Finite Group DatabasebetaFinite Group DatabasebetaOrder 9344.
578 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 68 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 | #17989654 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D73⋊#99 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #17989528 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #17989464 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #120×D73 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | D4×#64408 | — | 2 | 4 | no | — |
| 7 | #14556670 | — | 2 | 1 | no | — |
| 8 | C2×C22⋊#64408 | — | 2 | 4 | no | — |
| 9 | C2×C22⋊#64408 | — | 2 | 1 | no | — |
| 10 | C2×C22⋊#64408 | C22 | 1 | 2 | yes | — |
| 11 | C2×C22⋊#64408 | C22 | 1 | 2 | yes | — |
| 12 | #14556650 | — | 2 | 1 | no | — |
| 13 | #14556649 | — | 2 | 1 | no | — |
| 14 | C24×D73 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×D4×D73 | — | 2 | 1 | no | — |
| 16 | C2×D4×D73 | C22 | 1 | 1 | yes | characteristic |
| 17 | #14556561 | — | 2 | 1 | no | — |
| 18 | #14556561 | C4 | 1 | 1 | yes | characteristic, complemented |
| 19 | #14556560 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | #14556559 | — | 2 | 1 | no | — |
| 21 | #14556559 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | C22⋊C4×D73 | — | 2 | 1 | no | — |
| 23 | C22⋊C4×D73 | C22 | 1 | 1 | yes | characteristic |
| 24 | #120×C73 | C4 | 1 | 1 | yes | characteristic, complemented |
| 25 | C22×#64408 | — | 2 | 2 | no | — |
| 26 | C22×#64408 | — | 2 | 2 | no | — |
| 27 | C22×#64408 | D4 | 1 | 4 | yes | complemented |
| 28 | C4⋊#64408 | — | 2 | 2 | no | — |
| 29 | C22⋊#64408 | — | 2 | 2 | no | — |
| 30 | C22⋊#64408 | — | 2 | 2 | no | — |
| 31 | C22⋊#64408 | — | 2 | 2 | no | — |
| 32 | C22⋊#64408 | — | 2 | 2 | no | — |
| 33 | C22⋊#64408 | — | 2 | 2 | no | — |
| 34 | C22⋊#64408 | — | 4 | 8 | no | — |
| 35 | C4×#64408 | — | 2 | 2 | no | — |
| 36 | C23×D73 | — | 2 | 1 | no | — |
| 37 | C23×D73 | — | 2 | 2 | no | — |
| 38 | C23×D73 | — | 2 | 1 | no | — |
| 39 | C23×D73 | — | 2 | 2 | no | — |
| 40 | C23×D73 | — | 4 | 4 | no | — |
| 41 | C23×D73 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 42 | C23×D73 | D4 | 1 | 1 | yes | characteristic |
| 43 | C23×D73 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×D292 | — | 2 | 1 | no | — |
| 45 | C2×D292 | D4 | 1 | 1 | yes | characteristic |
| 46 | C2×C22⋊D73 | — | 2 | 1 | no | — |
| 47 | C2×C22⋊D73 | — | 2 | 1 | no | — |
| 48 | C2×C22⋊D73 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C2×C22⋊D73 | D4 | 1 | 1 | yes | characteristic |
| 50 | D4×D73 | — | 2 | 2 | no | — |
| 51 | D4×D73 | — | 2 | 2 | no | — |
| 52 | D4×D73 | — | 4 | 4 | no | — |
| 53 | C2×C4×D73 | — | 2 | 1 | no | — |
| 54 | C2×C4×D73 | D4 | 1 | 1 | yes | characteristic |
| 55 | C22⋊Dic73 | — | 2 | 1 | no | — |
| 56 | C22⋊Dic73 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 57 | #11924005 | — | 2 | 1 | no | — |
