Finite Group DatabasebetaFinite Group DatabasebetaOrder 56448.
1298 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 1298 conjugacy classes of subgroups — too many to draw. Only its 48 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 | D3×#11924306 | — | 8 | 2 | no | maximal |
| 2 | D3×#90632⋊C3 | — | 7 | 3 | no | maximal |
| 3 | D3×#13062401 | — | 7 | 3 | no | maximal |
| 4 | C2×D3×C22⋊AΓL(1,𝔽8) | — | 7 | 2 | no | maximal |
| 5 | D3×C22⋊AΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 6 | D3×C22⋊AΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 7 | D3×AGL(1,𝔽8)2 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | D3×C7×AGL(1,𝔽8) | — | 8 | 2 | no | — |
| 9 | D3×#90632 | ΓL(1,𝔽8) | 1 | 3 | yes | complemented |
| 10 | D3×#90631 | ΓL(1,𝔽8) | 1 | 3 | yes | complemented |
| 11 | C23×D3×AGL(1,𝔽8) | ΓL(1,𝔽8) | 1 | 2 | yes | complemented |
| 12 | C3×#72214179 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | C3×#11924306 | — | 8 | 2 | no | — |
| 14 | C3×#90632⋊C3 | — | 7 | 3 | no | — |
| 15 | C3×#13062401 | — | 7 | 3 | no | — |
| 16 | C2×C3×C22⋊AΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 17 | C3×C22⋊AΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 18 | C3×AGL(1,𝔽8)2 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×D3×AΓL(1,𝔽8) | — | 28 | 2 | no | — |
| 20 | C2×#72214179 | — | 3 | 1 | no | maximal |
| 21 | C2×#11924306 | — | 24 | 2 | no | — |
| 22 | C22×C22⋊AΓL(1,𝔽8) | — | 21 | 2 | no | — |
| 23 | C2×AGL(1,𝔽8)2 | — | 3 | 1 | no | Hall |
| 24 | #72214179 | — | 2 | 1 | no | — |
| 25 | #72214179 | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 26 | AGL(1,𝔽8)2 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 27 | C2×#90632⋊C3 | — | 21 | 3 | no | — |
| 28 | C2×#13062401 | — | 21 | 3 | no | — |
| 29 | C3×#90632 | C2×ΓL(1,𝔽8) | 1 | 3 | yes | complemented |
| 30 | C3×#90631 | C2×ΓL(1,𝔽8) | 1 | 3 | yes | complemented |
| 31 | C23×C3×AGL(1,𝔽8) | C2×ΓL(1,𝔽8) | 1 | 2 | yes | complemented |
| 32 | C22×D3×AGL(1,𝔽8) | — | 7 | 2 | no | — |
| 33 | C22×D3×AGL(1,𝔽8) | — | 7 | 2 | no | — |
| 34 | #90632⋊C3 | — | 7 | 3 | no | — |
| 35 | #90632⋊C3 | — | 14 | 3 | no | — |
| 36 | #13062401 | — | 7 | 3 | no | — |
| 37 | #13062401 | — | 14 | 3 | no | — |
| 38 | C2×C22⋊AΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 39 | C2×C22⋊AΓL(1,𝔽8) | — | 14 | 2 | no | — |
| 40 | C2×C22⋊AΓL(1,𝔽8) | — | 21 | 2 | no | — |
| 41 | C2×C22⋊AΓL(1,𝔽8) | — | 21 | 2 | no | — |
| 42 | C3×C7×AGL(1,𝔽8) | — | 8 | 2 | no | — |
| 43 | #11924306 | — | 8 | 2 | no | — |
| 44 | #11924306 | — | 16 | 2 | no | — |
| 45 | C22×D3×C22⋊A4 | — | 49 | 1 | no | Hall |
| 46 | C2×D3×C22⋊ΓL(1,𝔽8) | — | 56 | 2 | no | — |
| 47 | C2×C3×AΓL(1,𝔽8) | — | 28 | 2 | no | — |
| 48 | D3×AΓL(1,𝔽8) | — | 28 | 2 | no | — |
| 49 | D3×AΓL(1,𝔽8) | — | 28 | 2 | no | — |
| 50 | D3×AΓL(1,𝔽8) | — | 56 | 6 | no | — |
