Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
476 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 55 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 | #22626940 | C2 | 1 | 2 | yes | maximal |
| 2 | #22625121 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C22⋊#10596842 | — | 2 | 2 | no | — |
| 4 | C2×(C2×#64408)⋊C4 | — | 2 | 2 | no | — |
| 5 | C2×(C2×#64408)⋊C4 | C4 | 1 | 1 | yes | characteristic |
| 6 | C2×(C2×#64408)⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×Q8×D73 | C4 | 1 | 1 | yes | characteristic |
| 8 | C22×#10596842 | — | 4 | 2 | no | — |
| 9 | C22⋊#10596842 | — | 2 | 4 | no | — |
| 10 | C22⋊#10596842 | — | 4 | 2 | no | — |
| 11 | C23×#64408 | — | 2 | 1 | no | — |
| 12 | C23×#64408 | D4 | 1 | 2 | yes | — |
| 13 | (C2×#64408)⋊C4 | — | 4 | 1 | no | — |
| 14 | (C2×#64408)⋊C4 | — | 4 | 4 | no | — |
| 15 | (C2×#64408)⋊C4 | — | 4 | 1 | no | — |
| 16 | C22×Dic146 | — | 2 | 1 | no | — |
| 17 | C22×Dic146 | C8 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×Q8×D73 | — | 4 | 4 | no | — |
| 19 | C2×Q8×D73 | — | 4 | 1 | no | — |
| 20 | C2×Q8×D73 | — | 4 | 1 | no | — |
| 21 | C22×C4×D73 | — | 2 | 1 | no | — |
| 22 | C22×C4×D73 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C22×Q8×C73 | C8 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×#10596842 | — | 4 | 4 | no | — |
| 25 | C2×#10596842 | — | 8 | 2 | no | — |
| 26 | C2×#10596842 | — | 8 | 2 | no | — |
| 27 | C22×#64408 | — | 2 | 2 | no | — |
| 28 | C22×#64408 | — | 2 | 2 | no | — |
| 29 | C22×#64408 | — | 2 | 2 | no | — |
| 30 | C22×#64408 | — | 4 | 2 | no | — |
| 31 | C22×#64408 | — | 4 | 2 | no | — |
| 32 | C22×#64408 | — | 4 | 1 | no | — |
| 33 | C22×#64408 | — | 4 | 2 | no | — |
| 34 | C22×#64408 | — | 4 | 1 | no | — |
| 35 | C22×#64408 | — | 4 | 2 | no | — |
| 36 | C22×#64408 | SD16 | 1 | 2 | yes | — |
| 37 | C22×#64408 | Q16 | 1 | 2 | yes | — |
| 38 | C23×D73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 39 | C2×Dic146 | — | 4 | 2 | no | — |
| 40 | C2×Dic146 | — | 4 | 2 | no | — |
| 41 | C2×Dic146 | — | 4 | 2 | no | — |
| 42 | C2×Dic146 | — | 4 | 1 | no | — |
| 43 | C2×Dic146 | — | 4 | 2 | no | — |
| 44 | Q8×D73 | — | 4 | 4 | no | — |
| 45 | Q8×D73 | — | 4 | 4 | no | — |
| 46 | Q8×D73 | — | 4 | 4 | no | — |
| 47 | Q8×D73 | — | 4 | 4 | no | — |
| 48 | C22×Dic73 | — | 2 | 1 | no | — |
| 49 | C22×Dic73 | M16 | 1 | 1 | yes | characteristic |
| 50 | C2×C4×D73 | — | 2 | 4 | no | — |
| 51 | C2×C4×D73 | — | 2 | 1 | no | — |
| 52 | C2×C4×D73 | — | 2 | 1 | no | — |
| 53 | C2×C4×D73 | — | 4 | 2 | no | — |
| 54 | C2×C4×D73 | — | 4 | 1 | no | — |
| 55 | C2×C4×D73 | — | 4 | 1 | no | — |
| 56 | C2×C4×D73 | — | 4 | 2 | no | — |
| 57 | C2×Q8×C73 | — | 4 | 2 | no | — |
| 58 | C2×Q8×C73 | — | 4 | 1 | no | — |
| 59 | C22×C4×C73 | — | 2 | 1 | no | — |
| 60 | C22×C4×C73 | C2×C8 | 1 | 1 | yes | characteristic |
