Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
468 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 49 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 | C22×C7⋊S4 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C7×D4×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10621506 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C4⋊(C7⋊S4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C4×C7⋊S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | D4×C22⋊D7 | — | 3 | 1 | no | maximal, Hall |
| 7 | C22×C7×A4 | C22 | 1 | 2 | yes | complemented |
| 8 | C2×C7⋊S4 | — | 2 | 4 | no | — |
| 9 | C2×C7⋊S4 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C7⋊S4 | C22 | 1 | 2 | yes | complemented |
| 11 | D4×D21 | — | 4 | 1 | no | maximal |
| 12 | C4×C7×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C7⋊SL(2,ℤ/4) | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22×C7×D4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C22×C22⋊D7 | — | 3 | 2 | no | — |
| 16 | C2×D4×D7 | — | 3 | 1 | no | — |
| 17 | #6516 | — | 3 | 2 | no | — |
| 18 | #6503 | — | 3 | 1 | no | — |
| 19 | #6502 | — | 3 | 2 | no | — |
| 20 | #6501 | — | 3 | 1 | no | — |
| 21 | #6500 | — | 3 | 2 | no | — |
| 22 | D4×Dic7 | — | 3 | 1 | no | — |
| 23 | #6493 | — | 3 | 1 | no | — |
| 24 | C4×C22⋊D7 | — | 3 | 1 | no | — |
| 25 | D4×S4 | — | 7 | 1 | no | maximal, Hall |
| 26 | C22×D21 | — | 4 | 2 | no | — |
| 27 | C2×C7×A4 | — | 2 | 2 | no | — |
| 28 | C2×C7×A4 | C23 | 1 | 1 | yes | characteristic |
| 29 | C7⋊S4 | — | 2 | 2 | no | — |
| 30 | C7⋊S4 | D4 | 1 | 2 | yes | direct product factor |
| 31 | C3×C7×D4 | — | 4 | 1 | no | — |
| 32 | #4268 | — | 4 | 2 | no | — |
| 33 | D84 | — | 4 | 1 | no | — |
| 34 | C4×D21 | — | 4 | 1 | no | — |
| 35 | C24×C7 | C2×D3 | 1 | 2 | yes | complemented |
| 36 | C23×D7 | — | 3 | 2 | no | — |
| 37 | C2×C7×D4 | — | 3 | 1 | no | — |
| 38 | C2×C7×D4 | — | 3 | 2 | no | — |
| 39 | C2×C7×D4 | — | 3 | 1 | no | — |
| 40 | C22×C4×C7 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 41 | C2×C22⋊D7 | — | 3 | 1 | no | — |
| 42 | C2×C22⋊D7 | — | 3 | 2 | no | — |
| 43 | C2×C22⋊D7 | — | 3 | 2 | no | — |
| 44 | C2×C22⋊D7 | — | 3 | 2 | no | — |
| 45 | C2×C22⋊D7 | — | 3 | 2 | no | — |
| 46 | C2×C22⋊D7 | — | 3 | 2 | no | — |
| 47 | C2×C22⋊D7 | — | 6 | 4 | no | — |
| 48 | C2×C22⋊D7 | — | 6 | 4 | no | — |
| 49 | C22×Dic7 | — | 3 | 2 | no | — |
| 50 | D4×D7 | — | 6 | 2 | no | — |
| 51 | D4×D7 | — | 6 | 1 | no | — |
| 52 | D4×D7 | — | 6 | 1 | no | — |
| 53 | C2×D28 | — | 3 | 1 | no | — |
| 54 | C2×C4×D7 | — | 3 | 1 | no | — |
