Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
568 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #11739454 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11736625 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11729185 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11728410 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C7×#1384 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#6467 | C22 | 1 | 2 | yes | — |
| 7 | C22×C8×D7 | C22 | 1 | 1 | yes | characteristic |
| 8 | #90215 | — | 2 | 2 | no | — |
| 9 | C2×C7×M32 | C22 | 1 | 1 | yes | characteristic |
| 10 | C7×#427 | C22 | 1 | 2 | yes | — |
| 11 | C2×#6386 | C22 | 1 | 1 | yes | characteristic |
| 12 | #89774 | — | 2 | 2 | no | — |
| 13 | #89681 | — | 2 | 2 | no | — |
| 14 | #89680 | — | 2 | 2 | no | — |
| 15 | #89676 | — | 2 | 2 | no | — |
| 16 | #89659 | — | 2 | 2 | no | — |
| 17 | #89655 | — | 2 | 2 | no | — |
| 18 | #89636 | — | 2 | 4 | no | — |
| 19 | (C2×D7)⋊C16 | — | 2 | 2 | no | — |
| 20 | #89283 | — | 2 | 4 | no | — |
| 21 | #13309 | — | 7 | 1 | no | maximal, Sylow |
| 22 | C2×#1157 | — | 2 | 2 | no | — |
| 23 | C22×C4×D7 | D4 | 1 | 1 | yes | characteristic |
| 24 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 25 | C7×#117 | — | 2 | 2 | no | — |
| 26 | C22×C7⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×Dic28 | D4 | 1 | 2 | yes | complemented |
| 28 | #6467 | — | 2 | 2 | no | — |
| 29 | #6467 | — | 2 | 2 | no | — |
| 30 | #6467 | — | 2 | 2 | no | — |
| 31 | #6467 | — | 2 | 2 | no | — |
| 32 | C2×D56 | D4 | 1 | 2 | yes | complemented |
| 33 | C2×#1132 | — | 2 | 2 | no | — |
| 34 | C2×C8×D7 | — | 2 | 2 | no | — |
| 35 | C2×C8×D7 | — | 2 | 2 | no | — |
| 36 | C2×C8×D7 | — | 2 | 1 | no | — |
| 37 | C2×C8×D7 | — | 2 | 1 | no | — |
| 38 | #6460 | — | 2 | 4 | no | — |
| 39 | #6458 | — | 2 | 4 | no | — |
| 40 | #6449 | — | 4 | 2 | no | — |
| 41 | #6445 | — | 4 | 2 | no | — |
| 42 | C4×D28 | — | 4 | 4 | no | — |
| 43 | C7×M32 | — | 2 | 2 | no | — |
| 44 | C2×C7×C16 | D4 | 1 | 2 | yes | complemented |
| 45 | C7×#107 | — | 2 | 2 | no | — |
| 46 | C7×#106 | D4 | 1 | 4 | yes | complemented |
| 47 | #6386 | — | 2 | 2 | no | — |
| 48 | C2×C7⋊C16 | — | 2 | 1 | no | — |
| 49 | #6384 | — | 2 | 4 | no | — |
| 50 | #6383 | — | 2 | 4 | no | — |
| 51 | Dic56 | — | 2 | 2 | no | — |
| 52 | #6374 | — | 2 | 4 | no | — |
| 53 | D112 | — | 2 | 2 | no | — |
| 54 | #3414 | — | 7 | 1 | no | — |
| 55 | #3050 | — | 7 | 2 | no | — |
| 56 | #2145 | — | 7 | 2 | no | — |
| 57 | #2112 | — | 7 | 1 | no | — |
| 58 | #1384 | D7 | 1 | 1 | yes | characteristic, complemented |
| 59 | C23×D7 | D8 | 1 | 1 | yes | characteristic |
| 60 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 61 | C2×C22⋊D7 | — | 4 | 2 | no | — |
| 62 | C22×Dic7 | D8 | 1 | 1 | yes | characteristic |
| 63 | #1157 | — | 4 | 2 | no | — |
| 64 | #1157 | — | 4 | 2 | no | — |
| 65 | #1157 | — | 4 | 2 | no | — |
| 66 | #1157 | — | 4 | 2 | no | — |
| 67 | C2×D28 | — | 2 | 2 | no | — |
| 68 | C2×D28 | — | 4 | 4 | no | — |
| 69 | C2×C4×D7 | — | 2 | 2 | no | — |