| 51 | D3×AΓL(1,𝔽8) | — | 56 | 6 | no | — |
| 52 | D3×AΓL(1,𝔽8) | — | 56 | 3 | no | — |
| 53 | C2×#90632 | — | 3 | 3 | no | — |
| 54 | C2×#90631 | — | 3 | 3 | no | — |
| 55 | C24×AGL(1,𝔽8) | — | 3 | 2 | no | — |
| 56 | D3×#3861 | — | 64 | 1 | no | — |
| 57 | C2×C7×AGL(1,𝔽8) | — | 24 | 2 | no | — |
| 58 | C22×C3×AGL(1,𝔽8) | — | 7 | 2 | no | — |
| 59 | C2×D3×AGL(1,𝔽8) | — | 7 | 2 | no | — |
| 60 | C2×D3×AGL(1,𝔽8) | — | 21 | 2 | no | — |
| 61 | C22⋊AΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 62 | C22⋊AΓL(1,𝔽8) | — | 14 | 2 | no | — |
| 63 | C22×AΓL(1,𝔽8) | — | 84 | 2 | no | — |
| 64 | C22×C3×C22⋊A4 | — | 49 | 1 | no | — |
| 65 | C2×D3×C22⋊A4 | — | 49 | 2 | no | — |
| 66 | C2×D3×C22⋊A4 | — | 49 | 2 | no | — |
| 67 | C2×D3×C22⋊A4 | — | 49 | 1 | no | — |
| 68 | C2×D3×C22⋊A4 | — | 49 | 1 | no | — |
| 69 | C2×C3×C22⋊ΓL(1,𝔽8) | — | 56 | 2 | no | — |
| 70 | D3×C22⋊ΓL(1,𝔽8) | — | 56 | 2 | no | — |
| 71 | D3×C22⋊ΓL(1,𝔽8) | — | 56 | 2 | no | — |
| 72 | C3×AΓL(1,𝔽8) | — | 28 | 2 | no | — |
| 73 | C3×AΓL(1,𝔽8) | — | 56 | 6 | no | — |
| 74 | C3×AΓL(1,𝔽8) | — | 56 | 6 | no | — |
| 75 | C3×AΓL(1,𝔽8) | — | 56 | 3 | no | — |
| 76 | #90632 | D3×ΓL(1,𝔽8) | 1 | 3 | yes | complemented |
| 77 | #90631 | D3×ΓL(1,𝔽8) | 1 | 3 | yes | complemented |
| 78 | C23×AGL(1,𝔽8) | — | 21 | 2 | no | — |
| 79 | C23×AGL(1,𝔽8) | — | 21 | 2 | no | — |
| 80 | C23×AGL(1,𝔽8) | D3×ΓL(1,𝔽8) | 1 | 2 | yes | complemented |
| 81 | C3×#3861 | — | 64 | 1 | no | Hall |
| 82 | C7×AGL(1,𝔽8) | — | 8 | 2 | no | — |
| 83 | C26×D3 | #3861 | 1 | 1 | yes | characteristic, complemented |
| 84 | C23×C22⋊A4 | — | 147 | 1 | no | — |
| 85 | C23×D3×C7 | — | 8 | 2 | no | — |
| 86 | C22×C22⋊ΓL(1,𝔽8) | — | 168 | 2 | no | — |
| 87 | C2×C3×AGL(1,𝔽8) | — | 7 | 2 | no | — |
| 88 | D3×AGL(1,𝔽8) | — | 7 | 2 | no | — |
| 89 | D3×AGL(1,𝔽8) | — | 8 | 6 | no | — |
| 90 | D3×AGL(1,𝔽8) | — | 8 | 6 | no | — |
| 91 | D3×AGL(1,𝔽8) | — | 56 | 3 | no | — |
| 92 | D3×AGL(1,𝔽8) | AΓL(1,𝔽8) | 1 | 2 | yes | complemented |
| 93 | C2×AΓL(1,𝔽8) | — | 28 | 2 | no | — |
| 94 | C2×AΓL(1,𝔽8) | — | 56 | 2 | no | — |
| 95 | C2×AΓL(1,𝔽8) | — | 84 | 2 | no | — |
| 96 | C2×AΓL(1,𝔽8) | — | 84 | 2 | no | — |
| 97 | C2×AΓL(1,𝔽8) | — | 168 | 6 | no | — |
| 98 | C2×AΓL(1,𝔽8) | — | 168 | 6 | no | — |
| 99 | C2×AΓL(1,𝔽8) | — | 168 | 3 | no | — |
| 100 | D3×C72 | — | 64 | 1 | no | — |
| 101 | C2×#3861 | — | 192 | 1 | no | — |
| 102 | C2×C3×C22⋊A4 | — | 49 | 2 | no | — |
| 103 | C2×C3×C22⋊A4 | — | 49 | 1 | no | — |
| 104 | D3×C22⋊A4 | — | 49 | 1 | no | — |
| 105 | D3×C22⋊A4 | — | 49 | 2 | no | — |
| 106 | D3×C22⋊A4 | — | 49 | 1 | no | — |
| 107 | C22×D3×A4 | — | 196 | 2 | no | — |
| 108 | C22×D3×A4 | — | 196 | 3 | no | — |
| 109 | C3×C22⋊ΓL(1,𝔽8) | — | 56 | 2 | no | — |
| 110 | C2×D3×ΓL(1,𝔽8) | — | 224 | 2 | no | — |
| 111 | C22×AGL(1,𝔽8) | — | 7 | 2 | no | — |
