4⋊C4 | — | 18 | 2 | no | — |
| 229 | C4⋊C4 | — | 18 | 2 | no | — |
| 230 | C22⋊C4 | — | 9 | 1 | no | — |
| 231 | C22⋊C4 | — | 9 | 1 | no | — |
| 232 | C22⋊C4 | — | 18 | 2 | no | — |
| 233 | C22⋊C4 | — | 18 | 2 | no | — |
| 234 | C42 | — | 9 | 1 | no | — |
| 235 | C42 | — | 18 | 2 | no | — |
| 236 | C22×C3 | — | 3 | 1 | no | — |
| 237 | C22×C3 | — | 3 | 1 | no | — |
| 238 | C22×C3 | D3×Q16 | 1 | 1 | yes | characteristic |
| 239 | C2×D3 | — | 3 | 2 | no | — |
| 240 | C2×D3 | — | 6 | 1 | no | — |
| 241 | C2×D3 | — | 6 | 1 | no | — |
| 242 | C3×C4 | — | 2 | 2 | no | — |
| 243 | C3×C4 | — | 3 | 1 | no | — |
| 244 | C3×C4 | — | 6 | 2 | no | — |
| 245 | C3×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 246 | Dic3 | — | 3 | 2 | no | — |
| 247 | Dic3 | — | 6 | 1 | no | — |
| 248 | Dic3 | — | 6 | 1 | no | — |
| 249 | Dic3 | — | 12 | 2 | no | — |
| 250 | Dic3 | — | 12 | 2 | no | — |
| 251 | C9 | — | 4 | 1 | no | Sylow |
| 252 | C23 | — | 9 | 1 | no | — |
| 253 | C23 | D4×D9 | 1 | 1 | yes | characteristic |
| 254 | Q8 | — | 3 | 2 | no | — |
| 255 | Q8 | — | 6 | 2 | no | — |
| 256 | Q8 | — | 6 | 2 | no | — |
| 257 | Q8 | — | 18 | 2 | no | — |
| 258 | Q8 | — | 18 | 2 | no | — |
| 259 | Q8 | — | 36 | 2 | no | — |
| 260 | Q8 | — | 36 | 2 | no | — |
| 261 | Q8 | C2×(C22⋊C9)⋊C2 | 1 | 2 | yes | — |
| 262 | D4 | — | 9 | 2 | no | — |
| 263 | D4 | — | 9 | 2 | no | — |
| 264 | C2×C4 | — | 3 | 1 | no | — |
| 265 | C2×C4 | — | 3 | 1 | no | — |
| 266 | C2×C4 | — | 6 | 2 | no | — |
| 267 | C2×C4 | — | 6 | 2 | no | — |
| 268 | C2×C4 | — | 9 | 1 | no | — |
| 269 | C2×C4 | — | 9 | 1 | no | — |
| 270 | C2×C4 | — | 9 | 1 | no | — |
| 271 | C2×C4 | — | 18 | 1 | no | — |
| 272 | C2×C4 | — | 18 | 1 | no | — |
| 273 | C2×C4 | — | 18 | 2 | no | — |
| 274 | C2×C4 | — | 18 | 2 | no | — |
| 275 | C2×C4 | — | 36 | 2 | no | — |
| 276 | C2×C4 | — | 36 | 2 | no | — |
| 277 | C8 | — | 3 | 1 | no | — |
| 278 | C8 | — | 18 | 1 | no | — |
| 279 | C8 | — | 18 | 1 | no | — |
| 280 | C8 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 281 | C2×C3 | — | 3 | 1 | no | — |
| 282 | C2×C3 | — | 3 | 1 | no | — |
| 283 | C2×C3 | D4×S4 | 1 | 1 | yes | characteristic |
| 284 | D3 | — | 6 | 2 | no | — |
| 285 | C22 | — | 3 | 1 | no | — |
| 286 | C22 | — | 3 | 1 | no | — |
| 287 | C22 | — | 9 | 2 | no | — |
| 288 | C22 | — | 18 | 1 | no | — |
| 289 | C22 | — | 18 | 1 | no | — |
| 290 | C22 | Q16×D9 | 1 | 1 | yes | characteristic, complemented |
| 291 | C4 | — | 2 | 2 | no | — |
| 292 | C4 | — | 3 | 1 | no | — |
| 293 | C4 | — | 6 | 2 | no | — |
| 294 | C4 | — | 9 | 2 | no | — |
| 295 | C4 | — | 18 | 1 | no | — |
| 296 | C4 | — | 18 | 1 | no | — |
| 297 | C4 | — | 36 | 2 | no | — |
| 298 | C4 | — | 36 | 2 | no | — |
| 299 | C4 | C22×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 300 | C3 | Q16×S4 | 1 | 1 | yes | characteristic |
| 301 | C2 | — | 3 | 1 | no | — |
| 302 | C2 | — | 3 | 1 | no | — |
| 303 | C2 | — | 18 | 2 | no | — |
| 304 | C2 | D4×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q16×C22⋊C9 | C2 | semidirect |
| 3. #10589518 | C2 | semidirect |
| 5. #10589500 | C2 | semidirect |
| 68. (C22⋊C9)⋊C2 | Q16 | direct (N × H) |
| 77. C22×Q16 | D9 | semidirect |
| 129. C22⋊C9 | C2×Q16 | semidirect |
| 216. Q16 | (C22⋊C9)⋊C2 | direct (N × H) |
| 290. C22 | Q16×D9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×(C22⋊C9)⋊C2 | 2 |
| Q16×C22⋊C9 | 2 |
| #10589518 | 2 |
| C8×(C22⋊C9)⋊C2 | 2 |
| #10589500 | 2 |
| Q16×#67 | 3 |
| Q16×D9 | 4 |