| 55 | C22⋊Dic7 | — | 3 | 2 | no | — |
| 56 | C22⋊Dic7 | — | 3 | 2 | no | — |
| 57 | C22⋊Dic7 | — | 3 | 1 | no | — |
| 58 | (C2×D7)⋊C4 | — | 3 | 1 | no | — |
| 59 | (C2×D7)⋊C4 | — | 3 | 2 | no | — |
| 60 | C4⋊Dic7 | — | 3 | 1 | no | — |
| 61 | Dic7⋊C4 | — | 3 | 1 | no | — |
| 62 | C4×Dic7 | — | 3 | 1 | no | — |
| 63 | C22×S4 | — | 7 | 2 | no | — |
| 64 | D4×A4 | D7 | 1 | 1 | yes | characteristic, complemented |
| 65 | GL(2,ℤ/4) | — | 7 | 2 | no | — |
| 66 | C4⋊S4 | — | 7 | 1 | no | — |
| 67 | C4×S4 | — | 7 | 1 | no | — |
| 68 | C22×C3×C7 | — | 4 | 2 | no | — |
| 69 | C2×D21 | — | 4 | 1 | no | — |
| 70 | C2×D21 | — | 4 | 2 | no | — |
| 71 | C2×D21 | — | 8 | 4 | no | — |
| 72 | C7×A4 | C2×D4 | 1 | 1 | yes | characteristic, complemented |
| 73 | C3×C4×C7 | — | 4 | 1 | no | — |
| 74 | Dic21 | — | 4 | 1 | no | — |
| 75 | D42 | — | 21 | 1 | no | Sylow |
| 76 | C23×C7 | — | 2 | 2 | no | — |
| 77 | C23×C7 | — | 3 | 2 | no | — |
| 78 | C23×C7 | — | 3 | 2 | no | — |
| 79 | C23×C7 | — | 6 | 2 | no | — |
| 80 | C23×C7 | C22×D3 | 1 | 1 | yes | characteristic |
| 81 | C22×D7 | — | 3 | 1 | no | — |
| 82 | C22×D7 | — | 3 | 2 | no | — |
| 83 | C22×D7 | — | 6 | 4 | no | — |
| 84 | C22×D7 | — | 6 | 2 | no | — |
| 85 | C22×D7 | — | 6 | 2 | no | — |
| 86 | C22×D7 | — | 12 | 2 | no | — |
| 87 | C7×D4 | — | 3 | 2 | no | — |
| 88 | C7×D4 | — | 3 | 1 | no | — |
| 89 | C7×D4 | — | 6 | 1 | no | — |
| 90 | C7×D4 | S4 | 1 | 1 | yes | characteristic, complemented |
| 91 | C2×C4×C7 | — | 3 | 1 | no | — |
| 92 | C2×C4×C7 | — | 3 | 1 | no | — |
| 93 | C22⋊D7 | — | 3 | 2 | no | — |
| 94 | C22⋊D7 | — | 3 | 2 | no | — |
| 95 | C22⋊D7 | — | 6 | 4 | no | — |
| 96 | C22⋊D7 | — | 6 | 4 | no | — |
| 97 | C22⋊D7 | — | 6 | 2 | no | — |
| 98 | C22⋊D7 | — | 6 | 2 | no | — |
| 99 | C22⋊D7 | — | 6 | 2 | no | — |
| 100 | C22⋊D7 | — | 6 | 2 | no | — |
| 101 | C22⋊D7 | — | 12 | 2 | no | — |
| 102 | C2×Dic7 | — | 3 | 1 | no | — |
| 103 | C2×Dic7 | — | 3 | 1 | no | — |
| 104 | C2×Dic7 | — | 3 | 1 | no | — |
| 105 | C2×Dic7 | — | 3 | 2 | no | — |
| 106 | C2×Dic7 | — | 6 | 4 | no | — |
| 107 | D28 | — | 6 | 1 | no | — |
| 108 | D28 | — | 6 | 1 | no | — |
| 109 | C4×D7 | — | 6 | 1 | no | — |
| 110 | C4×D7 | — | 6 | 1 | no | — |
| 111 | C22×A4 | C2×D7 | 1 | 2 | yes | complemented |
| 112 | C2×S4 | — | 7 | 1 | no | — |
| 113 | C2×S4 | — | 7 | 2 | no | — |
| 114 | C2×S4 | — | 14 | 4 | no | — |
| 115 | D3×D4 | — | 28 | 1 | no | — |
| 116 | C4×A4 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 117 | SL(2,ℤ/4) | — | 7 | 1 | no | — |
| 118 | C2×C3×C7 | — | 4 | 1 | no | — |
| 119 | C2×C3×C7 | — | 8 | 2 | no | — |