| 70 | C2×C4×D7 | — | 2 | 1 | no | — |
| 71 | C2×C4×D7 | — | 2 | 1 | no | — |
| 72 | C2×C4×D7 | — | 2 | 2 | no | — |
| 73 | C2×C4×D7 | — | 4 | 2 | no | — |
| 74 | C2×Dic14 | — | 2 | 2 | no | — |
| 75 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 76 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 78 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 79 | C7×C4⋊C4 | — | 2 | 4 | no | — |
| 80 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 81 | C42×C7 | — | 4 | 2 | no | — |
| 82 | (C2×D7)⋊C4 | — | 4 | 4 | no | — |
| 83 | (C2×D7)⋊C4 | — | 4 | 4 | no | — |
| 84 | C4⋊Dic7 | — | 4 | 4 | no | — |
| 85 | C2×C7⋊C8 | — | 2 | 2 | no | — |
| 86 | C2×C7⋊C8 | — | 2 | 1 | no | — |
| 87 | Dic28 | — | 2 | 2 | no | — |
| 88 | Dic28 | — | 2 | 2 | no | — |
| 89 | D56 | — | 2 | 2 | no | — |
| 90 | D56 | — | 2 | 2 | no | — |
| 91 | #1132 | — | 4 | 2 | no | — |
| 92 | #1132 | — | 4 | 2 | no | — |
| 93 | C8×D7 | — | 4 | 1 | no | — |
| 94 | C8×D7 | — | 4 | 2 | no | — |
| 95 | C8×D7 | — | 4 | 1 | no | — |
| 96 | C7×C16 | — | 2 | 2 | no | — |
| 97 | C7⋊C16 | — | 4 | 1 | no | — |
| 98 | C2×#135 | — | 7 | 2 | no | — |
| 99 | C23×C8 | — | 7 | 1 | no | — |
| 100 | #525 | — | 14 | 2 | no | — |
| 101 | #510 | — | 14 | 2 | no | — |
| 102 | #509 | — | 14 | 2 | no | — |
| 103 | #508 | — | 14 | 2 | no | — |
| 104 | C2×M32 | — | 7 | 1 | no | — |
| 105 | C2×M32 | C2×D7 | 1 | 1 | yes | characteristic |
| 106 | #467 | — | 14 | 2 | no | — |
| 107 | #466 | — | 14 | 2 | no | — |
| 108 | #465 | — | 14 | 4 | no | — |
| 109 | #427 | C2×D7 | 1 | 2 | yes | — |
| 110 | #416 | — | 14 | 2 | no | — |
| 111 | #360 | — | 14 | 4 | no | — |
| 112 | C22⋊C16 | — | 14 | 2 | no | — |
| 113 | C23×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 114 | C22×D7 | — | 2 | 2 | no | — |
| 115 | C22×D7 | — | 2 | 1 | no | — |
| 116 | C22×D7 | — | 4 | 2 | no | — |
| 117 | C22×D7 | — | 4 | 2 | no | — |
| 118 | C2×C4×C7 | — | 4 | 4 | no | — |
| 119 | C2×C4×C7 | #120 | 1 | 2 | yes | — |
| 120 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C7 | #135 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C7 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 123 | C22⋊D7 | — | 8 | 2 | no | — |
| 124 | C22⋊D7 | — | 8 | 2 | no | — |
| 125 | C2×Dic7 | — | 2 | 2 | no | — |
| 126 | C2×Dic7 | — | 2 | 1 | no | — |
| 127 | C2×Dic7 | — | 4 | 2 | no | — |
| 128 | D28 | — | 4 | 2 | no | — |
| 129 | D28 | — | 4 | 2 | no | — |
| 130 | D28 | — | 8 | 8 | no | — |
| 131 | C4×D7 | — | 4 | 1 | no | — |
| 132 | C4×D7 | — | 4 | 2 | no | — |
| 133 | C4×D7 | — | 4 | 1 | no | — |
| 134 | C4×D7 | — | 8 | 2 | no | — |
| 135 | C4×D7 | — | 8 | 2 | no | — |
| 136 | Dic14 | — | 4 | 2 | no | — |
| 137 | Dic14 | — | 4 | 2 | no | — |
| 138 | C7×C8 | #120 | 1 | 1 | yes | characteristic |
| 139 | C7×C8 | #121 | 1 | 2 | yes | — |
| 140 | C7×C8 | #127 | 1 | 1 | yes | characteristic |
| 141 | C7⋊C8 | — | 4 | 1 | no | — |
| 142 | C2×#41 | — | 14 | 2 | no | — |
| 143 | C23×C4 | — | 7 | 1 | no | — |
| 144 | #135 | — | 14 | 2 | no | — |
| 145 | #135 | — | 14 | 2 | no | — |
