Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
288 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 37 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 | D7×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10621503 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C7×#985 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10621475 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10621474 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10621472 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10621469 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #90459 | — | 3 | 1 | no | maximal, Hall |
| 9 | C7×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #66675 | — | 4 | 1 | no | maximal |
| 11 | D7×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | #66662 | C22 | 1 | 1 | yes | characteristic |
| 13 | C7⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C7⋊#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C7×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C7×#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | D7×#41 | — | 3 | 1 | no | — |
| 18 | D7×#41 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C7×#135 | — | 3 | 1 | no | — |
| 20 | #6513 | — | 3 | 1 | no | — |
| 21 | #6513 | — | 3 | 1 | no | — |
| 22 | #6481 | — | 3 | 1 | no | — |
| 23 | #6481 | — | 3 | 1 | no | — |
| 24 | #6478 | — | 3 | 1 | no | — |
| 25 | #6478 | — | 3 | 1 | no | — |
| 26 | #6477 | — | 3 | 1 | no | — |
| 27 | #6477 | — | 3 | 1 | no | — |
| 28 | #6474 | — | 3 | 1 | no | — |
| 29 | #6474 | — | 3 | 1 | no | — |
| 30 | #6467 | — | 3 | 1 | no | — |
| 31 | C2×#1131 | — | 3 | 1 | no | — |
| 32 | #5923 | — | 7 | 1 | no | maximal, Hall |
| 33 | C4×D21 | — | 4 | 1 | no | — |
| 34 | C4×D3×C7 | — | 4 | 1 | no | — |
| 35 | C3×C4×D7 | — | 4 | 1 | no | — |
| 36 | C7×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 37 | #4248 | — | 4 | 1 | no | — |
| 38 | #4247 | — | 4 | 1 | no | — |
| 39 | #4246 | — | 4 | 1 | no | — |
| 40 | #4245 | — | 4 | 1 | no | — |
| 41 | C7×#41 | — | 3 | 1 | no | — |
| 42 | C7×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | #1161 | — | 3 | 1 | no | — |
| 44 | #1161 | C2×D3 | 1 | 1 | yes | characteristic |
| 45 | Q8×D7 | — | 3 | 1 | no | — |
| 46 | Q8×D7 | C2×D3 | 1 | 1 | yes | characteristic |
| 47 | #1159 | — | 3 | 1 | no | — |
| 48 | #1159 | — | 3 | 1 | no | — |
| 49 | #1159 | — | 6 | 1 | no | — |
| 50 | D4×D7 | — | 3 | 1 | no | — |
| 51 | D4×D7 | — | 3 | 1 | no | — |
| 52 | D4×D7 | — | 6 | 1 | no | — |