| 120 | D21 | — | 4 | 2 | no | — |
| 121 | D21 | — | 8 | 2 | no | — |
| 122 | C22×D4 | — | 21 | 1 | no | — |
| 123 | C22×D4 | — | 21 | 2 | no | — |
| 124 | C22×D4 | D21 | 1 | 1 | yes | characteristic, complemented |
| 125 | #127 | — | 21 | 1 | no | — |
| 126 | #121 | — | 21 | 1 | no | — |
| 127 | #121 | — | 21 | 2 | no | — |
| 128 | #121 | — | 21 | 1 | no | — |
| 129 | #120 | — | 21 | 2 | no | — |
| 130 | #120 | — | 21 | 2 | no | — |
| 131 | C4×D4 | — | 21 | 1 | no | — |
| 132 | C4×D4 | — | 21 | 1 | no | — |
| 133 | C22×C7 | — | 3 | 1 | no | — |
| 134 | C22×C7 | — | 3 | 1 | no | — |
| 135 | C22×C7 | — | 3 | 2 | no | — |
| 136 | C22×C7 | — | 6 | 2 | no | — |
| 137 | C22×C7 | — | 6 | 2 | no | — |
| 138 | C22×C7 | — | 6 | 2 | no | — |
| 139 | C22×C7 | — | 6 | 2 | no | — |
| 140 | C22×C7 | D3×D4 | 1 | 1 | yes | characteristic, complemented |
| 141 | C22×C7 | C2×S4 | 1 | 2 | yes | complemented |
| 142 | C2×D7 | — | 3 | 2 | no | — |
| 143 | C2×D7 | — | 6 | 1 | no | — |
| 144 | C2×D7 | — | 6 | 1 | no | — |
| 145 | C2×D7 | — | 6 | 2 | no | — |
| 146 | C2×D7 | — | 6 | 2 | no | — |
| 147 | C2×D7 | — | 6 | 2 | no | — |
| 148 | C2×D7 | — | 12 | 4 | no | — |
| 149 | C2×D7 | — | 12 | 4 | no | — |
| 150 | C4×C7 | — | 3 | 1 | no | — |
| 151 | C4×C7 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 152 | Dic7 | — | 3 | 2 | no | — |
| 153 | Dic7 | — | 6 | 2 | no | — |
| 154 | Dic7 | — | 6 | 1 | no | — |
| 155 | Dic7 | — | 6 | 1 | no | — |
| 156 | C22×D3 | — | 28 | 2 | no | — |
| 157 | C2×A4 | — | 2 | 2 | no | — |
| 158 | C2×A4 | C22×D7 | 1 | 1 | yes | characteristic |
| 159 | S4 | — | 7 | 2 | no | — |
| 160 | S4 | — | 14 | 2 | no | — |
| 161 | C3×D4 | — | 4 | 1 | no | — |
| 162 | #67 | — | 28 | 2 | no | — |
| 163 | D12 | — | 28 | 1 | no | — |
| 164 | C4×D3 | — | 28 | 1 | no | — |
| 165 | C3×C7 | — | 4 | 1 | no | Hall |
| 166 | C24 | — | 21 | 2 | no | — |
| 167 | C24 | C2×D21 | 1 | 2 | yes | complemented |
| 168 | C2×D4 | — | 3 | 1 | no | — |
| 169 | C2×D4 | — | 3 | 2 | no | — |
| 170 | C2×D4 | — | 3 | 1 | no | — |
| 171 | C2×D4 | — | 21 | 1 | no | — |
| 172 | C2×D4 | — | 21 | 1 | no | — |
| 173 | C2×D4 | — | 21 | 2 | no | — |
| 174 | C2×D4 | — | 21 | 2 | no | — |
| 175 | C2×D4 | — | 21 | 2 | no | — |
| 176 | C2×D4 | — | 21 | 2 | no | — |
| 177 | C2×D4 | — | 21 | 2 | no | — |
| 178 | C2×D4 | — | 42 | 2 | no | — |
| 179 | C2×D4 | — | 42 | 1 | no | — |
| 180 | C2×D4 | — | 42 | 4 | no | — |
| 181 | C2×D4 | — | 42 | 4 | no | — |
| 182 | C2×D4 | — | 42 | 1 | no | — |
| 183 | C22×C4 | — | 21 | 1 | no | — |
| 184 | C22×C4 | — | 21 | 2 | no | — |