| 58 | #11924005 | — | 2 | 1 | no | — |
| 59 | #11924005 | — | 2 | 1 | no | — |
| 60 | C24×C73 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 61 | C2×D4×C73 | — | 2 | 1 | no | — |
| 62 | C2×D4×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 63 | C22⋊C4×C73 | — | 2 | 1 | no | — |
| 64 | C22⋊C4×C73 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 65 | C23×C73 | — | 2 | 1 | no | — |
| 66 | C23×C73 | — | 2 | 2 | no | — |
| 67 | C23×C73 | — | 4 | 2 | no | — |
| 68 | C23×C73 | C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 69 | C23×C73 | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | C22×D73 | — | 2 | 4 | no | — |
| 71 | C22×D73 | — | 2 | 8 | no | — |
| 72 | C22×D73 | — | 2 | 1 | no | — |
| 73 | C22×D73 | — | 2 | 2 | no | — |
| 74 | C22×D73 | — | 2 | 1 | no | — |
| 75 | C22×D73 | — | 2 | 1 | no | — |
| 76 | C22×D73 | — | 4 | 2 | no | — |
| 77 | C22×D73 | — | 4 | 2 | no | — |
| 78 | C22×D73 | — | 4 | 2 | no | — |
| 79 | C22×D73 | — | 4 | 4 | no | — |
| 80 | C22×D73 | — | 4 | 2 | no | — |
| 81 | C22×D73 | — | 4 | 4 | no | — |
| 82 | C22×D73 | — | 4 | 2 | no | — |
| 83 | C22×D73 | — | 4 | 2 | no | — |
| 84 | C22×D73 | — | 4 | 4 | no | — |
| 85 | C22×D73 | — | 4 | 1 | no | — |
| 86 | C22×D73 | — | 4 | 2 | no | — |
| 87 | C22×D73 | — | 4 | 1 | no | — |
| 88 | C22×D73 | C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 89 | C22×D73 | C2×D4 | 1 | 2 | yes | — |
| 90 | C22×D73 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C22×D73 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C22×D73 | #41 | 1 | 2 | yes | — |
| 93 | C2×#64408 | — | 2 | 4 | no | — |
| 94 | C2×#64408 | — | 4 | 4 | no | — |
| 95 | C2×#64408 | — | 4 | 4 | no | — |
| 96 | C2×#64408 | — | 4 | 2 | no | — |
| 97 | C2×#64408 | — | 8 | 2 | no | — |
| 98 | D4×C73 | — | 2 | 2 | no | — |
| 99 | D4×C73 | — | 4 | 2 | no | — |
| 100 | C2×C4×C73 | — | 2 | 1 | no | — |
| 101 | C2×C4×C73 | C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 102 | C22⋊D73 | — | 2 | 2 | no | — |
| 103 | C22⋊D73 | — | 2 | 2 | no | — |
| 104 | C22⋊D73 | — | 4 | 2 | no | — |
| 105 | C22⋊D73 | — | 4 | 2 | no | — |
| 106 | C2×Dic73 | — | 2 | 1 | no | — |
| 107 | C2×Dic73 | C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 108 | D292 | — | 2 | 2 | no | — |
| 109 | D292 | — | 4 | 2 | no | — |
| 110 | C4×D73 | — | 2 | 1 | no | — |
| 111 | C4×D73 | — | 2 | 1 | no | — |
| 112 | C4×D73 | — | 4 | 2 | no | — |
| 113 | C22×C73 | — | 2 | 2 | no | — |
| 114 | C22×C73 | — | 2 | 4 | no | — |
| 115 | C22×C73 | — | 2 | 1 | no | — |
| 116 | C22×C73 | — | 4 | 1 | no | — |
| 117 | C22×C73 | — | 4 | 1 | no | — |
| 118 | C22×C73 | — | 4 | 1 | no | — |
| 119 | C22×C73 | — | 4 | 2 | no | — |
| 120 | C22×C73 | — | 4 | 1 | no | — |
| 121 | C22×C73 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 122 | C22×C73 | C4×D4 | 1 | 2 | yes | — |
| 123 | C2×D73 | — | 2 | 2 | no | — |
| 124 | C2×D73 | — | 2 | 4 | no | — |