| 112 | C22×AGL(1,𝔽8) | — | 21 | 2 | no | — |
| 113 | C22×AGL(1,𝔽8) | — | 63 | 2 | no | — |
| 114 | C26×C3 | C2×#3861 | 1 | 1 | yes | characteristic, complemented |
| 115 | C25×D3 | — | 7 | 2 | no | — |
| 116 | C25×D3 | — | 7 | 2 | no | — |
| 117 | C25×D3 | — | 49 | 1 | no | — |
| 118 | C25×D3 | — | 49 | 1 | no | — |
| 119 | C22×C22⋊A4 | — | 49 | 1 | no | — |
| 120 | C22×C22⋊A4 | — | 98 | 1 | no | — |
| 121 | C22×C22⋊A4 | — | 147 | 2 | no | — |
| 122 | C22×C22⋊A4 | — | 147 | 2 | no | — |
| 123 | C22×C22⋊A4 | — | 147 | 1 | no | — |
| 124 | C22×C22⋊A4 | — | 147 | 1 | no | — |
| 125 | C23×C3×C7 | — | 8 | 2 | no | — |
| 126 | C22×D3×C7 | — | 56 | 2 | no | — |
| 127 | C22×D3×C7 | — | 56 | 2 | no | — |
| 128 | C2×C22⋊ΓL(1,𝔽8) | — | 56 | 2 | no | — |
| 129 | C2×C22⋊ΓL(1,𝔽8) | — | 112 | 2 | no | — |
| 130 | C2×C22⋊ΓL(1,𝔽8) | — | 168 | 2 | no | — |
| 131 | C2×C22⋊ΓL(1,𝔽8) | — | 168 | 2 | no | — |
| 132 | C3×AGL(1,𝔽8) | — | 8 | 6 | no | — |
| 133 | C3×AGL(1,𝔽8) | — | 8 | 6 | no | — |
| 134 | C3×AGL(1,𝔽8) | — | 56 | 3 | no | — |
| 135 | C3×AGL(1,𝔽8) | C2×AΓL(1,𝔽8) | 1 | 2 | yes | complemented |
| 136 | AΓL(1,𝔽8) | — | 28 | 2 | no | — |
| 137 | AΓL(1,𝔽8) | — | 56 | 6 | no | — |
| 138 | AΓL(1,𝔽8) | — | 56 | 6 | no | — |
| 139 | AΓL(1,𝔽8) | — | 56 | 3 | no | — |
| 140 | AΓL(1,𝔽8) | — | 56 | 2 | no | — |
| 141 | AΓL(1,𝔽8) | — | 112 | 6 | no | — |
| 142 | AΓL(1,𝔽8) | — | 112 | 6 | no | — |
| 143 | AΓL(1,𝔽8) | — | 112 | 3 | no | — |
| 144 | C3×C72 | — | 64 | 1 | no | — |
| 145 | #3861 | — | 64 | 1 | no | — |
| 146 | #3861 | — | 128 | 1 | no | — |
| 147 | C3×C22⋊A4 | — | 49 | 1 | no | — |
| 148 | C22×C3×A4 | — | 196 | 2 | no | — |
| 149 | C22×C3×A4 | — | 196 | 3 | no | — |
| 150 | C2×D3×A4 | — | 196 | 2 | no | — |
| 151 | C2×D3×A4 | — | 196 | 3 | no | — |
| 152 | C2×D3×A4 | — | 196 | 2 | no | — |
| 153 | C2×D3×A4 | — | 196 | 2 | no | — |
| 154 | C2×D3×A4 | — | 196 | 2 | no | — |
| 155 | C2×D3×A4 | — | 196 | 2 | no | — |
| 156 | C2×D3×A4 | — | 196 | 3 | no | — |
| 157 | C2×D3×A4 | — | 196 | 6 | no | — |
| 158 | C2×D3×A4 | — | 196 | 6 | no | — |
| 159 | C2×D3×A4 | — | 196 | 2 | no | — |
| 160 | C27 | — | 3 | 1 | no | Sylow |
| 161 | C2×C3×ΓL(1,𝔽8) | — | 224 | 2 | no | — |
| 162 | D3×ΓL(1,𝔽8) | — | 448 | 3 | no | — |
| 163 | D3×ΓL(1,𝔽8) | — | 448 | 3 | no | — |
| 164 | D3×ΓL(1,𝔽8) | — | 224 | 2 | no | — |
| 165 | D3×ΓL(1,𝔽8) | — | 224 | 2 | no | — |
| 166 | C24×C7 | — | 24 | 2 | no | — |
| 167 | C2×AGL(1,𝔽8) | — | 3 | 2 | no | — |
| 168 | C2×AGL(1,𝔽8) | — | 7 | 2 | no | — |
| 169 | C2×AGL(1,𝔽8) | — | 21 | 2 | no | — |
| 170 | C2×AGL(1,𝔽8) | — | 24 | 6 | no | — |
| 171 | C2×AGL(1,𝔽8) | — | 24 | 6 | no | — |
| 172 | C2×AGL(1,𝔽8) | — | 168 | 3 | no | — |
| 173 | C2×C72 | — | 192 | 1 | no | — |
| 174 | C25×C3 | — | 7 | 2 | no | — |
| 175 | C25×C3 | — | 49 | 1 | no | — |