| 61 | C23×C73 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 62 | C22×D73 | — | 2 | 2 | no | — |
| 63 | C22×D73 | — | 2 | 1 | no | — |
| 64 | C22×D73 | — | 2 | 1 | no | — |
| 65 | C22×D73 | #99 | 1 | 1 | yes | characteristic |
| 66 | C22×D73 | #100 | 1 | 1 | yes | characteristic |
| 67 | C22×D73 | #103 | 1 | 2 | yes | — |
| 68 | C22×D73 | #104 | 1 | 2 | yes | — |
| 69 | C2×#64408 | — | 2 | 2 | no | — |
| 70 | C2×#64408 | — | 2 | 2 | no | — |
| 71 | C2×#64408 | — | 4 | 2 | no | — |
| 72 | C2×#64408 | — | 4 | 2 | no | — |
| 73 | C2×#64408 | — | 4 | 2 | no | — |
| 74 | C2×#64408 | — | 4 | 2 | no | — |
| 75 | C2×#64408 | — | 4 | 2 | no | — |
| 76 | C2×#64408 | — | 4 | 4 | no | — |
| 77 | C2×#64408 | — | 8 | 2 | no | — |
| 78 | C2×#64408 | — | 8 | 1 | no | — |
| 79 | C2×#64408 | — | 8 | 2 | no | — |
| 80 | C2×#64408 | — | 8 | 1 | no | — |
| 81 | #10596842 | — | 8 | 4 | no | — |
| 82 | Q8×C73 | — | 4 | 2 | no | — |
| 83 | Q8×C73 | — | 4 | 2 | no | — |
| 84 | C2×C4×C73 | — | 2 | 2 | no | — |
| 85 | C2×C4×C73 | — | 2 | 1 | no | — |
| 86 | C2×C4×C73 | — | 4 | 1 | no | — |
| 87 | C2×C4×C73 | — | 4 | 1 | no | — |
| 88 | C2×C4×C73 | — | 4 | 1 | no | — |
| 89 | C2×Dic73 | — | 2 | 2 | no | — |
| 90 | C2×Dic73 | — | 2 | 1 | no | — |
| 91 | C2×Dic73 | — | 4 | 1 | no | — |
| 92 | C2×Dic73 | — | 4 | 1 | no | — |
| 93 | C2×Dic73 | — | 4 | 1 | no | — |
| 94 | C4×D73 | — | 4 | 2 | no | — |
| 95 | C4×D73 | — | 4 | 4 | no | — |
| 96 | C4×D73 | — | 4 | 4 | no | — |
| 97 | C4×D73 | — | 4 | 2 | no | — |
| 98 | Dic146 | — | 4 | 8 | no | — |
| 99 | Dic146 | — | 4 | 2 | no | — |
| 100 | Dic146 | — | 4 | 2 | no | — |
| 101 | C22×C73 | — | 2 | 1 | no | — |
| 102 | C22×C73 | — | 2 | 1 | no | — |
| 103 | C22×C73 | #323 | 1 | 1 | yes | characteristic |
| 104 | C22×C73 | Q8⋊C8 | 1 | 2 | yes | — |
| 105 | C2×D73 | — | 2 | 1 | no | — |
| 106 | C2×D73 | — | 2 | 1 | no | — |
| 107 | C2×D73 | — | 2 | 2 | no | — |
| 108 | C2×D73 | — | 2 | 2 | no | — |
| 109 | C2×D73 | — | 2 | 2 | no | — |
| 110 | C2×D73 | — | 2 | 2 | no | — |
| 111 | C2×D73 | #328 | 1 | 2 | yes | — |
| 112 | C2×D73 | #329 | 1 | 1 | yes | characteristic |
| 113 | C2×D73 | #333 | 1 | 1 | yes | characteristic |
| 114 | C2×D73 | AΣL(1,𝔽16) | 1 | 2 | yes | — |
| 115 | C2×D73 | #354 | 1 | 1 | yes | characteristic |
| 116 | C2×D73 | #356 | 1 | 1 | yes | characteristic |
| 117 | #64408 | — | 4 | 2 | no | — |
| 118 | #64408 | — | 4 | 2 | no | — |
| 119 | #64408 | — | 8 | 2 | no | — |
| 120 | C4×C73 | — | 4 | 2 | no | — |
| 121 | C4×C73 | — | 4 | 1 | no | — |
| 122 | Dic73 | — | 4 | 1 | no | — |
| 123 | Dic73 | — | 4 | 2 | no | — |
| 124 | #7131 | — | 73 | 1 | no | maximal, Sylow |
| 125 | C2×C73 | — | 2 | 2 | no | — |
| 126 | C2×C73 | #1272 | 1 | 1 | yes | characteristic |
| 127 | C2×C73 | #1316 | 1 | 1 | yes | characteristic |