| 146 | #135 | — | 14 | 2 | no | — |
| 147 | #135 | — | 14 | 2 | no | — |
| 148 | C2×Q16 | — | 7 | 2 | no | — |
| 149 | C2×SD16 | — | 14 | 2 | no | — |
| 150 | C2×D8 | — | 7 | 2 | no | — |
| 151 | C22×C8 | — | 7 | 1 | no | — |
| 152 | C22×C8 | — | 14 | 1 | no | — |
| 153 | C22×C8 | — | 14 | 2 | no | — |
| 154 | C22×C8 | — | 14 | 2 | no | — |
| 155 | C22×C8 | — | 14 | 1 | no | — |
| 156 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 157 | #123 | — | 28 | 2 | no | — |
| 158 | C22⋊Q8 | — | 14 | 4 | no | — |
| 159 | #121 | — | 14 | 4 | no | — |
| 160 | #120 | — | 28 | 2 | no | — |
| 161 | C4×D4 | — | 28 | 4 | no | — |
| 162 | #117 | — | 2 | 2 | no | — |
| 163 | Q32 | — | 14 | 2 | no | — |
| 164 | SD32 | — | 14 | 4 | no | — |
| 165 | D16 | — | 14 | 2 | no | — |
| 166 | M32 | — | 2 | 2 | no | — |
| 167 | M32 | — | 14 | 2 | no | — |
| 168 | C2×C16 | — | 14 | 1 | no | — |
| 169 | C2×C16 | D28 | 1 | 2 | yes | complemented |
| 170 | #107 | — | 2 | 2 | no | — |
| 171 | #106 | D28 | 1 | 4 | yes | complemented |
| 172 | #103 | — | 14 | 4 | no | — |
| 173 | #102 | — | 14 | 4 | no | — |
| 174 | C22×C7 | — | 4 | 1 | no | — |
| 175 | C22×C7 | #447 | 1 | 2 | yes | — |
| 176 | C22×C7 | #466 | 1 | 1 | yes | characteristic |
| 177 | C2×D7 | — | 2 | 4 | no | — |
| 178 | C2×D7 | — | 2 | 2 | no | — |
| 179 | C2×D7 | — | 4 | 2 | no | — |
| 180 | C2×D7 | — | 4 | 1 | no | — |
| 181 | C2×D7 | — | 4 | 1 | no | — |
| 182 | C2×D7 | — | 8 | 4 | no | — |
| 183 | C2×D7 | — | 8 | 2 | no | — |
| 184 | C4×C7 | — | 8 | 4 | no | — |
| 185 | C4×C7 | #392 | 1 | 1 | yes | characteristic |
| 186 | C4×C7 | #449 | 1 | 2 | yes | — |
| 187 | C4×C7 | #469 | 1 | 1 | yes | characteristic |
| 188 | Dic7 | — | 4 | 1 | no | — |
| 189 | Dic7 | — | 8 | 2 | no | — |
| 190 | C24 | — | 7 | 1 | no | — |
| 191 | #41 | — | 28 | 2 | no | — |
| 192 | #41 | — | 28 | 2 | no | — |
| 193 | #41 | — | 28 | 2 | no | — |
| 194 | #41 | — | 28 | 2 | no | — |
| 195 | C2×Q8 | — | 14 | 2 | no | — |
| 196 | C2×D4 | — | 14 | 2 | no | — |
| 197 | C2×D4 | — | 28 | 4 | no | — |
| 198 | C2×D4 | — | 28 | 2 | no | — |
| 199 | C22×C4 | — | 7 | 1 | no | — |
| 200 | C22×C4 | — | 14 | 2 | no | — |
| 201 | C22×C4 | — | 14 | 1 | no | — |
| 202 | C22×C4 | — | 14 | 2 | no | — |
| 203 | C22×C4 | — | 14 | 1 | no | — |
| 204 | C22×C4 | — | 28 | 2 | no | — |
| 205 | C22×C4 | D4×D7 | 1 | 1 | yes | characteristic |
| 206 | Q16 | — | 14 | 2 | no | — |
| 207 | Q16 | — | 14 | 2 | no | — |
| 208 | SD16 | — | 28 | 2 | no | — |
| 209 | SD16 | — | 28 | 2 | no | — |
| 210 | D8 | — | 14 | 2 | no | — |
| 211 | D8 | — | 14 | 2 | no | — |
| 212 | C2×C8 | — | 14 | 2 | no | — |
| 213 | C2×C8 | — | 14 | 1 | no | — |
| 214 | C2×C8 | — | 28 | 1 | no | — |
| 215 | C2×C8 | — | 28 | 1 | no | — |
| 216 | C2×C8 | — | 28 | 2 | no | — |
| 217 | C2×C8 | C2×D28 | 1 | 1 | yes | characteristic |
| 218 | C2×C8 | C2×D28 | 1 | 2 | yes | — |
| 219 | C2×C8 | D4×D7 | 1 | 2 | yes | — |
| 220 | C2×C8 | #1161 | 1 | 1 | yes | characteristic |
| 221 | C4⋊C4 | — | 2 | 4 | no | — |
| 222 | C4⋊C4 | — | 28 | 4 | no | — |
| 223 | C22⋊C4 | — | 4 | 2 | no | — |