| 176 | C24×D3 | — | 7 | 2 | no | — |
| 177 | C24×D3 | — | 21 | 2 | no | — |
| 178 | C24×D3 | — | 49 | 3 | no | — |
| 179 | C24×D3 | — | 49 | 1 | no | — |
| 180 | C24×D3 | — | 49 | 2 | no | — |
| 181 | C24×D3 | — | 49 | 1 | no | — |
| 182 | C24×D3 | — | 147 | 1 | no | — |
| 183 | C24×D3 | — | 147 | 2 | no | — |
| 184 | C24×D3 | — | 147 | 2 | no | — |
| 185 | C24×D3 | — | 147 | 3 | no | — |
| 186 | C24×D3 | — | 147 | 2 | no | — |
| 187 | C24×D3 | — | 147 | 3 | no | — |
| 188 | C24×D3 | — | 147 | 2 | no | — |
| 189 | C2×C22⋊A4 | — | 49 | 2 | no | — |
| 190 | C2×C22⋊A4 | — | 49 | 1 | no | — |
| 191 | C2×C22⋊A4 | — | 98 | 2 | no | — |
| 192 | C2×C22⋊A4 | — | 98 | 1 | no | — |
| 193 | C2×C22⋊A4 | — | 147 | 1 | no | — |
| 194 | C2×C22⋊A4 | — | 147 | 2 | no | — |
| 195 | C2×C22⋊A4 | — | 147 | 1 | no | — |
| 196 | C23×A4 | — | 588 | 2 | no | — |
| 197 | C23×A4 | — | 588 | 3 | no | — |
| 198 | C22×C3×C7 | — | 56 | 2 | no | — |
| 199 | C2×D3×C7 | — | 56 | 2 | no | — |
| 200 | C2×D3×C7 | — | 168 | 2 | no | — |
| 201 | C22⋊ΓL(1,𝔽8) | — | 56 | 2 | no | — |
| 202 | C22⋊ΓL(1,𝔽8) | — | 112 | 2 | no | — |
| 203 | C22×ΓL(1,𝔽8) | — | 672 | 2 | no | — |
| 204 | C22×C3×D3 | — | 784 | 1 | no | — |
| 205 | C2×C3×A4 | — | 196 | 3 | no | — |
| 206 | C2×C3×A4 | — | 196 | 2 | no | — |
| 207 | C2×C3×A4 | — | 196 | 6 | no | — |
| 208 | C2×C3×A4 | — | 196 | 2 | no | — |
| 209 | C2×C3×A4 | — | 196 | 2 | no | — |
| 210 | D3×A4 | — | 196 | 2 | no | — |
| 211 | D3×A4 | — | 196 | 2 | no | — |
| 212 | D3×A4 | — | 196 | 2 | no | — |
| 213 | D3×A4 | — | 196 | 6 | no | — |
| 214 | D3×A4 | — | 196 | 2 | no | — |
| 215 | D3×A4 | — | 196 | 3 | no | — |
| 216 | D3×A4 | — | 196 | 3 | no | — |
| 217 | C26 | — | 21 | 2 | no | — |
| 218 | C26 | — | 21 | 2 | no | — |
| 219 | C26 | — | 147 | 1 | no | — |
| 220 | C26 | — | 147 | 1 | no | — |
| 221 | C26 | D3×#3861 | 1 | 1 | yes | characteristic, complemented |
| 222 | C3×ΓL(1,𝔽8) | — | 448 | 3 | no | — |
| 223 | C3×ΓL(1,𝔽8) | — | 448 | 3 | no | — |
| 224 | C3×ΓL(1,𝔽8) | — | 224 | 2 | no | — |
| 225 | C23×C7 | — | 8 | 2 | no | — |
| 226 | C23×C7 | — | 168 | 2 | no | — |
| 227 | C23×C7 | — | 168 | 2 | no | — |
| 228 | AGL(1,𝔽8) | — | 8 | 6 | no | — |
| 229 | AGL(1,𝔽8) | — | 8 | 6 | no | — |
| 230 | AGL(1,𝔽8) | — | 56 | 3 | no | — |
| 231 | AGL(1,𝔽8) | D3×AΓL(1,𝔽8) | 1 | 2 | yes | complemented |
| 232 | C72 | — | 64 | 1 | no | Sylow |
| 233 | C24×C3 | — | 7 | 2 | no | — |
| 234 | C24×C3 | — | 49 | 3 | no | — |
| 235 | C24×C3 | — | 49 | 1 | no | — |
| 236 | C24×C3 | — | 147 | 1 | no | — |
| 237 | C24×C3 | — | 147 | 2 | no | — |
| 238 | C23×D3 | — | 7 | 3 | no | — |
| 239 | C23×D3 | — | 7 | 2 | no | — |
| 240 | C23×D3 | — | 49 | 2 | no | — |
| 241 | C23×D3 | — | 49 | 2 | no | — |
| 242 | C23×D3 | — | 49 | 2 | no | — |
| 243 | C23×D3 | — | 49 | 2 | no | — |
| 244 | C23×D3 | — | 49 | 3 | no | — |