| 53 | #1157 | — | 3 | 1 | no | — |
| 54 | #1157 | — | 3 | 1 | no | — |
| 55 | #1157 | — | 6 | 1 | no | — |
| 56 | C2×C4×D7 | — | 3 | 1 | no | — |
| 57 | C2×C4×D7 | — | 6 | 1 | no | — |
| 58 | C7×Q16 | — | 3 | 1 | no | — |
| 59 | C7×SD16 | — | 3 | 1 | no | — |
| 60 | C7×SD16 | — | 3 | 1 | no | — |
| 61 | C7×D8 | — | 3 | 1 | no | — |
| 62 | C2×C7×C8 | — | 3 | 1 | no | — |
| 63 | #1144 | — | 3 | 1 | no | — |
| 64 | #1144 | — | 3 | 1 | no | — |
| 65 | #1143 | — | 3 | 1 | no | — |
| 66 | #1143 | — | 3 | 1 | no | — |
| 67 | #1142 | — | 3 | 1 | no | — |
| 68 | #1142 | — | 3 | 1 | no | — |
| 69 | #1141 | — | 3 | 1 | no | — |
| 70 | #1141 | — | 3 | 1 | no | — |
| 71 | C2×C7⋊C8 | — | 3 | 1 | no | — |
| 72 | Dic28 | — | 3 | 1 | no | — |
| 73 | D56 | — | 3 | 1 | no | — |
| 74 | #1132 | — | 3 | 1 | no | — |
| 75 | #1132 | — | 3 | 1 | no | — |
| 76 | #1131 | — | 3 | 1 | no | — |
| 77 | #1131 | — | 3 | 1 | no | — |
| 78 | #1131 | — | 3 | 1 | no | — |
| 79 | #1131 | — | 3 | 1 | no | — |
| 80 | C2×#41⋊C3 | — | 7 | 1 | no | — |
| 81 | #986 | — | 7 | 1 | no | — |
| 82 | #985 | — | 7 | 1 | no | — |
| 83 | #985 | D7 | 1 | 1 | yes | characteristic, complemented |
| 84 | #984 | — | 7 | 1 | no | — |
| 85 | #983 | — | 7 | 1 | no | — |
| 86 | #983 | — | 7 | 1 | no | — |
| 87 | C2×D21 | — | 4 | 1 | no | — |
| 88 | C2×D3×C7 | — | 4 | 1 | no | — |
| 89 | C2×C3×D7 | — | 4 | 1 | no | — |
| 90 | C3×C4×C7 | — | 4 | 1 | no | — |
| 91 | Dic21 | — | 4 | 1 | no | — |
| 92 | C3×Dic7 | — | 4 | 1 | no | — |
| 93 | C7×Dic3 | — | 4 | 1 | no | — |
| 94 | #575 | — | 21 | 1 | no | Sylow |
| 95 | C22×D7 | — | 3 | 1 | no | — |
| 96 | C22×D7 | — | 6 | 1 | no | — |
| 97 | C7×Q8 | — | 3 | 1 | no | — |
| 98 | C7×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 99 | C7×D4 | — | 3 | 1 | no | — |
| 100 | C7×D4 | — | 3 | 1 | no | — |
| 101 | C7×D4 | — | 6 | 1 | no | — |
| 102 | C2×C4×C7 | — | 3 | 1 | no | — |
| 103 | C2×C4×C7 | — | 6 | 1 | no | — |
| 104 | C22⋊D7 | — | 6 | 1 | no | — |
| 105 | C22⋊D7 | — | 6 | 1 | no | — |
| 106 | C22⋊D7 | — | 6 | 1 | no | — |
| 107 | C22⋊D7 | — | 6 | 1 | no | — |
| 108 | C2×Dic7 | — | 3 | 1 | no | — |
| 109 | C2×Dic7 | — | 6 | 1 | no | — |
| 110 | D28 | — | 3 | 1 | no | — |
| 111 | D28 | — | 3 | 1 | no | — |
| 112 | D28 | — | 6 | 1 | no | — |
| 113 | C4×D7 | — | 3 | 1 | no | — |
| 114 | C4×D7 | — | 3 | 1 | no | — |
| 115 | C4×D7 | — | 3 | 1 | no | — |
| 116 | C4×D7 | — | 6 | 1 | no | — |
| 117 | C4×D7 | — | 6 | 1 | no | — |
| 118 | C4×D7 | — | 6 | 1 | no | — |
