Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
2154 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 2154 conjugacy classes of subgroups — too many to draw. Only its 82 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 | C7×#5950 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C7×#997 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C7×H(7,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 4 | C7×#997 | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | C7×2−1+4⋊C3 | — | 4 | 2 | no | maximal |
| 6 | C7×D4×A4 | — | 4 | 3 | no | maximal |
| 7 | C7×#585 | — | 3 | 4 | no | — |
| 8 | C7×#585 | — | 3 | 2 | no | — |
| 9 | C7×#585 | — | 3 | 3 | no | — |
| 10 | C7×#585 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×C7×H(5,𝔽2) | — | 3 | 2 | no | — |
| 12 | C2×C7×H(5,𝔽2) | — | 3 | 3 | no | — |
| 13 | C2×C7×H(5,𝔽2) | — | 3 | 6 | no | — |
| 14 | C2×C7×H(5,𝔽2) | C2×C3 | 1 | 2 | yes | complemented |
| 15 | #87547 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 16 | C22×C7×A4 | — | 4 | 6 | no | — |
| 17 | C7×#41⋊C3 | — | 4 | 2 | no | — |
| 18 | C2×C7×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 19 | C4×C7×A4 | — | 4 | 3 | no | — |
| 20 | C7×2−1+4 | — | 3 | 6 | no | — |
| 21 | C7×2−1+4 | — | 3 | 6 | no | — |
| 22 | C7×2−1+4 | — | 3 | 4 | no | — |
| 23 | C7×2−1+4 | — | 3 | 2 | no | — |
| 24 | C7×2−1+4 | A4 | 1 | 2 | yes | complemented |
| 25 | C7×H(5,𝔽2) | — | 3 | 4 | no | — |
| 26 | C7×H(5,𝔽2) | — | 3 | 12 | no | — |
| 27 | C7×H(5,𝔽2) | — | 3 | 6 | no | — |
| 28 | C7×H(5,𝔽2) | — | 3 | 12 | no | — |
| 29 | C7×H(5,𝔽2) | — | 3 | 12 | no | — |
| 30 | C7×H(5,𝔽2) | — | 3 | 24 | no | — |
| 31 | C7×H(5,𝔽2) | — | 3 | 6 | no | — |
| 32 | C7×H(5,𝔽2) | — | 3 | 6 | no | — |
| 33 | C7×H(5,𝔽2) | — | 3 | 2 | no | — |
| 34 | C7×H(5,𝔽2) | — | 3 | 3 | no | — |
| 35 | C7×H(5,𝔽2) | — | 3 | 6 | no | — |
| 36 | C7×H(5,𝔽2) | C22×C3 | 1 | 1 | yes | characteristic |
| 37 | C2×C7×#41 | — | 3 | 24 | no | — |
| 38 | C2×C7×#41 | — | 3 | 12 | no | — |
| 39 | C2×C7×#41 | — | 3 | 4 | no | — |
| 40 | C2×C7×#41 | — | 3 | 6 | no | — |
| 41 | C2×C7×#41 | — | 3 | 3 | no | — |
| 42 | C2×C7×#41 | — | 3 | 6 | no | — |
| 43 | C2×C7×#41 | — | 3 | 3 | no | — |
| 44 | C2×C7×#41 | — | 3 | 2 | no | — |
| 45 | C2×C7×#41 | — | 3 | 4 | no | — |
| 46 | C2×C7×#41 | — | 3 | 3 | no | — |
| 47 | C2×C7×#41 | — | 3 | 3 | no | — |
| 48 | C22×C7×D4 | — | 3 | 12 | no | — |
| 49 | C22×C7×D4 | — | 3 | 3 | no | — |
| 50 | C22×C7×D4 | — | 3 | 6 | no | — |
| 51 | C22×C7×D4 | — | 3 | 1 | no | — |
| 52 | C22×C7×D4 | — | 3 | 6 | no | — |
| 53 | C22×C7×D4 | — | 3 | 6 | no | — |
| 54 | C22×C7×D4 | A4 | 1 | 3 | yes | complemented |