| 125 | C2×D73 | — | 2 | 2 | no | — |
| 126 | C2×D73 | — | 2 | 4 | no | — |
| 127 | C2×D73 | — | 2 | 1 | no | — |
| 128 | C2×D73 | — | 4 | 1 | no | — |
| 129 | C2×D73 | — | 4 | 2 | no | — |
| 130 | C2×D73 | — | 4 | 8 | no | — |
| 131 | C2×D73 | — | 4 | 4 | no | — |
| 132 | C2×D73 | — | 4 | 1 | no | — |
| 133 | C2×D73 | — | 4 | 2 | no | — |
| 134 | C2×D73 | — | 4 | 1 | no | — |
| 135 | C2×D73 | — | 4 | 2 | no | — |
| 136 | C2×D73 | — | 4 | 4 | no | — |
| 137 | C2×D73 | — | 4 | 1 | no | — |
| 138 | C2×D73 | — | 8 | 1 | no | — |
| 139 | C2×D73 | #120 | 1 | 1 | yes | characteristic |
| 140 | C2×D73 | #121 | 1 | 2 | yes | — |
| 141 | C2×D73 | #123 | 1 | 1 | yes | characteristic |
| 142 | #64408 | — | 4 | 4 | no | — |
| 143 | #64408 | — | 8 | 2 | no | — |
| 144 | C4×C73 | — | 2 | 1 | no | — |
| 145 | C4×C73 | — | 4 | 1 | no | — |
| 146 | Dic73 | — | 2 | 1 | no | — |
| 147 | Dic73 | — | 4 | 1 | no | — |
| 148 | C2×C73 | — | 2 | 1 | no | — |
| 149 | C2×C73 | — | 2 | 2 | no | — |
| 150 | C2×C73 | — | 4 | 2 | no | — |
| 151 | C2×C73 | — | 4 | 1 | no | — |
| 152 | C2×C73 | #386 | 1 | 1 | yes | characteristic |
| 153 | D73 | — | 2 | 2 | no | — |
| 154 | D73 | — | 2 | 1 | no | — |
| 155 | D73 | — | 4 | 2 | no | — |
| 156 | D73 | — | 4 | 1 | no | — |
| 157 | D73 | B(4,𝔽2) | 1 | 2 | yes | — |
| 158 | #1889 | — | 73 | 1 | no | maximal, Sylow |
| 159 | C73 | #1889 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 160 | C2×#120 | — | 73 | 1 | no | — |
| 161 | #518 | — | 73 | 1 | no | — |
| 162 | C2×#99 | — | 73 | 1 | no | — |
| 163 | #379 | — | 73 | 2 | no | — |
| 164 | #342 | — | 73 | 2 | no | — |
| 165 | C25 | — | 73 | 1 | no | — |
| 166 | C22×D4 | — | 73 | 1 | no | — |
| 167 | C22×D4 | — | 146 | 1 | no | — |
| 168 | #120 | — | 73 | 1 | no | — |
| 169 | #120 | — | 73 | 1 | no | — |
| 170 | #120 | — | 73 | 1 | no | — |
| 171 | #120 | — | 146 | 1 | no | — |
| 172 | #120 | — | 146 | 1 | no | — |
| 173 | #120 | #64408 | 1 | 1 | yes | characteristic, complemented |
| 174 | C4×D4 | — | 146 | 4 | no | — |
| 175 | #117 | — | 146 | 1 | no | — |
| 176 | C2×C22⋊C4 | — | 73 | 1 | no | — |
| 177 | C2×C22⋊C4 | — | 73 | 2 | no | — |
| 178 | C2×C22⋊C4 | — | 73 | 2 | no | — |
| 179 | C2×C22⋊C4 | — | 146 | 1 | no | — |
| 180 | C2×C22⋊C4 | — | 146 | 4 | no | — |
| 181 | C2×C22⋊C4 | — | 146 | 1 | no | — |
| 182 | #99 | — | 146 | 1 | no | — |
| 183 | #99 | — | 146 | 1 | no | — |
| 184 | C24 | — | 292 | 4 | no | — |
| 185 | C24 | — | 73 | 1 | no | — |
| 186 | C24 | — | 73 | 1 | no | — |
| 187 | C24 | — | 73 | 1 | no | — |
| 188 | C24 | — | 146 | 1 | no | — |
| 189 | C24 | — | 146 | 2 | no | — |
| 190 | C24 | — | 146 | 2 | no | — |
| 191 | C24 | — | 146 | 1 | no | — |
| 192 | C24 | C2×#64408 | 1 | 1 | yes | characteristic, complemented |
| 193 | C2×D4 | — | 2 | 1 | no | — |
| 194 | C2×D4 | — | 292 | 4 | no | — |
| 195 | C2×D4 | — | 73 | 1 | no | — |
| 196 | C2×D4 | — | 73 | 1 | no | — |
