Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #11763441 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11762997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11762995 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11762979 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11762601 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11762597 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C17×AΣL(1,𝔽16) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×C22⋊D17 | — | 2 | 1 | no | — |
| 9 | #10587691 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×D4×D17 | — | 2 | 1 | no | — |
| 11 | C17×#120 | C22 | 1 | 1 | yes | characteristic |
| 12 | #10587639 | — | 2 | 1 | no | — |
| 13 | #10587626 | C22 | 1 | 1 | yes | characteristic |
| 14 | #10587625 | — | 2 | 1 | no | — |
| 15 | #10587623 | C22 | 1 | 1 | yes | characteristic |
| 16 | #10587594 | — | 2 | 1 | no | — |
| 17 | #10587570 | — | 2 | 1 | no | — |
| 18 | #10587568 | — | 2 | 1 | no | — |
| 19 | #10587568 | — | 2 | 1 | no | — |
| 20 | #10587567 | — | 2 | 1 | no | — |
| 21 | C17×#100 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | C17×#99 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | #10587505 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #10587496 | — | 2 | 1 | no | — |
| 25 | #10587477 | — | 2 | 1 | no | — |
| 26 | #10587477 | — | 2 | 1 | no | — |
| 27 | C24×C17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 28 | C23×D17 | — | 2 | 1 | no | — |
| 29 | C23×D17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 30 | C2×D4×C17 | — | 2 | 1 | no | — |
| 31 | C2×D4×C17 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 33 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 34 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 35 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 36 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 37 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 38 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 39 | C2×C22⋊D17 | — | 4 | 4 | no | — |
| 40 | C2×C22⋊D17 | — | 4 | 1 | no | — |
| 41 | C22×Dic17 | — | 2 | 1 | no | — |
| 42 | #63254 | — | 4 | 1 | no | — |
| 43 | D4×D17 | — | 4 | 1 | no | — |
| 44 | D4×D17 | — | 4 | 4 | no | — |
| 45 | #63252 | — | 2 | 1 | no | — |
| 46 | C2×D68 | — | 2 | 1 | no | — |
| 47 | C2×D68 | — | 4 | 1 | no | — |
| 48 | C2×D68 | D4 | 1 | 1 | yes | characteristic, complemented |
| 49 | C2×C4×D17 | — | 2 | 1 | no | — |
| 50 | M16×C17 | — | 2 | 1 | no | — |
| 51 | C22⋊C4×C17 | — | 2 | 1 | no | — |
| 52 | C22⋊C4×C17 | — | 2 | 1 | no | — |
| 53 | C22⋊C4×C17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 54 | C22⋊Dic17 | — | 2 | 1 | no | — |
| 55 | C22⋊Dic17 | — | 4 | 1 | no | — |
| 56 | C22⋊Dic17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 57 | #63227 | — | 2 | 1 | no | — |
| 58 | #63227 | — | 2 | 1 | no | — |
| 59 | #63227 | — | 4 | 1 | no | — |
| 60 | #63225 | — | 2 | 1 | no | — |
| 61 | C4×Dic17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 62 | D136 | — | 4 | 1 | no | — |
| 63 | #63219 | — | 4 | 1 | no | — |
| 64 | C23×C17 | — | 2 | 1 | no | — |
| 65 | C23×C17 | — | 4 | 1 | no | — |
| 66 | C23×C17 | — | 4 | 2 | no | — |