| 55 | #5950 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 56 | C2×#997 | C2×C7 | 1 | 2 | yes | complemented |
| 57 | C2×C7×A4 | — | 4 | 3 | no | — |
| 58 | C2×C7×A4 | — | 8 | 6 | no | — |
| 59 | C3×C7×D4 | — | 16 | 1 | no | — |
| 60 | C7×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 61 | H(7,𝔽2) | C3×C7 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 62 | C24×C7 | — | 3 | 6 | no | — |
| 63 | C24×C7 | — | 3 | 2 | no | — |
| 64 | C24×C7 | C2×A4 | 1 | 6 | yes | complemented |
| 65 | C7×#41 | — | 3 | 6 | no | — |
| 66 | C7×#41 | — | 3 | 12 | no | — |
| 67 | C7×#41 | — | 3 | 12 | no | — |
| 68 | C7×#41 | — | 3 | 12 | no | — |
| 69 | C7×#41 | — | 3 | 24 | no | — |
| 70 | C7×#41 | — | 3 | 24 | no | — |
| 71 | C7×#41 | — | 3 | 12 | no | — |
| 72 | C7×#41 | — | 3 | 12 | no | — |
| 73 | C7×#41 | — | 3 | 12 | no | — |
| 74 | C7×#41 | — | 3 | 6 | no | — |
| 75 | C7×#41 | — | 3 | 12 | no | — |
| 76 | C7×#41 | — | 3 | 12 | no | — |
| 77 | C7×#41 | — | 3 | 12 | no | — |
| 78 | C7×#41 | — | 3 | 6 | no | — |
| 79 | C7×#41 | — | 3 | 2 | no | — |
| 80 | C7×#41 | — | 3 | 4 | no | — |
| 81 | C7×#41 | — | 3 | 2 | no | — |
| 82 | C7×#41 | — | 3 | 4 | no | — |
| 83 | C7×#41 | C2×A4 | 1 | 2 | yes | complemented |
| 84 | C2×C7×Q8 | — | 3 | 12 | no | — |
| 85 | C2×C7×Q8 | — | 3 | 3 | no | — |
| 86 | C2×C7×Q8 | — | 3 | 3 | no | — |
| 87 | C2×C7×Q8 | — | 3 | 2 | no | — |
| 88 | C2×C7×Q8 | — | 3 | 2 | no | — |
| 89 | C2×C7×Q8 | C2×A4 | 1 | 4 | yes | complemented |
| 90 | C2×C7×D4 | — | 3 | 12 | no | — |
| 91 | C2×C7×D4 | — | 3 | 6 | no | — |
| 92 | C2×C7×D4 | — | 3 | 12 | no | — |
| 93 | C2×C7×D4 | — | 3 | 24 | no | — |
| 94 | C2×C7×D4 | — | 3 | 12 | no | — |
| 95 | C2×C7×D4 | — | 3 | 24 | no | — |
| 96 | C2×C7×D4 | — | 3 | 6 | no | — |
| 97 | C2×C7×D4 | — | 3 | 24 | no | — |
| 98 | C2×C7×D4 | — | 3 | 6 | no | — |
| 99 | C2×C7×D4 | — | 3 | 6 | no | — |
| 100 | C2×C7×D4 | — | 3 | 24 | no | — |
| 101 | C2×C7×D4 | — | 3 | 6 | no | — |
| 102 | C2×C7×D4 | — | 3 | 12 | no | — |
| 103 | C2×C7×D4 | — | 3 | 6 | no | — |
| 104 | C2×C7×D4 | — | 3 | 3 | no | — |
| 105 | C2×C7×D4 | — | 3 | 3 | no | — |
| 106 | C2×C7×D4 | — | 3 | 3 | no | — |
| 107 | C2×C7×D4 | — | 3 | 6 | no | — |
| 108 | C2×C7×D4 | — | 3 | 3 | no | — |
| 109 | C2×C7×D4 | — | 3 | 3 | no | — |
| 110 | C2×C7×D4 | — | 3 | 3 | no | — |
| 111 | C2×C7×D4 | — | 3 | 6 | no | — |
| 112 | C22×C4×C7 | — | 3 | 12 | no | — |
| 113 | C22×C4×C7 | — | 3 | 6 | no | — |
| 114 | C22×C4×C7 | — | 3 | 6 | no | — |
| 115 | C22×C4×C7 | — | 3 | 6 | no | — |
| 116 | C22×C4×C7 | — | 3 | 3 | no | — |
| 117 | C22×C4×C7 | — | 3 | 1 | no | — |
| 118 | C22×C4×C7 | C2×A4 | 1 | 3 | yes | complemented |
| 119 | #997 | C22×C7 | 1 | 1 | yes | characteristic, complemented |