| 245 | C23×D3 | — | 49 | 6 | no | — |
| 246 | C23×D3 | — | 49 | 6 | no | — |
| 247 | C23×D3 | — | 147 | 2 | no | — |
| 248 | C23×D3 | — | 147 | 2 | no | — |
| 249 | C23×D3 | — | 147 | 1 | no | — |
| 250 | C23×D3 | — | 147 | 3 | no | — |
| 251 | C23×D3 | — | 147 | 3 | no | — |
| 252 | C23×D3 | — | 147 | 2 | no | — |
| 253 | C23×D3 | — | 147 | 2 | no | — |
| 254 | C23×D3 | — | 147 | 6 | no | — |
| 255 | C23×D3 | — | 147 | 3 | no | — |
| 256 | C23×D3 | — | 147 | 6 | no | — |
| 257 | C23×D3 | — | 147 | 3 | no | — |
| 258 | C23×D3 | — | 147 | 6 | no | — |
| 259 | C23×D3 | — | 147 | 2 | no | — |
| 260 | C23×D3 | — | 147 | 2 | no | — |
| 261 | C23×D3 | — | 147 | 6 | no | — |
| 262 | C23×D3 | — | 147 | 6 | no | — |
| 263 | C23×D3 | — | 147 | 1 | no | — |
| 264 | C23×D3 | — | 147 | 3 | no | — |
| 265 | C23×D3 | — | 147 | 6 | no | — |
| 266 | C23×D3 | — | 147 | 3 | no | — |
| 267 | C23×D3 | #11924306 | 1 | 2 | yes | complemented |
| 268 | C22⋊A4 | — | 49 | 1 | no | — |
| 269 | C22⋊A4 | — | 98 | 1 | no | — |
| 270 | C22×A4 | — | 588 | 2 | no | — |
| 271 | C22×A4 | — | 588 | 2 | no | — |
| 272 | C22×A4 | — | 588 | 3 | no | — |
| 273 | C22×A4 | — | 588 | 2 | no | — |
| 274 | C22×A4 | — | 588 | 6 | no | — |
| 275 | C22×A4 | — | 588 | 3 | no | — |
| 276 | C22×A4 | — | 588 | 2 | no | — |
| 277 | C22×A4 | — | 588 | 2 | no | — |
| 278 | C22×A4 | — | 588 | 2 | no | — |
| 279 | C22×A4 | — | 588 | 6 | no | — |
| 280 | C22×A4 | — | 392 | 2 | no | — |
| 281 | C22×A4 | — | 392 | 3 | no | — |
| 282 | C22×A4 | — | 196 | 2 | no | — |
| 283 | C22×A4 | — | 196 | 3 | no | — |
| 284 | C2×C3×C7 | — | 56 | 2 | no | — |
| 285 | D3×C7 | — | 8 | 2 | no | — |
| 286 | D3×C7 | — | 56 | 2 | no | — |
| 287 | D3×C7 | — | 64 | 3 | no | — |
| 288 | D3×C7 | — | 64 | 3 | no | — |
| 289 | C2×ΓL(1,𝔽8) | — | 1344 | 3 | no | — |
| 290 | C2×ΓL(1,𝔽8) | — | 1344 | 3 | no | — |
| 291 | C2×ΓL(1,𝔽8) | — | 672 | 2 | no | — |
| 292 | C2×ΓL(1,𝔽8) | — | 672 | 2 | no | — |
| 293 | C2×ΓL(1,𝔽8) | — | 448 | 2 | no | — |
| 294 | C2×ΓL(1,𝔽8) | — | 224 | 2 | no | — |
| 295 | C22×C32 | — | 784 | 1 | no | — |
| 296 | C2×C3×D3 | — | 784 | 2 | no | — |
| 297 | C2×C3×D3 | — | 784 | 1 | no | — |
| 298 | C2×C3×D3 | — | 784 | 2 | no | — |
| 299 | C2×C3×D3 | — | 784 | 1 | no | — |
| 300 | C3×A4 | — | 196 | 2 | no | — |
| 301 | C3×A4 | — | 196 | 3 | no | — |
| 302 | C25 | — | 7 | 2 | no | — |
| 303 | C25 | — | 21 | 2 | no | — |
| 304 | C25 | — | 49 | 1 | no | — |
| 305 | C25 | — | 63 | 2 | no | — |
| 306 | C25 | — | 147 | 3 | no | — |
| 307 | C25 | — | 147 | 1 | no | — |
| 308 | C25 | — | 147 | 2 | no | — |
| 309 | C25 | — | 147 | 1 | no | — |
| 310 | C25 | — | 441 | 3 | no | — |
| 311 | C25 | — | 441 | 1 | no | — |
| 312 | C25 | — | 441 | 2 | no | — |
| 313 | C25 | — | 441 | 2 | no | — |
| 314 | C25 | — | 441 | 2 | no | — |