| 185 | C22×C4 | C2×D21 | 1 | 1 | yes | characteristic, complemented |
| 186 | C4⋊C4 | — | 21 | 1 | no | — |
| 187 | C4⋊C4 | — | 21 | 1 | no | — |
| 188 | C22⋊C4 | — | 21 | 1 | no | — |
| 189 | C22⋊C4 | — | 21 | 2 | no | — |
| 190 | C22⋊C4 | — | 21 | 2 | no | — |
| 191 | C22⋊C4 | — | 21 | 2 | no | — |
| 192 | C22⋊C4 | — | 21 | 1 | no | — |
| 193 | C42 | — | 21 | 1 | no | — |
| 194 | C2×C7 | — | 2 | 2 | no | — |
| 195 | C2×C7 | — | 3 | 1 | no | — |
| 196 | C2×C7 | — | 3 | 1 | no | — |
| 197 | C2×C7 | — | 6 | 2 | no | — |
| 198 | C2×C7 | C22×S4 | 1 | 1 | yes | characteristic |
| 199 | D7 | — | 6 | 2 | no | — |
| 200 | D7 | — | 12 | 2 | no | — |
| 201 | C22×C3 | — | 4 | 2 | no | — |
| 202 | C2×D3 | — | 28 | 1 | no | — |
| 203 | C2×D3 | — | 28 | 2 | no | — |
| 204 | C2×D3 | — | 56 | 4 | no | — |
| 205 | A4 | D4×D7 | 1 | 1 | yes | characteristic, complemented |
| 206 | C3×C4 | — | 4 | 1 | no | — |
| 207 | Dic3 | — | 28 | 1 | no | — |
| 208 | C23 | — | 2 | 2 | no | — |
| 209 | C23 | — | 3 | 2 | no | — |
| 210 | C23 | — | 3 | 2 | no | — |
| 211 | C23 | — | 6 | 2 | no | — |
| 212 | C23 | — | 21 | 1 | no | — |
| 213 | C23 | — | 21 | 2 | no | — |
| 214 | C23 | — | 42 | 2 | no | — |
| 215 | C23 | — | 42 | 4 | no | — |
| 216 | C23 | — | 42 | 2 | no | — |
| 217 | C23 | — | 84 | 2 | no | — |
| 218 | C23 | C22×D21 | 1 | 1 | yes | characteristic |
| 219 | D4 | — | 3 | 2 | no | — |
| 220 | D4 | — | 3 | 1 | no | — |
| 221 | D4 | — | 6 | 1 | no | — |
| 222 | D4 | — | 21 | 2 | no | — |
| 223 | D4 | — | 21 | 2 | no | — |
| 224 | D4 | — | 42 | 1 | no | — |
| 225 | D4 | — | 42 | 4 | no | — |
| 226 | D4 | — | 42 | 2 | no | — |
| 227 | D4 | — | 42 | 2 | no | — |
| 228 | D4 | — | 42 | 4 | no | — |
| 229 | D4 | — | 42 | 2 | no | — |
| 230 | D4 | — | 42 | 2 | no | — |
| 231 | D4 | — | 42 | 1 | no | — |
| 232 | D4 | — | 84 | 2 | no | — |
| 233 | D4 | C7⋊S4 | 1 | 1 | yes | characteristic, direct product factor |
| 234 | C2×C4 | — | 3 | 1 | no | — |
| 235 | C2×C4 | — | 3 | 1 | no | — |
| 236 | C2×C4 | — | 21 | 1 | no | — |
| 237 | C2×C4 | — | 21 | 1 | no | — |
| 238 | C2×C4 | — | 21 | 2 | no | — |
| 239 | C2×C4 | — | 21 | 1 | no | — |
| 240 | C2×C4 | — | 42 | 1 | no | — |
| 241 | C2×C4 | — | 42 | 4 | no | — |
| 242 | C2×C4 | — | 42 | 1 | no | — |
| 243 | C7 | D4×S4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 244 | C2×C3 | — | 4 | 1 | no | — |
| 245 | C2×C3 | — | 8 | 2 | no | — |
| 246 | D3 | — | 28 | 2 | no | — |
| 247 | D3 | — | 56 | 2 | no | — |
| 248 | C22 | — | 3 | 1 | no | — |
| 249 | C22 | — | 3 | 1 | no | — |
| 250 | C22 | — | 3 | 2 | no | — |