| 120 | 2−1+4⋊C3 | — | 4 | 2 | no | — |
| 121 | D4×A4 | — | 4 | 3 | no | — |
| 122 | C22×C3×C7 | — | 16 | 2 | no | — |
| 123 | C7×A4 | — | 4 | 3 | no | — |
| 124 | C3×C4×C7 | — | 16 | 1 | no | — |
| 125 | #585 | — | 3 | 4 | no | — |
| 126 | #585 | — | 3 | 2 | no | — |
| 127 | #585 | — | 3 | 3 | no | — |
| 128 | #585 | C2×C3×C7 | 1 | 1 | yes | characteristic, complemented |
| 129 | C2×H(5,𝔽2) | — | 3 | 2 | no | — |
| 130 | C2×H(5,𝔽2) | — | 3 | 3 | no | — |
| 131 | C2×H(5,𝔽2) | — | 3 | 6 | no | — |
| 132 | C2×H(5,𝔽2) | C2×C3×C7 | 1 | 2 | yes | complemented |
| 133 | C23×C7 | — | 3 | 6 | no | — |
| 134 | C23×C7 | — | 3 | 12 | no | — |
| 135 | C23×C7 | — | 3 | 1 | no | — |
| 136 | C23×C7 | — | 3 | 3 | no | — |
| 137 | C23×C7 | — | 3 | 6 | no | — |
| 138 | C23×C7 | — | 3 | 6 | no | — |
| 139 | C23×C7 | — | 8 | 6 | no | — |
| 140 | C23×C7 | — | 24 | 2 | no | — |
| 141 | C23×C7 | — | 24 | 6 | no | — |
| 142 | C23×C7 | C22×A4 | 1 | 3 | yes | — |
| 143 | C7×Q8 | — | 3 | 4 | no | — |
| 144 | C7×Q8 | — | 3 | 6 | no | — |
| 145 | C7×Q8 | — | 3 | 6 | no | — |
| 146 | C7×Q8 | — | 3 | 2 | no | — |
| 147 | C7×Q8 | C22×A4 | 1 | 2 | yes | — |
| 148 | C7×D4 | — | 3 | 2 | no | — |
| 149 | C7×D4 | — | 3 | 24 | no | — |
| 150 | C7×D4 | — | 3 | 12 | no | — |
| 151 | C7×D4 | — | 3 | 12 | no | — |
| 152 | C7×D4 | — | 3 | 6 | no | — |
| 153 | C7×D4 | — | 3 | 6 | no | — |
| 154 | C7×D4 | — | 3 | 6 | no | — |
| 155 | C7×D4 | — | 3 | 12 | no | — |
| 156 | C7×D4 | — | 3 | 4 | no | — |
| 157 | C7×D4 | — | 3 | 6 | no | — |
| 158 | C7×D4 | — | 3 | 3 | no | — |
| 159 | C7×D4 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 160 | C2×C4×C7 | — | 3 | 6 | no | — |
| 161 | C2×C4×C7 | — | 3 | 4 | no | — |
| 162 | C2×C4×C7 | — | 3 | 12 | no | — |
| 163 | C2×C4×C7 | — | 3 | 24 | no | — |
| 164 | C2×C4×C7 | — | 3 | 3 | no | — |
| 165 | C2×C4×C7 | — | 3 | 6 | no | — |
| 166 | C2×C4×C7 | — | 3 | 3 | no | — |
| 167 | C2×C4×C7 | — | 3 | 4 | no | — |
| 168 | C2×C4×C7 | — | 3 | 2 | no | — |
| 169 | C2×C4×C7 | — | 3 | 3 | no | — |
| 170 | C2×C4×C7 | — | 3 | 3 | no | — |
| 171 | C22×A4 | — | 4 | 6 | no | — |
| 172 | #41⋊C3 | — | 4 | 2 | no | — |
| 173 | C2×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 174 | C4×A4 | — | 4 | 3 | no | — |
| 175 | C2×C3×C7 | — | 16 | 1 | no | — |
| 176 | C2×C3×C7 | — | 32 | 2 | no | — |
| 177 | 2−1+4 | — | 3 | 6 | no | — |
| 178 | 2−1+4 | — | 3 | 6 | no | — |
| 179 | 2−1+4 | — | 3 | 4 | no | — |
| 180 | 2−1+4 | — | 3 | 2 | no | — |
| 181 | 2−1+4 | C7×A4 | 1 | 2 | yes | complemented |
| 182 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 183 | H(5,𝔽2) | — | 3 | 12 | no | — |
| 184 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 185 | H(5,𝔽2) | — | 3 | 12 | no | — |