| 197 | C2×D4 | — | 73 | 1 | no | — |
| 198 | C2×D4 | — | 146 | 1 | no | — |
| 199 | C2×D4 | — | 146 | 2 | no | — |
| 200 | C2×D4 | — | 146 | 2 | no | — |
| 201 | C2×D4 | — | 146 | 1 | no | — |
| 202 | C2×D4 | — | 146 | 1 | no | — |
| 203 | C2×D4 | C2×#64408 | 1 | 1 | yes | characteristic |
| 204 | C22×C4 | — | 73 | 4 | no | — |
| 205 | C22×C4 | — | 73 | 1 | no | — |
| 206 | C22×C4 | — | 146 | 1 | no | — |
| 207 | C22×C4 | — | 146 | 2 | no | — |
| 208 | C22×C4 | — | 146 | 2 | no | — |
| 209 | C4⋊C4 | — | 146 | 2 | no | — |
| 210 | C22⋊C4 | — | 2 | 1 | no | — |
| 211 | C22⋊C4 | — | 292 | 8 | no | — |
| 212 | C22⋊C4 | — | 73 | 1 | no | — |
| 213 | C22⋊C4 | — | 146 | 1 | no | — |
| 214 | C22⋊C4 | — | 146 | 1 | no | — |
| 215 | C22⋊C4 | — | 146 | 2 | no | — |
| 216 | C22⋊C4 | — | 146 | 2 | no | — |
| 217 | C22⋊C4 | — | 146 | 1 | no | — |
| 218 | C22⋊C4 | — | 146 | 2 | no | — |
| 219 | C22⋊C4 | — | 146 | 2 | no | — |
| 220 | C22⋊C4 | — | 146 | 2 | no | — |
| 221 | C22⋊C4 | — | 146 | 1 | no | — |
| 222 | C22⋊C4 | C2×#64408 | 1 | 1 | yes | characteristic, complemented |
| 223 | C42 | — | 146 | 2 | no | — |
| 224 | C23 | — | 2 | 1 | no | — |
| 225 | C23 | — | 2 | 2 | no | — |
| 226 | C23 | — | 4 | 2 | no | — |
| 227 | C23 | — | 292 | 2 | no | — |
| 228 | C23 | — | 292 | 4 | no | — |
| 229 | C23 | — | 292 | 1 | no | — |
| 230 | C23 | — | 292 | 2 | no | — |
| 231 | C23 | — | 292 | 1 | no | — |
| 232 | C23 | — | 292 | 4 | no | — |
| 233 | C23 | — | 292 | 2 | no | — |
| 234 | C23 | — | 292 | 2 | no | — |
| 235 | C23 | — | 292 | 4 | no | — |
| 236 | C23 | — | 292 | 2 | no | — |
| 237 | C23 | — | 292 | 2 | no | — |
| 238 | C23 | — | 292 | 2 | no | — |
| 239 | C23 | — | 73 | 1 | no | — |
| 240 | C23 | — | 73 | 2 | no | — |
| 241 | C23 | — | 73 | 1 | no | — |
| 242 | C23 | — | 73 | 1 | no | — |
| 243 | C23 | — | 73 | 2 | no | — |
| 244 | C23 | — | 146 | 1 | no | — |
| 245 | C23 | — | 146 | 8 | no | — |
| 246 | C23 | — | 146 | 4 | no | — |
| 247 | C23 | — | 146 | 1 | no | — |
| 248 | C23 | — | 146 | 1 | no | — |
| 249 | C23 | — | 146 | 2 | no | — |
| 250 | C23 | C22⋊#64408 | 1 | 1 | yes | characteristic, complemented |
| 251 | C23 | C22×#64408 | 1 | 1 | yes | characteristic |
| 252 | D4 | — | 2 | 2 | no | — |
| 253 | D4 | — | 4 | 2 | no | — |
| 254 | D4 | — | 292 | 2 | no | — |
| 255 | D4 | — | 292 | 2 | no | — |
| 256 | D4 | — | 292 | 2 | no | — |
| 257 | D4 | — | 146 | 2 | no | — |
| 258 | D4 | — | 146 | 2 | no | — |
| 259 | D4 | — | 146 | 2 | no | — |
| 260 | C2×C4 | — | 2 | 1 | no | — |
| 261 | C2×C4 | — | 292 | 2 | no | — |
| 262 | C2×C4 | — | 292 | 4 | no | — |
| 263 | C2×C4 | — | 292 | 2 | no | — |
| 264 | C2×C4 | — | 292 | 4 | no | — |
| 265 | C2×C4 | — | 584 | 2 | no | — |
| 266 | C2×C4 | — | 73 | 1 | no | — |
| 267 | C2×C4 | — | 146 | 1 | no | — |
| 268 | C2×C4 | — | 146 | 1 | no | — |
| 269 | C2×C4 | — | 146 | 4 | no | — |
| 270 | C2×C4 | — | 146 | 1 | no | — |