| 67 | C23×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | C23×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C22×D17 | — | 2 | 1 | no | — |
| 70 | C22×D17 | — | 2 | 1 | no | — |
| 71 | C22×D17 | — | 2 | 1 | no | — |
| 72 | C22×D17 | — | 2 | 1 | no | — |
| 73 | C22×D17 | — | 4 | 2 | no | — |
| 74 | C22×D17 | — | 4 | 2 | no | — |
| 75 | C22×D17 | — | 4 | 1 | no | — |
| 76 | C22×D17 | — | 4 | 2 | no | — |
| 77 | C22×D17 | — | 8 | 1 | no | — |
| 78 | D4×C17 | — | 4 | 1 | no | — |
| 79 | D4×C17 | — | 4 | 2 | no | — |
| 80 | C2×C4×C17 | — | 2 | 1 | no | — |
| 81 | C2×C4×C17 | — | 4 | 1 | no | — |
| 82 | C2×C4×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C22⋊D17 | — | 2 | 4 | no | — |
| 84 | C22⋊D17 | — | 4 | 2 | no | — |
| 85 | C22⋊D17 | — | 4 | 2 | no | — |
| 86 | C22⋊D17 | — | 4 | 2 | no | — |
| 87 | C22⋊D17 | — | 4 | 1 | no | — |
| 88 | C22⋊D17 | — | 4 | 1 | no | — |
| 89 | C22⋊D17 | — | 4 | 2 | no | — |
| 90 | C22⋊D17 | — | 4 | 1 | no | — |
| 91 | C22⋊D17 | — | 8 | 2 | no | — |
| 92 | C2×Dic17 | — | 2 | 1 | no | — |
| 93 | C2×Dic17 | — | 2 | 1 | no | — |
| 94 | C2×Dic17 | — | 2 | 1 | no | — |
| 95 | C2×Dic17 | — | 2 | 1 | no | — |
| 96 | C2×Dic17 | — | 4 | 2 | no | — |
| 97 | C2×Dic17 | — | 4 | 1 | no | — |
| 98 | D68 | — | 2 | 1 | no | — |
| 99 | D68 | — | 4 | 1 | no | — |
| 100 | D68 | — | 4 | 2 | no | — |
| 101 | D68 | — | 8 | 2 | no | — |
| 102 | C4×D17 | — | 4 | 2 | no | — |
| 103 | C4×D17 | — | 4 | 1 | no | — |
| 104 | Dic34 | — | 2 | 1 | no | — |
| 105 | C8×C17 | — | 4 | 1 | no | — |
| 106 | #2196 | — | 17 | 1 | no | maximal, Sylow |
| 107 | C22×C17 | — | 2 | 2 | no | — |
| 108 | C22×C17 | — | 2 | 1 | no | — |
| 109 | C22×C17 | — | 2 | 1 | no | — |
| 110 | C22×C17 | — | 4 | 1 | no | — |
| 111 | C22×C17 | — | 4 | 1 | no | — |
| 112 | C22×C17 | — | 4 | 1 | no | — |
| 113 | C22×C17 | — | 4 | 2 | no | — |
| 114 | C22×C17 | — | 4 | 1 | no | — |
| 115 | C22×C17 | — | 8 | 1 | no | — |
| 116 | C22×C17 | #120 | 1 | 1 | yes | characteristic |
| 117 | C2×D17 | — | 2 | 2 | no | — |
| 118 | C2×D17 | — | 4 | 1 | no | — |
| 119 | C2×D17 | — | 4 | 1 | no | — |
| 120 | C2×D17 | — | 4 | 2 | no | — |
| 121 | C2×D17 | — | 4 | 1 | no | — |
| 122 | C2×D17 | — | 4 | 1 | no | — |
| 123 | C2×D17 | — | 4 | 1 | no | — |
| 124 | C2×D17 | — | 4 | 1 | no | — |
| 125 | C2×D17 | — | 8 | 1 | no | — |
| 126 | C2×D17 | — | 8 | 4 | no | — |
| 127 | C2×D17 | — | 8 | 1 | no | — |
| 128 | C2×D17 | — | 8 | 1 | no | — |
| 129 | C4×C17 | — | 2 | 1 | no | — |
| 130 | C4×C17 | — | 4 | 1 | no | — |
| 131 | C4×C17 | — | 8 | 1 | no | — |
| 132 | Dic17 | — | 2 | 2 | no | — |
| 133 | Dic17 | — | 4 | 1 | no | — |
| 134 | Dic17 | — | 4 | 1 | no | — |
| 135 | Dic17 | — | 8 | 1 | no | — |
| 136 | D42 | — | 17 | 1 | no | — |
| 137 | B(4,𝔽2) | — | 17 | 1 | no | — |
| 138 | B(4,𝔽2) | — | 17 | 1 | no | — |
| 139 | #453 | — | 17 | 1 | no | — |
| 140 | #353 | — | 17 | 1 | no | — |
| 141 | AΣL(1,𝔽16) | — | 17 | 1 | no | — |
| 142 | AΣL(1,𝔽16) | D17 | 1 | 1 | yes | characteristic, complemented |
| 143 | C2×C17 | — | 2 | 1 | no | — |
| 144 | C2×C17 | — | 4 | 1 | no | — |
| 145 | C2×C17 | — | 4 | 1 | no | — |
| 146 | C2×C17 | — | 4 | 2 | no | — |