| 128 | C2×C73 | #1319 | 1 | 1 | yes | characteristic |
| 129 | D73 | — | 2 | 1 | no | — |
| 130 | D73 | — | 2 | 1 | no | — |
| 131 | D73 | #1351 | 1 | 2 | yes | — |
| 132 | D73 | #1354 | 1 | 2 | yes | — |
| 133 | #1446 | — | 73 | 1 | no | — |
| 134 | #1280 | — | 73 | 2 | no | — |
| 135 | C73 | #7131 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 136 | C23×Q8 | — | 73 | 1 | no | — |
| 137 | C2×C22⋊C8 | — | 146 | 2 | no | — |
| 138 | C2×#95 | — | 73 | 1 | no | — |
| 139 | C2×#95 | — | 73 | 1 | no | — |
| 140 | C2×#95 | — | 146 | 2 | no | — |
| 141 | C22×Q8 | — | 292 | 4 | no | — |
| 142 | C22×Q8 | — | 292 | 1 | no | — |
| 143 | C22×Q8 | — | 292 | 1 | no | — |
| 144 | C22×Q8 | — | 73 | 1 | no | — |
| 145 | C22×Q8 | — | 146 | 1 | no | — |
| 146 | C22×Q8 | #10596842 | 1 | 1 | yes | characteristic, complemented |
| 147 | C23×C4 | — | 73 | 2 | no | — |
| 148 | C23×C4 | — | 73 | 1 | no | — |
| 149 | C23×C4 | — | 146 | 1 | no | — |
| 150 | C23×C4 | — | 146 | 1 | no | — |
| 151 | C22×C8 | — | 292 | 2 | no | — |
| 152 | C22⋊C8 | — | 292 | 2 | no | — |
| 153 | C22⋊C8 | — | 146 | 4 | no | — |
| 154 | #95 | — | 292 | 4 | no | — |
| 155 | #95 | — | 292 | 1 | no | — |
| 156 | #95 | — | 292 | 1 | no | — |
| 157 | C24 | — | 73 | 1 | no | — |
| 158 | C2×Q8 | — | 4 | 2 | no | — |
| 159 | C2×Q8 | — | 4 | 1 | no | — |
| 160 | C2×Q8 | — | 292 | 2 | no | — |
| 161 | C2×Q8 | — | 292 | 1 | no | — |
| 162 | C2×Q8 | — | 292 | 2 | no | — |
| 163 | C2×Q8 | — | 292 | 2 | no | — |
| 164 | C2×Q8 | — | 292 | 2 | no | — |
| 165 | C2×Q8 | — | 292 | 4 | no | — |
| 166 | C2×Q8 | — | 292 | 4 | no | — |
| 167 | C2×Q8 | — | 292 | 4 | no | — |
| 168 | C2×Q8 | — | 292 | 4 | no | — |
| 169 | C22×C4 | — | 2 | 1 | no | — |
| 170 | C22×C4 | — | 292 | 1 | no | — |
| 171 | C22×C4 | — | 292 | 1 | no | — |
| 172 | C22×C4 | — | 292 | 2 | no | — |
| 173 | C22×C4 | — | 292 | 2 | no | — |
| 174 | C22×C4 | — | 292 | 2 | no | — |
| 175 | C22×C4 | — | 292 | 2 | no | — |
| 176 | C22×C4 | — | 292 | 1 | no | — |
| 177 | C22×C4 | — | 292 | 2 | no | — |
| 178 | C22×C4 | — | 292 | 2 | no | — |
| 179 | C22×C4 | — | 292 | 1 | no | — |
| 180 | C22×C4 | — | 73 | 2 | no | — |
| 181 | C22×C4 | — | 73 | 2 | no | — |
| 182 | C22×C4 | — | 73 | 1 | no | — |
| 183 | C22×C4 | — | 146 | 1 | no | — |
| 184 | C22×C4 | — | 146 | 1 | no | — |
| 185 | C22×C4 | — | 146 | 4 | no | — |
| 186 | C22×C4 | — | 146 | 2 | no | — |
| 187 | C22×C4 | — | 146 | 2 | no | — |
| 188 | C22×C4 | — | 146 | 1 | no | — |
| 189 | C22×C4 | — | 146 | 2 | no | — |
| 190 | C22×C4 | C2×#10596842 | 1 | 1 | yes | characteristic |
| 191 | C2×C8 | — | 292 | 4 | no | — |
| 192 | C2×C8 | — | 584 | 2 | no | — |
| 193 | C2×C8 | — | 584 | 2 | no | — |
| 194 | C23 | — | 73 | 1 | no | — |
| 195 | C23 | — | 73 | 1 | no | — |
| 196 | C23 | — | 73 | 2 | no | — |
| 197 | C23 | — | 73 | 2 | no | — |