| 224 | C22⋊C4 | — | 28 | 4 | no | — |
| 225 | C22⋊C4 | — | 28 | 4 | no | — |
| 226 | C42 | — | 4 | 2 | no | — |
| 227 | C16 | — | 2 | 2 | no | — |
| 228 | C16 | — | 28 | 1 | no | — |
| 229 | C2×C7 | — | 2 | 2 | no | — |
| 230 | C2×C7 | — | 2 | 1 | no | — |
| 231 | C2×C7 | #1999 | 1 | 1 | yes | characteristic |
| 232 | D7 | — | 4 | 2 | no | — |
| 233 | D7 | — | 16 | 2 | no | — |
| 234 | C23 | — | 14 | 1 | no | — |
| 235 | C23 | — | 14 | 2 | no | — |
| 236 | C23 | — | 28 | 2 | no | — |
| 237 | C23 | — | 28 | 2 | no | — |
| 238 | C23 | D7×D8 | 1 | 1 | yes | characteristic |
| 239 | Q8 | — | 28 | 2 | no | — |
| 240 | Q8 | — | 28 | 2 | no | — |
| 241 | D4 | — | 28 | 2 | no | — |
| 242 | D4 | — | 28 | 2 | no | — |
| 243 | D4 | — | 56 | 8 | no | — |
| 244 | D4 | — | 56 | 2 | no | — |
| 245 | D4 | — | 56 | 2 | no | — |
| 246 | C2×C4 | — | 4 | 4 | no | — |
| 247 | C2×C4 | — | 14 | 1 | no | — |
| 248 | C2×C4 | — | 14 | 2 | no | — |
| 249 | C2×C4 | — | 28 | 1 | no | — |
| 250 | C2×C4 | — | 28 | 1 | no | — |
| 251 | C2×C4 | — | 28 | 2 | no | — |
| 252 | C2×C4 | — | 28 | 2 | no | — |
| 253 | C2×C4 | — | 56 | 2 | no | — |
| 254 | C2×C4 | — | 56 | 2 | no | — |
| 255 | C2×C4 | #6445 | 1 | 2 | yes | — |
| 256 | C2×C4 | #6458 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | #6477 | 1 | 2 | yes | — |
| 258 | C2×C4 | #6482 | 1 | 1 | yes | characteristic |
| 259 | C8 | — | 28 | 1 | no | — |
| 260 | C8 | #6437 | 1 | 1 | yes | characteristic |
| 261 | C8 | #6445 | 1 | 1 | yes | characteristic |
| 262 | C8 | #6458 | 1 | 2 | yes | — |
| 263 | C7 | #13309 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 264 | C22 | — | 4 | 1 | no | — |
| 265 | C22 | — | 14 | 4 | no | — |
| 266 | C22 | — | 14 | 2 | no | — |
| 267 | C22 | — | 28 | 2 | no | — |
| 268 | C22 | — | 28 | 1 | no | — |
| 269 | C22 | — | 28 | 1 | no | — |
| 270 | C22 | — | 56 | 4 | no | — |
| 271 | C22 | — | 56 | 2 | no | — |
| 272 | C22 | #89545 | 1 | 2 | yes | — |
| 273 | C22 | #89655 | 1 | 1 | yes | characteristic |
| 274 | C4 | — | 8 | 4 | no | — |
| 275 | C4 | — | 28 | 1 | no | — |
| 276 | C4 | — | 56 | 2 | no | — |
| 277 | C4 | #89443 | 1 | 1 | yes | characteristic |
| 278 | C4 | #89583 | 1 | 2 | yes | — |
| 279 | C4 | #89637 | 1 | 1 | yes | characteristic |
| 280 | C2 | — | 2 | 2 | no | — |
| 281 | C2 | — | 2 | 1 | no | — |
| 282 | C2 | — | 28 | 2 | no | — |
| 283 | C2 | — | 112 | 2 | no | — |
| 284 | C2 | #11726640 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11739454 | C2 | semidirect |
| 2. #11736625 | C2 | semidirect |
| 3. #11729185 | C2 | semidirect |
| 4. #11728410 | C2 | semidirect |
| 5. C7×#1384 | C2 | semidirect |
| 27. C2×Dic28 | D4 | semidirect |
| 32. C2×D56 | D4 | semidirect |
| 44. C2×C7×C16 | D4 | semidirect |
| 46. C7×#106 | D4 | semidirect |
| 58. #1384 | D7 | semidirect |
| 169. C2×C16 | D28 | semidirect |
| 171. #106 | D28 | semidirect |
| 263. C7 | #13309 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11739454 | 2 |
| #11736625 | 2 |
| #11729185 | 2 |
| #11728410 | 2 |
| C7×#1384 | 2 |
| #13309 | 7 |