| 147 | C2×C17 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 148 | D17 | — | 4 | 2 | no | — |
| 149 | D17 | — | 8 | 1 | no | — |
| 150 | D17 | — | 8 | 1 | no | — |
| 151 | H(5,𝔽2) | — | 17 | 1 | no | — |
| 152 | C22×D4 | — | 34 | 1 | no | — |
| 153 | C22×D4 | — | 34 | 1 | no | — |
| 154 | AGL(1,ℤ/8) | — | 34 | 1 | no | — |
| 155 | #127 | — | 17 | 1 | no | — |
| 156 | #121 | — | 34 | 1 | no | — |
| 157 | #121 | — | 34 | 1 | no | — |
| 158 | #120 | — | 17 | 1 | no | — |
| 159 | #120 | — | 34 | 1 | no | — |
| 160 | #120 | — | 34 | 1 | no | — |
| 161 | #120 | — | 34 | 1 | no | — |
| 162 | #120 | C2×D17 | 1 | 1 | yes | characteristic |
| 163 | C4×D4 | — | 34 | 1 | no | — |
| 164 | #104 | — | 34 | 1 | no | — |
| 165 | #100 | C2×D17 | 1 | 1 | yes | characteristic, complemented |
| 166 | #99 | — | 17 | 1 | no | — |
| 167 | #99 | — | 34 | 1 | no | — |
| 168 | #99 | — | 34 | 1 | no | — |
| 169 | #99 | C2×D17 | 1 | 1 | yes | characteristic, complemented |
| 170 | C17 | #2196 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 171 | C24 | — | 17 | 1 | no | — |
| 172 | C24 | — | 34 | 1 | no | — |
| 173 | C24 | D68 | 1 | 1 | yes | characteristic, complemented |
| 174 | #41 | — | 34 | 1 | no | — |
| 175 | #41 | — | 68 | 1 | no | — |
| 176 | C2×D4 | — | 2 | 1 | no | — |
| 177 | C2×D4 | — | 17 | 1 | no | — |
| 178 | C2×D4 | — | 34 | 1 | no | — |
| 179 | C2×D4 | — | 34 | 1 | no | — |
| 180 | C2×D4 | — | 34 | 1 | no | — |
| 181 | C2×D4 | — | 34 | 1 | no | — |
| 182 | C2×D4 | — | 34 | 1 | no | — |
| 183 | C2×D4 | — | 34 | 1 | no | — |
| 184 | C2×D4 | — | 34 | 1 | no | — |
| 185 | C2×D4 | — | 34 | 1 | no | — |
| 186 | C2×D4 | — | 68 | 1 | no | — |
| 187 | C2×D4 | — | 68 | 1 | no | — |
| 188 | C2×D4 | — | 68 | 4 | no | — |
| 189 | C2×D4 | — | 68 | 4 | no | — |
| 190 | C2×D4 | — | 68 | 1 | no | — |
| 191 | C2×D4 | C22×D17 | 1 | 1 | yes | characteristic |
| 192 | C22×C4 | — | 34 | 1 | no | — |
| 193 | C22×C4 | — | 34 | 1 | no | — |
| 194 | SD16 | — | 68 | 1 | no | — |
| 195 | D8 | — | 68 | 1 | no | — |
| 196 | M16 | — | 2 | 1 | no | — |
| 197 | C4⋊C4 | — | 34 | 1 | no | — |
| 198 | C22⋊C4 | — | 2 | 1 | no | — |
| 199 | C22⋊C4 | — | 2 | 1 | no | — |
| 200 | C22⋊C4 | — | 17 | 1 | no | — |
| 201 | C22⋊C4 | — | 34 | 1 | no | — |
| 202 | C22⋊C4 | — | 34 | 1 | no | — |
| 203 | C22⋊C4 | — | 34 | 1 | no | — |
| 204 | C22⋊C4 | — | 68 | 1 | no | — |
| 205 | C22⋊C4 | — | 68 | 1 | no | — |
| 206 | C22⋊C4 | D68 | 1 | 1 | yes | characteristic, complemented |
| 207 | C42 | — | 17 | 1 | no | — |
| 208 | C23 | — | 2 | 1 | no | — |
| 209 | C23 | — | 4 | 1 | no | — |
| 210 | C23 | — | 4 | 2 | no | — |
| 211 | C23 | — | 34 | 1 | no | — |
| 212 | C23 | — | 34 | 1 | no | — |
| 213 | C23 | — | 34 | 1 | no | — |
| 214 | C23 | — | 34 | 1 | no | — |
| 215 | C23 | — | 68 | 1 | no | — |
| 216 | C23 | — | 68 | 2 | no | — |
| 217 | C23 | — | 68 | 2 | no | — |
| 218 | C23 | — | 68 | 2 | no | — |
| 219 | C23 | — | 136 | 1 | no | — |
| 220 | C23 | C2×D68 | 1 | 1 | yes | characteristic |
| 221 | C23 | D4×D17 | 1 | 1 | yes | characteristic |
| 222 | Q8 | — | 34 | 1 | no | — |
| 223 | D4 | — | 4 | 1 | no | — |
| 224 | D4 | — | 4 | 2 | no | — |
| 225 | D4 | — | 34 | 1 | no | — |
| 226 | D4 | — | 34 | 4 | no | — |
| 227 | D4 | — | 68 | 1 | no | — |