| 315 | C25 | — | 441 | 3 | no | — |
| 316 | C25 | — | 441 | 2 | no | — |
| 317 | C22×C7 | — | 56 | 2 | no | — |
| 318 | C22×C7 | — | 168 | 2 | no | — |
| 319 | C22×C7 | — | 504 | 2 | no | — |
| 320 | C23×C3 | — | 7 | 3 | no | — |
| 321 | C23×C3 | — | 2352 | 1 | no | — |
| 322 | C23×C3 | — | 49 | 2 | no | — |
| 323 | C23×C3 | — | 49 | 2 | no | — |
| 324 | C23×C3 | — | 49 | 6 | no | — |
| 325 | C23×C3 | — | 147 | 1 | no | — |
| 326 | C23×C3 | — | 147 | 2 | no | — |
| 327 | C23×C3 | — | 147 | 3 | no | — |
| 328 | C23×C3 | C2×#11924306 | 1 | 2 | yes | complemented |
| 329 | C22×D3 | — | 7 | 2 | no | — |
| 330 | C22×D3 | — | 7 | 2 | no | — |
| 331 | C22×D3 | — | 49 | 1 | no | — |
| 332 | C22×D3 | — | 49 | 2 | no | — |
| 333 | C22×D3 | — | 49 | 3 | no | — |
| 334 | C22×D3 | — | 49 | 6 | no | — |
| 335 | C22×D3 | — | 49 | 2 | no | — |
| 336 | C22×D3 | — | 49 | 3 | no | — |
| 337 | C22×D3 | — | 147 | 2 | no | — |
| 338 | C22×D3 | — | 147 | 2 | no | — |
| 339 | C22×D3 | — | 147 | 2 | no | — |
| 340 | C22×D3 | — | 147 | 2 | no | — |
| 341 | C22×D3 | — | 147 | 2 | no | — |
| 342 | C22×D3 | — | 147 | 3 | no | — |
| 343 | C22×D3 | — | 147 | 3 | no | — |
| 344 | C22×D3 | — | 147 | 6 | no | — |
| 345 | C22×D3 | — | 147 | 6 | no | — |
| 346 | C22×D3 | — | 147 | 1 | no | — |
| 347 | C22×D3 | — | 147 | 1 | no | — |
| 348 | C22×D3 | — | 147 | 6 | no | — |
| 349 | C22×D3 | — | 147 | 3 | no | — |
| 350 | C22×D3 | — | 147 | 6 | no | — |
| 351 | C22×D3 | — | 147 | 6 | no | — |
| 352 | C22×D3 | — | 147 | 6 | no | — |
| 353 | C22×D3 | — | 147 | 3 | no | — |
| 354 | C22×D3 | — | 147 | 2 | no | — |
| 355 | C22×D3 | — | 147 | 3 | no | — |
| 356 | C2×A4 | — | 588 | 2 | no | — |
| 357 | C2×A4 | — | 588 | 2 | no | — |
| 358 | C2×A4 | — | 588 | 3 | no | — |
| 359 | C2×A4 | — | 588 | 2 | no | — |
| 360 | C2×A4 | — | 588 | 6 | no | — |
| 361 | C2×A4 | — | 588 | 2 | no | — |
| 362 | C2×A4 | — | 588 | 3 | no | — |
| 363 | C2×A4 | — | 392 | 3 | no | — |
| 364 | C2×A4 | — | 392 | 2 | no | — |
| 365 | C2×A4 | — | 392 | 6 | no | — |
| 366 | C2×A4 | — | 392 | 2 | no | — |
| 367 | C2×A4 | — | 392 | 2 | no | — |
| 368 | C2×A4 | — | 196 | 3 | no | — |
| 369 | C2×A4 | — | 196 | 2 | no | — |
| 370 | C2×A4 | — | 196 | 6 | no | — |
| 371 | C2×A4 | — | 196 | 2 | no | — |
| 372 | C2×A4 | — | 196 | 2 | no | — |
| 373 | C3×C7 | — | 8 | 2 | no | — |
| 374 | C3×C7 | — | 64 | 3 | no | — |
| 375 | C3×C7 | — | 64 | 3 | no | — |
| 376 | ΓL(1,𝔽8) | — | 896 | 3 | no | — |
| 377 | ΓL(1,𝔽8) | — | 896 | 3 | no | — |
| 378 | ΓL(1,𝔽8) | — | 448 | 3 | no | — |
| 379 | ΓL(1,𝔽8) | — | 448 | 3 | no | — |
| 380 | ΓL(1,𝔽8) | — | 448 | 2 | no | — |
| 381 | ΓL(1,𝔽8) | — | 224 | 2 | no | — |
| 382 | C2×C32 | — | 784 | 2 | no | — |
| 383 | C2×C32 | — | 784 | 1 | no | — |
| 384 | C3×D3 | — | 784 | 2 | no | — |
| 385 | C3×D3 | — | 784 | 1 | no | — |