| 119 | C4×D7 | S4 | 1 | 1 | yes | characteristic |
| 120 | Dic14 | — | 3 | 1 | no | — |
| 121 | Dic14 | — | 3 | 1 | no | — |
| 122 | Dic14 | — | 6 | 1 | no | — |
| 123 | C7×C8 | — | 3 | 1 | no | — |
| 124 | C7×C8 | — | 3 | 1 | no | — |
| 125 | C7⋊C8 | — | 3 | 1 | no | — |
| 126 | C7⋊C8 | — | 3 | 1 | no | — |
| 127 | #235 | — | 28 | 1 | no | — |
| 128 | #41⋊C3 | — | 7 | 1 | no | — |
| 129 | #41⋊C3 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 130 | C2×SL(2,𝔽3) | — | 7 | 1 | no | — |
| 131 | GL(2,𝔽3) | — | 7 | 1 | no | — |
| 132 | GL(2,𝔽3) | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 133 | #226 | — | 7 | 1 | no | — |
| 134 | #226 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 135 | C2×C3×C7 | — | 4 | 1 | no | — |
| 136 | D21 | — | 8 | 1 | no | — |
| 137 | C3×D7 | — | 8 | 1 | no | — |
| 138 | D3×C7 | — | 8 | 1 | no | — |
| 139 | C2×#41 | — | 7 | 1 | no | — |
| 140 | C2×#41 | — | 21 | 1 | no | — |
| 141 | #137 | — | 21 | 1 | no | — |
| 142 | #137 | — | 21 | 1 | no | — |
| 143 | #137 | — | 21 | 1 | no | — |
| 144 | #137 | — | 21 | 1 | no | — |
| 145 | AGL(1,ℤ/8) | — | 21 | 1 | no | — |
| 146 | AGL(1,ℤ/8) | — | 21 | 1 | no | — |
| 147 | AGL(1,ℤ/8) | — | 21 | 1 | no | — |
| 148 | AGL(1,ℤ/8) | — | 21 | 1 | no | — |
| 149 | #135 | — | 3 | 1 | no | — |
| 150 | #135 | — | 21 | 1 | no | — |
| 151 | #135 | — | 21 | 1 | no | — |
| 152 | #135 | — | 21 | 1 | no | — |
| 153 | C2×M16 | — | 21 | 1 | no | — |
| 154 | C22×C7 | — | 3 | 1 | no | — |
| 155 | C22×C7 | — | 6 | 1 | no | — |
| 156 | C2×D7 | — | 3 | 1 | no | — |
| 157 | C2×D7 | — | 6 | 1 | no | — |
| 158 | C2×D7 | — | 12 | 1 | no | — |
| 159 | C2×D7 | — | 12 | 2 | no | — |
| 160 | C2×D7 | C2×S4 | 1 | 1 | yes | characteristic |
| 161 | C4×C7 | — | 3 | 1 | no | — |
| 162 | C4×C7 | — | 6 | 1 | no | — |
| 163 | C4×C7 | C2×S4 | 1 | 1 | yes | characteristic |
| 164 | Dic7 | — | 3 | 1 | no | — |
| 165 | Dic7 | — | 6 | 1 | no | — |
| 166 | Dic7 | C2×S4 | 1 | 1 | yes | characteristic |
| 167 | C2×C3×C4 | — | 28 | 1 | no | — |
| 168 | #67 | — | 28 | 1 | no | — |
| 169 | #67 | — | 28 | 1 | no | — |
| 170 | D12 | — | 28 | 1 | no | — |
| 171 | C4×D3 | — | 4 | 1 | no | — |
| 172 | C4×D3 | — | 28 | 1 | no | — |
| 173 | Dic6 | — | 28 | 1 | no | — |
| 174 | SL(2,𝔽3) | C22×D7 | 1 | 1 | yes | characteristic |
| 175 | C3×C7 | — | 4 | 1 | no | Hall |
| 176 | #41 | — | 3 | 1 | no | — |
| 177 | #41 | — | 7 | 1 | no | — |
| 178 | #41 | — | 21 | 1 | no | — |
| 179 | #41 | — | 21 | 1 | no | — |