| 186 | H(5,𝔽2) | — | 3 | 12 | no | — |
| 187 | H(5,𝔽2) | — | 3 | 24 | no | — |
| 188 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 189 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 190 | H(5,𝔽2) | — | 3 | 2 | no | — |
| 191 | H(5,𝔽2) | — | 3 | 3 | no | — |
| 192 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 193 | H(5,𝔽2) | C22×C3×C7 | 1 | 1 | yes | characteristic |
| 194 | C2×#41 | — | 3 | 6 | no | — |
| 195 | C2×#41 | — | 3 | 24 | no | — |
| 196 | C2×#41 | — | 3 | 12 | no | — |
| 197 | C2×#41 | — | 3 | 4 | no | — |
| 198 | C2×#41 | — | 3 | 3 | no | — |
| 199 | C2×#41 | — | 3 | 6 | no | — |
| 200 | C2×#41 | — | 3 | 3 | no | — |
| 201 | C2×#41 | — | 3 | 2 | no | — |
| 202 | C2×#41 | — | 3 | 4 | no | — |
| 203 | C2×#41 | — | 3 | 3 | no | — |
| 204 | C2×#41 | — | 3 | 3 | no | — |
| 205 | C22×D4 | — | 3 | 12 | no | — |
| 206 | C22×D4 | — | 3 | 3 | no | — |
| 207 | C22×D4 | — | 3 | 6 | no | — |
| 208 | C22×D4 | — | 3 | 1 | no | — |
| 209 | C22×D4 | — | 3 | 6 | no | — |
| 210 | C22×D4 | — | 3 | 6 | no | — |
| 211 | C22×D4 | C7×A4 | 1 | 3 | yes | complemented |
| 212 | C22×C7 | — | 3 | 6 | no | — |
| 213 | C22×C7 | — | 3 | 3 | no | — |
| 214 | C22×C7 | — | 3 | 2 | no | — |
| 215 | C22×C7 | — | 4 | 3 | no | — |
| 216 | C22×C7 | — | 12 | 6 | no | — |
| 217 | C22×C7 | — | 12 | 6 | no | — |
| 218 | C22×C7 | — | 12 | 1 | no | — |
| 219 | C22×C7 | — | 12 | 6 | no | — |
| 220 | C22×C7 | — | 12 | 3 | no | — |
| 221 | C22×C7 | — | 12 | 12 | no | — |
| 222 | C22×C7 | C2×C22⋊A4 | 1 | 2 | yes | — |
| 223 | C4×C7 | — | 3 | 4 | no | — |
| 224 | C4×C7 | — | 3 | 2 | no | — |
| 225 | C4×C7 | — | 3 | 3 | no | — |
| 226 | C4×C7 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 227 | C2×A4 | — | 4 | 3 | no | — |
| 228 | C2×A4 | — | 8 | 6 | no | — |
| 229 | C3×D4 | — | 16 | 1 | no | — |
| 230 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 231 | C3×C7 | — | 16 | 1 | no | Hall |
| 232 | C24 | — | 3 | 6 | no | — |
| 233 | C24 | — | 3 | 2 | no | — |
| 234 | C24 | C2×C7×A4 | 1 | 6 | yes | complemented |
| 235 | #41 | — | 3 | 6 | no | — |
| 236 | #41 | — | 3 | 12 | no | — |
| 237 | #41 | — | 3 | 12 | no | — |
| 238 | #41 | — | 3 | 12 | no | — |
| 239 | #41 | — | 3 | 24 | no | — |
| 240 | #41 | — | 3 | 24 | no | — |
| 241 | #41 | — | 3 | 12 | no | — |
| 242 | #41 | — | 3 | 12 | no | — |
| 243 | #41 | — | 3 | 12 | no | — |
| 244 | #41 | — | 3 | 6 | no | — |
| 245 | #41 | — | 3 | 12 | no | — |
| 246 | #41 | — | 3 | 12 | no | — |
| 247 | #41 | — | 3 | 12 | no | — |
| 248 | #41 | — | 3 | 6 | no | — |
| 249 | #41 | — | 3 | 2 | no | — |
| 250 | #41 | — | 3 | 4 | no | — |
| 251 | #41 | — | 3 | 2 | no | — |
| 252 | #41 | — | 3 | 4 | no | — |