| 386 | C3×D3 | — | 784 | 1 | no | — |
| 387 | C24 | — | 3 | 2 | no | — |
| 388 | C24 | — | 7 | 2 | no | — |
| 389 | C24 | — | 21 | 3 | no | — |
| 390 | C24 | — | 21 | 2 | no | — |
| 391 | C24 | — | 49 | 3 | no | — |
| 392 | C24 | — | 49 | 1 | no | — |
| 393 | C24 | — | 147 | 3 | no | — |
| 394 | C24 | — | 147 | 2 | no | — |
| 395 | C24 | — | 147 | 2 | no | — |
| 396 | C24 | — | 147 | 2 | no | — |
| 397 | C24 | — | 147 | 2 | no | — |
| 398 | C24 | — | 147 | 1 | no | — |
| 399 | C24 | — | 147 | 2 | no | — |
| 400 | C24 | — | 147 | 6 | no | — |
| 401 | C24 | — | 147 | 6 | no | — |
| 402 | C24 | — | 441 | 2 | no | — |
| 403 | C24 | — | 441 | 2 | no | — |
| 404 | C24 | — | 441 | 2 | no | — |
| 405 | C24 | — | 441 | 2 | no | — |
| 406 | C24 | — | 441 | 6 | no | — |
| 407 | C24 | — | 441 | 6 | no | — |
| 408 | C24 | — | 441 | 1 | no | — |
| 409 | C24 | — | 441 | 2 | no | — |
| 410 | C24 | — | 441 | 2 | no | — |
| 411 | C24 | — | 441 | 3 | no | — |
| 412 | C24 | — | 441 | 3 | no | — |
| 413 | C24 | — | 441 | 3 | no | — |
| 414 | C24 | — | 441 | 3 | no | — |
| 415 | C24 | — | 441 | 6 | no | — |
| 416 | C24 | — | 441 | 6 | no | — |
| 417 | C24 | — | 441 | 6 | no | — |
| 418 | C24 | — | 441 | 3 | no | — |
| 419 | C24 | — | 441 | 6 | no | — |
| 420 | C24 | — | 441 | 3 | no | — |
| 421 | C24 | — | 441 | 1 | no | — |
| 422 | C2×C7 | — | 24 | 2 | no | — |
| 423 | C2×C7 | — | 56 | 2 | no | — |
| 424 | C2×C7 | — | 168 | 2 | no | — |
| 425 | C2×C7 | — | 192 | 3 | no | — |
| 426 | C2×C7 | — | 192 | 3 | no | — |
| 427 | C22×C3 | — | 7 | 2 | no | — |
| 428 | C22×C3 | — | 784 | 1 | no | — |
| 429 | C22×C3 | — | 1568 | 1 | no | — |
| 430 | C22×C3 | — | 2352 | 2 | no | — |
| 431 | C22×C3 | — | 2352 | 1 | no | — |
| 432 | C22×C3 | — | 2352 | 2 | no | — |
| 433 | C22×C3 | — | 2352 | 1 | no | — |
| 434 | C22×C3 | — | 49 | 1 | no | — |
| 435 | C22×C3 | — | 49 | 3 | no | — |
| 436 | C22×C3 | — | 147 | 2 | no | — |
| 437 | C22×C3 | — | 147 | 1 | no | — |
| 438 | C2×D3 | — | 7 | 2 | no | — |
| 439 | C2×D3 | — | 21 | 2 | no | — |
| 440 | C2×D3 | — | 49 | 1 | no | — |
| 441 | C2×D3 | — | 49 | 2 | no | — |
| 442 | C2×D3 | — | 49 | 1 | no | — |
| 443 | C2×D3 | — | 147 | 2 | no | — |
| 444 | C2×D3 | — | 147 | 2 | no | — |
| 445 | C2×D3 | — | 147 | 2 | no | — |
| 446 | C2×D3 | — | 147 | 3 | no | — |
| 447 | C2×D3 | — | 147 | 3 | no | — |
| 448 | A4 | — | 392 | 2 | no | — |
| 449 | A4 | — | 392 | 3 | no | — |
| 450 | A4 | — | 196 | 2 | no | — |
| 451 | A4 | — | 196 | 3 | no | — |
| 452 | C32 | — | 784 | 1 | no | Sylow |
| 453 | C23 | — | 7 | 3 | no | — |
| 454 | C23 | — | 21 | 2 | no | — |
| 455 | C23 | — | 21 | 2 | no | — |
| 456 | C23 | — | 49 | 2 | no | — |
| 457 | C23 | — | 49 | 2 | no | — |
| 458 | C23 | — | 49 | 6 | no | — |
| 459 | C23 | — | 147 | 2 | no | — |
| 460 | C23 | — | 147 | 2 | no | — |