| 228 | D4 | — | 68 | 2 | no | — |
| 229 | D4 | — | 68 | 2 | no | — |
| 230 | D4 | — | 68 | 2 | no | — |
| 231 | D4 | — | 68 | 2 | no | — |
| 232 | D4 | — | 68 | 1 | no | — |
| 233 | D4 | — | 68 | 1 | no | — |
| 234 | D4 | — | 68 | 2 | no | — |
| 235 | D4 | — | 68 | 1 | no | — |
| 236 | D4 | — | 136 | 2 | no | — |
| 237 | D4 | — | 136 | 2 | no | — |
| 238 | C2×C4 | — | 2 | 1 | no | — |
| 239 | C2×C4 | — | 4 | 1 | no | — |
| 240 | C2×C4 | — | 34 | 1 | no | — |
| 241 | C2×C4 | — | 34 | 1 | no | — |
| 242 | C2×C4 | — | 34 | 1 | no | — |
| 243 | C2×C4 | — | 34 | 1 | no | — |
| 244 | C2×C4 | — | 68 | 1 | no | — |
| 245 | C2×C4 | — | 68 | 2 | no | — |
| 246 | C2×C4 | — | 68 | 2 | no | — |
| 247 | C2×C4 | — | 68 | 1 | no | — |
| 248 | C2×C4 | D4×D17 | 1 | 1 | yes | characteristic |
| 249 | C8 | — | 4 | 1 | no | — |
| 250 | C22 | — | 2 | 2 | no | — |
| 251 | C22 | — | 2 | 1 | no | — |
| 252 | C22 | — | 2 | 1 | no | — |
| 253 | C22 | — | 4 | 1 | no | — |
| 254 | C22 | — | 4 | 1 | no | — |
| 255 | C22 | — | 4 | 1 | no | — |
| 256 | C22 | — | 4 | 2 | no | — |
| 257 | C22 | — | 4 | 1 | no | — |
| 258 | C22 | — | 8 | 1 | no | — |
| 259 | C22 | — | 34 | 2 | no | — |
| 260 | C22 | — | 68 | 1 | no | — |
| 261 | C22 | — | 68 | 1 | no | — |
| 262 | C22 | — | 68 | 1 | no | — |
| 263 | C22 | — | 68 | 2 | no | — |
| 264 | C22 | — | 68 | 1 | no | — |
| 265 | C22 | — | 68 | 1 | no | — |
| 266 | C22 | — | 68 | 1 | no | — |
| 267 | C22 | — | 136 | 1 | no | — |
| 268 | C22 | — | 136 | 4 | no | — |
| 269 | C22 | — | 136 | 1 | no | — |
| 270 | C22 | — | 136 | 1 | no | — |
| 271 | C22 | #10587568 | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 2 | 1 | no | — |
| 273 | C4 | — | 4 | 1 | no | — |
| 274 | C4 | — | 8 | 1 | no | — |
| 275 | C4 | — | 34 | 2 | no | — |
| 276 | C4 | — | 68 | 1 | no | — |
| 277 | C4 | — | 68 | 1 | no | — |
| 278 | C4 | — | 136 | 1 | no | — |
| 279 | C2 | — | 2 | 1 | no | — |
| 280 | C2 | — | 4 | 1 | no | — |
| 281 | C2 | — | 4 | 1 | no | — |
| 282 | C2 | — | 4 | 2 | no | — |
| 283 | C2 | — | 68 | 2 | no | — |
| 284 | C2 | — | 136 | 1 | no | — |
| 285 | C2 | — | 136 | 1 | no | — |
| 286 | C2 | #11762997 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11763441 | C2 | semidirect |
| 2. #11762997 | C2 | semidirect |
| 3. #11762995 | C2 | semidirect |
| 4. #11762979 | C2 | semidirect |
| 5. #11762601 | C2 | semidirect |
| 6. #11762597 | C2 | semidirect |
| 7. C17×AΣL(1,𝔽16) | C2 | semidirect |
| 9. #10587691 | C22 | semidirect |
| 21. C17×#100 | C22 | semidirect |
| 22. C17×#99 | C22 | semidirect |
| 23. #10587505 | C22 | semidirect |
| 27. C24×C17 | D4 | semidirect |
| 29. C23×D17 | D4 | semidirect |
| 48. C2×D68 | D4 | semidirect |
| 53. C22⋊C4×C17 | D4 | semidirect |
| 56. C22⋊Dic17 | D4 | semidirect |
| 61. C4×Dic17 | D4 | semidirect |
| 142. AΣL(1,𝔽16) | D17 | semidirect |
| 165. #100 | C2×D17 | semidirect |
| 169. #99 | C2×D17 | semidirect |
| 170. C17 | #2196 | semidirect |
| 173. C24 | D68 | semidirect |
| 206. C22⋊C4 | D68 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11763441 | 2 |
| #11762997 | 2 |
| #11762995 | 2 |
| #11762979 | 2 |
| #11762601 | 2 |
| #11762597 | 2 |
| C17×AΣL(1,𝔽16) | 2 |
| #2196 | 17 |