| 180 | #41 | — | 21 | 1 | no | — |
| 181 | #41 | — | 21 | 1 | no | — |
| 182 | #41 | — | 21 | 1 | no | — |
| 183 | #41 | — | 42 | 1 | no | — |
| 184 | #41 | — | 42 | 1 | no | — |
| 185 | #41 | D3×D7 | 1 | 1 | yes | characteristic |
| 186 | C2×Q8 | — | 7 | 1 | no | — |
| 187 | C2×Q8 | — | 21 | 1 | no | — |
| 188 | C2×D4 | — | 21 | 1 | no | — |
| 189 | C2×D4 | — | 21 | 1 | no | — |
| 190 | C2×D4 | — | 42 | 1 | no | — |
| 191 | C22×C4 | — | 21 | 1 | no | — |
| 192 | C22×C4 | — | 42 | 1 | no | — |
| 193 | Q16 | — | 3 | 1 | no | — |
| 194 | Q16 | — | 21 | 1 | no | — |
| 195 | Q16 | — | 21 | 1 | no | — |
| 196 | Q16 | — | 21 | 1 | no | — |
| 197 | SD16 | — | 3 | 1 | no | — |
| 198 | SD16 | — | 3 | 1 | no | — |
| 199 | SD16 | — | 21 | 1 | no | — |
| 200 | SD16 | — | 21 | 1 | no | — |
| 201 | SD16 | — | 21 | 1 | no | — |
| 202 | SD16 | — | 21 | 1 | no | — |
| 203 | SD16 | — | 21 | 1 | no | — |
| 204 | SD16 | — | 21 | 1 | no | — |
| 205 | D8 | — | 3 | 1 | no | — |
| 206 | D8 | — | 21 | 1 | no | — |
| 207 | D8 | — | 21 | 1 | no | — |
| 208 | D8 | — | 21 | 1 | no | — |
| 209 | M16 | — | 21 | 1 | no | — |
| 210 | M16 | — | 21 | 1 | no | — |
| 211 | M16 | — | 21 | 1 | no | — |
| 212 | M16 | — | 21 | 1 | no | — |
| 213 | C2×C8 | — | 3 | 1 | no | — |
| 214 | C2×C8 | — | 21 | 1 | no | — |
| 215 | C2×C7 | — | 6 | 1 | no | — |
| 216 | C2×C7 | — | 12 | 1 | no | — |
| 217 | C2×C7 | C22×S4 | 1 | 1 | yes | characteristic |
| 218 | D7 | — | 2 | 1 | no | — |
| 219 | D7 | — | 6 | 1 | no | — |
| 220 | D7 | — | 12 | 1 | no | — |
| 221 | C22×C3 | — | 28 | 1 | no | — |
| 222 | C2×D3 | — | 4 | 1 | no | — |
| 223 | C2×D3 | — | 28 | 1 | no | — |
| 224 | C3×C4 | — | 4 | 1 | no | — |
| 225 | C3×C4 | — | 28 | 1 | no | — |
| 226 | Dic3 | — | 4 | 1 | no | — |
| 227 | Dic3 | — | 28 | 1 | no | — |
| 228 | C23 | — | 21 | 1 | no | — |
| 229 | C23 | — | 42 | 1 | no | — |
| 230 | Q8 | — | 3 | 1 | no | — |
| 231 | Q8 | — | 21 | 1 | no | — |
| 232 | Q8 | — | 21 | 1 | no | — |
| 233 | Q8 | — | 42 | 1 | no | — |
| 234 | Q8 | C2×D3×D7 | 1 | 1 | yes | characteristic |
| 235 | D4 | — | 3 | 1 | no | — |
| 236 | D4 | — | 3 | 1 | no | — |
| 237 | D4 | — | 6 | 1 | no | — |
| 238 | D4 | — | 21 | 1 | no | — |
| 239 | D4 | — | 21 | 1 | no | — |
| 240 | D4 | — | 42 | 1 | no | — |
| 241 | D4 | — | 42 | 1 | no | — |
| 242 | D4 | — | 42 | 1 | no | — |
| 243 | D4 | — | 42 | 1 | no | — |
| 244 | D4 | — | 42 | 1 | no | — |
| 245 | C2×C4 | — | 3 | 1 | no | — |
| 246 | C2×C4 | — | 6 | 1 | no | — |
| 247 | C2×C4 | — | 7 | 1 | no | — |