| 251 | C22 | — | 6 | 2 | no | — |
| 252 | C22 | — | 6 | 2 | no | — |
| 253 | C22 | — | 6 | 2 | no | — |
| 254 | C22 | — | 6 | 2 | no | — |
| 255 | C22 | — | 21 | 2 | no | — |
| 256 | C22 | — | 42 | 1 | no | — |
| 257 | C22 | — | 42 | 2 | no | — |
| 258 | C22 | — | 42 | 2 | no | — |
| 259 | C22 | — | 42 | 1 | no | — |
| 260 | C22 | — | 42 | 2 | no | — |
| 261 | C22 | — | 84 | 4 | no | — |
| 262 | C22 | — | 84 | 4 | no | — |
| 263 | C22 | D4×D21 | 1 | 1 | yes | characteristic, complemented |
| 264 | C22 | C2×C7⋊S4 | 1 | 2 | yes | complemented |
| 265 | C4 | — | 3 | 1 | no | — |
| 266 | C4 | — | 21 | 2 | no | — |
| 267 | C4 | — | 42 | 2 | no | — |
| 268 | C4 | — | 42 | 1 | no | — |
| 269 | C4 | — | 42 | 1 | no | — |
| 270 | C4 | C2×C7⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 271 | C3 | — | 4 | 1 | no | Sylow |
| 272 | C2 | — | 2 | 2 | no | — |
| 273 | C2 | — | 3 | 1 | no | — |
| 274 | C2 | — | 3 | 1 | no | — |
| 275 | C2 | — | 6 | 2 | no | — |
| 276 | C2 | — | 42 | 2 | no | — |
| 277 | C2 | — | 84 | 2 | no | — |
| 278 | C2 | C22×C7⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C7⋊S4 |
| commutator, cosocle | C2×C7×A4 | C23 |
| Fitting | C22×C7×D4 | D3 |
| socle | C23×C7 | C22×D3 |
| hypercenter | D4 | C7⋊S4 |
| hypocenter | C7×A4 | C2×D4 |
| LCS | G ▷ C2×C7×A4 ▷ C7×A4 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7⋊S4 | C2 | semidirect |
| 2. C7×D4×A4 | C2 | semidirect |
| 3. #10621506 | C2 | semidirect |
| 4. C4⋊(C7⋊S4) | C2 | semidirect |
| 5. C4×C7⋊S4 | C2 | semidirect |
| 7. C22×C7×A4 | C22 | semidirect |
| 10. C2×C7⋊S4 | C22 | semidirect |
| 12. C4×C7×A4 | C22 | semidirect |
| 13. C7⋊SL(2,ℤ/4) | C22 | semidirect |
| 14. C22×C7×D4 | D3 | semidirect |
| 30. C7⋊S4 | D4 | direct (N × H) |
| 35. C24×C7 | C2×D3 | semidirect |
| 40. C22×C4×C7 | C2×D3 | semidirect |
| 64. D4×A4 | D7 | semidirect |
| 72. C7×A4 | C2×D4 | semidirect |
| 90. C7×D4 | S4 | semidirect |
| 111. C22×A4 | C2×D7 | semidirect |
| 116. C4×A4 | C2×D7 | semidirect |
| 124. C22×D4 | D21 | semidirect |
| 140. C22×C7 | D3×D4 | semidirect |
| 141. C22×C7 | C2×S4 | semidirect |
| 151. C4×C7 | C2×S4 | semidirect |
| 167. C24 | C2×D21 | semidirect |
| 185. C22×C4 | C2×D21 | semidirect |
| 205. A4 | D4×D7 | semidirect |
| 233. D4 | C7⋊S4 | direct (N × H) |
| 243. C7 | D4×S4 | semidirect |
| 263. C22 | D4×D21 | semidirect |
| 264. C22 | C2×C7⋊S4 | semidirect |
| 270. C4 | C2×C7⋊S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7⋊S4 | 2 |
| C7×D4×A4 | 2 |
| #10621506 | 2 |
| C4⋊(C7⋊S4) | 2 |
| C4×C7⋊S4 | 2 |
| D4×C22⋊D7 | 3 |
| D4×D21 | 4 |
| D4×S4 | 7 |