| 271 | C2×C4 | C22⋊#64408 | 1 | 1 | yes | characteristic, complemented |
| 272 | C22 | — | 2 | 2 | no | — |
| 273 | C22 | — | 2 | 4 | no | — |
| 274 | C22 | — | 2 | 1 | no | — |
| 275 | C22 | — | 4 | 1 | no | — |
| 276 | C22 | — | 4 | 1 | no | — |
| 277 | C22 | — | 4 | 1 | no | — |
| 278 | C22 | — | 4 | 2 | no | — |
| 279 | C22 | — | 4 | 1 | no | — |
| 280 | C22 | — | 292 | 1 | no | — |
| 281 | C22 | — | 292 | 1 | no | — |
| 282 | C22 | — | 292 | 2 | no | — |
| 283 | C22 | — | 292 | 1 | no | — |
| 284 | C22 | — | 292 | 2 | no | — |
| 285 | C22 | — | 292 | 2 | no | — |
| 286 | C22 | — | 292 | 4 | no | — |
| 287 | C22 | — | 292 | 8 | no | — |
| 288 | C22 | — | 292 | 4 | no | — |
| 289 | C22 | — | 292 | 1 | no | — |
| 290 | C22 | — | 584 | 1 | no | — |
| 291 | C22 | — | 73 | 1 | no | — |
| 292 | C22 | — | 73 | 2 | no | — |
| 293 | C22 | — | 73 | 1 | no | — |
| 294 | C22 | — | 146 | 2 | no | — |
| 295 | C22 | — | 146 | 4 | no | — |
| 296 | C22 | — | 146 | 4 | no | — |
| 297 | C22 | — | 146 | 2 | no | — |
| 298 | C22 | — | 146 | 1 | no | — |
| 299 | C22 | C2×C22⋊#64408 | 1 | 1 | yes | characteristic |
| 300 | C22 | D4×#64408 | 1 | 2 | yes | — |
| 301 | C4 | — | 2 | 1 | no | — |
| 302 | C4 | — | 4 | 1 | no | — |
| 303 | C4 | — | 292 | 1 | no | — |
| 304 | C4 | — | 292 | 4 | no | — |
| 305 | C4 | — | 584 | 2 | no | — |
| 306 | C4 | — | 146 | 1 | no | — |
| 307 | C2 | — | 2 | 1 | no | — |
| 308 | C2 | — | 2 | 2 | no | — |
| 309 | C2 | — | 4 | 2 | no | — |
| 310 | C2 | — | 4 | 1 | no | — |
| 311 | C2 | — | 292 | 2 | no | — |
| 312 | C2 | — | 292 | 1 | no | — |
| 313 | C2 | — | 73 | 2 | no | — |
| 314 | C2 | — | 146 | 2 | no | — |
| 315 | C2 | — | 146 | 1 | no | — |
| 316 | C2 | #17989535 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #17989535 |
| commutator | C23×C73 | C22×C4 |
| Frattini | C23 | C22×#64408 |
| Fitting | #120×C73 | C4 |
| socle | C2×C73 | #386 |
| hypercenter | #120 | #64408 |
| hypocenter | C73 | #1889 |
| LCS | G ▷ C23×C73 ▷ C2×C73 ▷ C73 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #17989654 | C2 | semidirect |
| 2. D73⋊#99 | C2 | semidirect |
| 3. #17989528 | C2 | semidirect |
| 4. #17989464 | C2 | semidirect |
| 18. #14556561 | C4 | semidirect |
| 19. #14556560 | C4 | semidirect |
| 21. #14556559 | C4 | semidirect |
| 24. #120×C73 | C4 | semidirect |
| 27. C22×#64408 | D4 | semidirect |
| 41. C23×D73 | C2×C4 | semidirect |
| 56. C22⋊Dic73 | C2×C4 | semidirect |
| 60. C24×C73 | C2×C4 | semidirect |
| 64. C22⋊C4×C73 | C2×C4 | semidirect |
| 68. C23×C73 | C22⋊C4 | semidirect |
| 88. C22×D73 | C22⋊C4 | semidirect |
| 101. C2×C4×C73 | C22⋊C4 | semidirect |
| 107. C2×Dic73 | C22⋊C4 | semidirect |
| 159. C73 | #1889 | semidirect |
| 173. #120 | #64408 | semidirect |
| 192. C24 | C2×#64408 | semidirect |
| 222. C22⋊C4 | C2×#64408 | semidirect |
| 250. C23 | C22⋊#64408 | semidirect |
| 271. C2×C4 | C22⋊#64408 | semidirect |