| 253 | #41 | C2×C7×A4 | 1 | 2 | yes | complemented |
| 254 | C2×Q8 | — | 3 | 2 | no | — |
| 255 | C2×Q8 | — | 3 | 12 | no | — |
| 256 | C2×Q8 | — | 3 | 3 | no | — |
| 257 | C2×Q8 | — | 3 | 3 | no | — |
| 258 | C2×Q8 | — | 3 | 2 | no | — |
| 259 | C2×Q8 | C2×C7×A4 | 1 | 4 | yes | complemented |
| 260 | C2×D4 | — | 3 | 12 | no | — |
| 261 | C2×D4 | — | 3 | 6 | no | — |
| 262 | C2×D4 | — | 3 | 12 | no | — |
| 263 | C2×D4 | — | 3 | 12 | no | — |
| 264 | C2×D4 | — | 3 | 24 | no | — |
| 265 | C2×D4 | — | 3 | 6 | no | — |
| 266 | C2×D4 | — | 3 | 24 | no | — |
| 267 | C2×D4 | — | 3 | 6 | no | — |
| 268 | C2×D4 | — | 3 | 24 | no | — |
| 269 | C2×D4 | — | 3 | 6 | no | — |
| 270 | C2×D4 | — | 3 | 24 | no | — |
| 271 | C2×D4 | — | 3 | 6 | no | — |
| 272 | C2×D4 | — | 3 | 12 | no | — |
| 273 | C2×D4 | — | 3 | 6 | no | — |
| 274 | C2×D4 | — | 3 | 3 | no | — |
| 275 | C2×D4 | — | 3 | 3 | no | — |
| 276 | C2×D4 | — | 3 | 6 | no | — |
| 277 | C2×D4 | — | 3 | 3 | no | — |
| 278 | C2×D4 | — | 3 | 3 | no | — |
| 279 | C2×D4 | — | 3 | 3 | no | — |
| 280 | C2×D4 | — | 3 | 3 | no | — |
| 281 | C2×D4 | — | 3 | 6 | no | — |
| 282 | C22×C4 | — | 3 | 12 | no | — |
| 283 | C22×C4 | — | 3 | 6 | no | — |
| 284 | C22×C4 | — | 3 | 6 | no | — |
| 285 | C22×C4 | — | 3 | 6 | no | — |
| 286 | C22×C4 | — | 3 | 3 | no | — |
| 287 | C22×C4 | — | 3 | 1 | no | — |
| 288 | C22×C4 | C2×C7×A4 | 1 | 3 | yes | complemented |
| 289 | C2×C7 | — | 2 | 2 | no | — |
| 290 | C2×C7 | — | 6 | 3 | no | — |
| 291 | C2×C7 | — | 6 | 6 | no | — |
| 292 | C2×C7 | — | 6 | 2 | no | — |
| 293 | C2×C7 | C22×C22⋊A4 | 1 | 1 | yes | characteristic |
| 294 | C22×C3 | — | 16 | 2 | no | — |
| 295 | A4 | — | 4 | 3 | no | — |
| 296 | C3×C4 | — | 16 | 1 | no | — |
| 297 | C23 | — | 3 | 1 | no | — |
| 298 | C23 | — | 3 | 6 | no | — |
| 299 | C23 | — | 3 | 3 | no | — |
| 300 | C23 | — | 3 | 12 | no | — |
| 301 | C23 | — | 3 | 6 | no | — |
| 302 | C23 | — | 3 | 6 | no | — |
| 303 | C23 | — | 8 | 6 | no | — |
| 304 | C23 | — | 24 | 2 | no | — |
| 305 | C23 | — | 24 | 6 | no | — |
| 306 | C23 | C22×C7×A4 | 1 | 3 | yes | — |
| 307 | Q8 | — | 3 | 4 | no | — |
| 308 | Q8 | — | 3 | 6 | no | — |
| 309 | Q8 | — | 3 | 6 | no | — |
| 310 | Q8 | — | 3 | 2 | no | — |
| 311 | Q8 | C22×C7×A4 | 1 | 2 | yes | — |
| 312 | D4 | — | 3 | 2 | no | — |
| 313 | D4 | — | 3 | 6 | no | — |
| 314 | D4 | — | 3 | 12 | no | — |
| 315 | D4 | — | 3 | 24 | no | — |
| 316 | D4 | — | 3 | 12 | no | — |
| 317 | D4 | — | 3 | 12 | no | — |
| 318 | D4 | — | 3 | 6 | no | — |
| 319 | D4 | — | 3 | 6 | no | — |
| 320 | D4 | — | 3 | 4 | no | — |
| 321 | D4 | — | 3 | 6 | no | — |
| 322 | D4 | — | 3 | 3 | no | — |
| 323 | D4 | C7×C22⋊A4 | 1 | 1 | yes | characteristic |