| 198 | C23 | — | 146 | 2 | no | — |
| 199 | C23 | — | 146 | 1 | no | — |
| 200 | C23 | — | 146 | 1 | no | — |
| 201 | C23 | C22⋊#10596842 | 1 | 1 | yes | characteristic |
| 202 | Q8 | — | 4 | 2 | no | — |
| 203 | Q8 | — | 4 | 2 | no | — |
| 204 | Q8 | — | 292 | 2 | no | — |
| 205 | Q8 | — | 292 | 8 | no | — |
| 206 | Q8 | — | 292 | 2 | no | — |
| 207 | C2×C4 | — | 2 | 2 | no | — |
| 208 | C2×C4 | — | 2 | 1 | no | — |
| 209 | C2×C4 | — | 4 | 1 | no | — |
| 210 | C2×C4 | — | 4 | 1 | no | — |
| 211 | C2×C4 | — | 4 | 1 | no | — |
| 212 | C2×C4 | — | 292 | 4 | no | — |
| 213 | C2×C4 | — | 292 | 2 | no | — |
| 214 | C2×C4 | — | 292 | 4 | no | — |
| 215 | C2×C4 | — | 292 | 2 | no | — |
| 216 | C2×C4 | — | 292 | 1 | no | — |
| 217 | C2×C4 | — | 292 | 2 | no | — |
| 218 | C2×C4 | — | 292 | 2 | no | — |
| 219 | C2×C4 | — | 292 | 1 | no | — |
| 220 | C2×C4 | — | 292 | 1 | no | — |
| 221 | C2×C4 | — | 292 | 2 | no | — |
| 222 | C2×C4 | — | 292 | 2 | no | — |
| 223 | C2×C4 | — | 292 | 4 | no | — |
| 224 | C2×C4 | — | 292 | 2 | no | — |
| 225 | C2×C4 | — | 584 | 2 | no | — |
| 226 | C2×C4 | — | 584 | 1 | no | — |
| 227 | C2×C4 | — | 584 | 2 | no | — |
| 228 | C2×C4 | — | 584 | 1 | no | — |
| 229 | C2×C4 | — | 146 | 2 | no | — |
| 230 | C2×C4 | — | 146 | 2 | no | — |
| 231 | C2×C4 | — | 146 | 2 | no | — |
| 232 | C2×C4 | — | 146 | 1 | no | — |
| 233 | C8 | — | 584 | 4 | no | — |
| 234 | C22 | — | 2 | 1 | no | — |
| 235 | C22 | — | 2 | 1 | no | — |
| 236 | C22 | — | 73 | 2 | no | — |
| 237 | C22 | — | 73 | 2 | no | — |
| 238 | C22 | — | 73 | 1 | no | — |
| 239 | C22 | — | 73 | 1 | no | — |
| 240 | C22 | — | 73 | 1 | no | — |
| 241 | C22 | — | 73 | 1 | no | — |
| 242 | C22 | — | 146 | 2 | no | — |
| 243 | C22 | — | 146 | 2 | no | — |
| 244 | C22 | — | 146 | 2 | no | — |
| 245 | C22 | — | 146 | 2 | no | — |
| 246 | C22 | — | 146 | 1 | no | — |
| 247 | C22 | — | 146 | 1 | no | — |
| 248 | C22 | #17989672 | 1 | 1 | yes | characteristic |
| 249 | C22 | Q8⋊#10596842 | 1 | 2 | yes | — |
| 250 | C4 | — | 4 | 2 | no | — |
| 251 | C4 | — | 4 | 1 | no | — |
| 252 | C4 | — | 292 | 2 | no | — |
| 253 | C4 | — | 292 | 2 | no | — |
| 254 | C4 | — | 292 | 2 | no | — |
| 255 | C4 | — | 292 | 1 | no | — |
| 256 | C4 | — | 584 | 2 | no | — |
| 257 | C2 | — | 2 | 2 | no | — |
| 258 | C2 | — | 73 | 2 | no | — |
| 259 | C2 | — | 73 | 2 | no | — |
| 260 | C2 | — | 146 | 1 | no | — |
| 261 | C2 | — | 146 | 1 | no | — |
| 262 | C2 | #22626932 | 1 | 1 | yes | characteristic |
| 263 | C2 | #22626956 | 1 | 1 | yes | characteristic |
| 264 | C2 | #22626959 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 17. C22×Dic146 | C8 | semidirect |
| 23. C22×Q8×C73 | C8 | semidirect |
| 135. C73 | #7131 | semidirect |
| 146. C22×Q8 | #10596842 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22626940 | 2 |
| #22625121 | 2 |
| #7131 | 73 |