| 248 | C2×C4 | — | 21 | 1 | no | — |
| 249 | C2×C4 | — | 21 | 1 | no | — |
| 250 | C2×C4 | — | 21 | 1 | no | — |
| 251 | C2×C4 | — | 21 | 1 | no | — |
| 252 | C2×C4 | — | 42 | 1 | no | — |
| 253 | C2×C4 | — | 42 | 1 | no | — |
| 254 | C2×C4 | — | 42 | 1 | no | — |
| 255 | C2×C4 | — | 42 | 1 | no | — |
| 256 | C8 | — | 3 | 1 | no | — |
| 257 | C8 | — | 3 | 1 | no | — |
| 258 | C8 | — | 21 | 1 | no | — |
| 259 | C8 | — | 21 | 1 | no | — |
| 260 | C7 | #5923 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 261 | C2×C3 | — | 4 | 1 | no | — |
| 262 | C2×C3 | — | 56 | 1 | no | — |
| 263 | D3 | — | 8 | 1 | no | — |
| 264 | D3 | — | 56 | 1 | no | — |
| 265 | C22 | — | 3 | 1 | no | — |
| 266 | C22 | — | 6 | 1 | no | — |
| 267 | C22 | — | 7 | 1 | no | — |
| 268 | C22 | — | 21 | 1 | no | — |
| 269 | C22 | — | 42 | 1 | no | — |
| 270 | C22 | — | 84 | 1 | no | — |
| 271 | C22 | — | 84 | 2 | no | — |
| 272 | C4 | — | 3 | 1 | no | — |
| 273 | C4 | — | 6 | 1 | no | — |
| 274 | C4 | — | 7 | 1 | no | — |
| 275 | C4 | — | 21 | 1 | no | — |
| 276 | C4 | — | 42 | 1 | no | — |
| 277 | C4 | D7×S4 | 1 | 1 | yes | characteristic |
| 278 | C3 | — | 4 | 1 | no | Sylow |
| 279 | C2 | — | 6 | 1 | no | — |
| 280 | C2 | — | 12 | 1 | no | — |
| 281 | C2 | — | 14 | 1 | no | — |
| 282 | C2 | — | 42 | 1 | no | — |
| 283 | C2 | — | 84 | 1 | no | — |
| 284 | C2 | C2×D7×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D7×S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C23 |
| Frattini | C2 | C2×D7×S4 |
| Fitting | C7×#41 | C2×D3 |
| socle | C2×C7 | C22×S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D7×#41⋊C3 | C2 | semidirect |
| 2. #10621503 | C2 | semidirect |
| 3. C7×#985 | C2 | semidirect |
| 4. #10621475 | C2 | semidirect |
| 5. #10621474 | C2 | semidirect |
| 6. #10621472 | C2 | semidirect |
| 7. #10621469 | C2 | semidirect |
| 9. C7×#41⋊C3 | C22 | semidirect |
| 11. D7×SL(2,𝔽3) | C22 | semidirect |
| 13. C7⋊GL(2,𝔽3) | C22 | semidirect |
| 14. C7⋊#226 | C22 | semidirect |
| 15. C7×GL(2,𝔽3) | C22 | semidirect |
| 16. C7×#226 | C22 | semidirect |
| 18. D7×#41 | D3 | semidirect |
| 83. #985 | D7 | semidirect |
| 129. #41⋊C3 | C2×D7 | semidirect |
| 132. GL(2,𝔽3) | C2×D7 | semidirect |
| 134. #226 | C2×D7 | semidirect |
| 260. C7 | #5923 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D7×#41⋊C3 | 2 |
| #10621503 | 2 |
| C7×#985 | 2 |
| #10621475 | 2 |
| #10621474 | 2 |
| #10621472 | 2 |
| #10621469 | 2 |
| #90459 | 3 |
| #66675 | 4 |
| #5923 | 7 |