| 324 | C2×C4 | — | 3 | 3 | no | — |
| 325 | C2×C4 | — | 3 | 12 | no | — |
| 326 | C2×C4 | — | 3 | 6 | no | — |
| 327 | C2×C4 | — | 3 | 6 | no | — |
| 328 | C2×C4 | — | 3 | 4 | no | — |
| 329 | C2×C4 | — | 3 | 24 | no | — |
| 330 | C2×C4 | — | 3 | 3 | no | — |
| 331 | C2×C4 | — | 3 | 4 | no | — |
| 332 | C2×C4 | — | 3 | 2 | no | — |
| 333 | C2×C4 | — | 3 | 3 | no | — |
| 334 | C2×C4 | — | 3 | 3 | no | — |
| 335 | C7 | #87547 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 336 | C2×C3 | — | 16 | 1 | no | — |
| 337 | C2×C3 | — | 32 | 2 | no | — |
| 338 | C22 | — | 3 | 3 | no | — |
| 339 | C22 | — | 3 | 6 | no | — |
| 340 | C22 | — | 3 | 2 | no | — |
| 341 | C22 | — | 4 | 3 | no | — |
| 342 | C22 | — | 12 | 6 | no | — |
| 343 | C22 | — | 12 | 6 | no | — |
| 344 | C22 | — | 12 | 1 | no | — |
| 345 | C22 | — | 12 | 3 | no | — |
| 346 | C22 | — | 12 | 6 | no | — |
| 347 | C22 | — | 12 | 12 | no | — |
| 348 | C22 | C2×C7×C22⋊A4 | 1 | 2 | yes | — |
| 349 | C4 | — | 3 | 2 | no | — |
| 350 | C4 | — | 3 | 4 | no | — |
| 351 | C4 | — | 3 | 3 | no | — |
| 352 | C4 | C2×C7×C22⋊A4 | 1 | 1 | yes | characteristic |
| 353 | C3 | — | 16 | 1 | no | Sylow |
| 354 | C2 | — | 2 | 2 | no | — |
| 355 | C2 | — | 6 | 3 | no | — |
| 356 | C2 | — | 6 | 2 | no | — |
| 357 | C2 | — | 6 | 6 | no | — |
| 358 | C2 | C22×C7×C22⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3×C7 |
| Frattini | C2 | C22×C7×C22⋊A4 |
| Fitting | C7×H(7,𝔽2) | C3 |
| hypercenter | C7×D4 | C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C7 ◁ C7×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#5950 | C2 | semidirect |
| 2. C2×C7×#997 | C2 | semidirect |
| 3. C7×H(7,𝔽2) | C3 | semidirect |
| 4. C7×#997 | C22 | semidirect |
| 10. C7×#585 | C2×C3 | semidirect |
| 14. C2×C7×H(5,𝔽2) | C2×C3 | semidirect |
| 15. #87547 | C7 | direct (N × H) |
| 24. C7×2−1+4 | A4 | semidirect |
| 54. C22×C7×D4 | A4 | semidirect |
| 55. #5950 | C2×C7 | semidirect |
| 56. C2×#997 | C2×C7 | semidirect |
| 61. H(7,𝔽2) | C3×C7 | semidirect |
| 64. C24×C7 | C2×A4 | semidirect |
| 83. C7×#41 | C2×A4 | semidirect |
| 89. C2×C7×Q8 | C2×A4 | semidirect |
| 118. C22×C4×C7 | C2×A4 | semidirect |
| 119. #997 | C22×C7 | semidirect |
| 128. #585 | C2×C3×C7 | semidirect |
| 132. C2×H(5,𝔽2) | C2×C3×C7 | semidirect |
| 181. 2−1+4 | C7×A4 | semidirect |
| 211. C22×D4 | C7×A4 | semidirect |
| 234. C24 | C2×C7×A4 | semidirect |
| 253. #41 | C2×C7×A4 | semidirect |
| 259. C2×Q8 | C2×C7×A4 | semidirect |
| 288. C22×C4 | C2×C7×A4 | semidirect |
| 335. C7 | #87547 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#5950 | 2 |
| C2×C7×#997 | 2 |
| C7×H(7,𝔽2) | 3 |
| C7×2−1+4⋊C3 | 4 |
| C7×D4×A4 | 4 |
| #87547 | 7 |