| 461 | C23 | — | 147 | 1 | no | — |
| 462 | C23 | — | 147 | 1 | no | — |
| 463 | C23 | — | 147 | 2 | no | — |
| 464 | C23 | — | 147 | 3 | no | — |
| 465 | C23 | — | 147 | 3 | no | — |
| 466 | C23 | — | 147 | 3 | no | — |
| 467 | C23 | — | 147 | 6 | no | — |
| 468 | C23 | — | 441 | 2 | no | — |
| 469 | C23 | — | 441 | 2 | no | — |
| 470 | C23 | — | 441 | 3 | no | — |
| 471 | C23 | — | 441 | 3 | no | — |
| 472 | C23 | — | 441 | 6 | no | — |
| 473 | C23 | — | 441 | 3 | no | — |
| 474 | C23 | — | 441 | 2 | no | — |
| 475 | C23 | — | 441 | 2 | no | — |
| 476 | C23 | — | 441 | 2 | no | — |
| 477 | C23 | — | 441 | 1 | no | — |
| 478 | C23 | — | 441 | 6 | no | — |
| 479 | C23 | — | 441 | 6 | no | — |
| 480 | C23 | — | 441 | 3 | no | — |
| 481 | C23 | — | 441 | 2 | no | — |
| 482 | C23 | — | 441 | 3 | no | — |
| 483 | C23 | — | 441 | 6 | no | — |
| 484 | C23 | — | 441 | 6 | no | — |
| 485 | C23 | — | 441 | 6 | no | — |
| 486 | C23 | — | 441 | 1 | no | — |
| 487 | C23 | D3×#11924306 | 1 | 2 | yes | complemented |
| 488 | C7 | — | 8 | 2 | no | — |
| 489 | C7 | — | 64 | 3 | no | — |
| 490 | C7 | — | 64 | 3 | no | — |
| 491 | C2×C3 | — | 7 | 2 | no | — |
| 492 | C2×C3 | — | 784 | 2 | no | — |
| 493 | C2×C3 | — | 784 | 1 | no | — |
| 494 | C2×C3 | — | 1568 | 2 | no | — |
| 495 | C2×C3 | — | 1568 | 1 | no | — |
| 496 | C2×C3 | — | 2352 | 1 | no | — |
| 497 | C2×C3 | — | 2352 | 2 | no | — |
| 498 | C2×C3 | — | 2352 | 1 | no | — |
| 499 | C2×C3 | — | 49 | 1 | no | — |
| 500 | D3 | — | 7 | 2 | no | — |
| 501 | D3 | — | 49 | 1 | no | — |
| 502 | D3 | #72214179 | 1 | 1 | yes | characteristic, direct product factor |
| 503 | C22 | — | 7 | 2 | no | — |
| 504 | C22 | — | 21 | 2 | no | — |
| 505 | C22 | — | 49 | 1 | no | — |
| 506 | C22 | — | 49 | 3 | no | — |
| 507 | C22 | — | 63 | 2 | no | — |
| 508 | C22 | — | 147 | 1 | no | — |
| 509 | C22 | — | 147 | 2 | no | — |
| 510 | C22 | — | 147 | 2 | no | — |
| 511 | C22 | — | 147 | 1 | no | — |
| 512 | C22 | — | 147 | 1 | no | — |
| 513 | C22 | — | 441 | 2 | no | — |
| 514 | C22 | — | 441 | 2 | no | — |
| 515 | C22 | — | 441 | 2 | no | — |
| 516 | C22 | — | 441 | 3 | no | — |
| 517 | C22 | — | 441 | 3 | no | — |
| 518 | C3 | — | 784 | 1 | no | — |
| 519 | C3 | — | 1568 | 1 | no | — |
| 520 | C3 | C2×#72214179 | 1 | 1 | yes | characteristic, complemented |
| 521 | C2 | — | 3 | 1 | no | — |
| 522 | C2 | — | 7 | 2 | no | — |
| 523 | C2 | — | 21 | 2 | no | — |
| 524 | C2 | — | 49 | 1 | no | — |
| 525 | C2 | — | 147 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×AGL(1,𝔽8)2 | C2×C3 |
| Fitting, socle | C26×C3 | C2×#3861 |
| LCS | G ▷ C3×AGL(1,𝔽8)2 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. D3×AGL(1,𝔽8)2 | C3 | semidirect |
| 9. D3×#90632 | ΓL(1,𝔽8) | semidirect |
| 10. D3×#90631 | ΓL(1,𝔽8) | semidirect |
| 11. C23×D3×AGL(1,𝔽8) | ΓL(1,𝔽8) | semidirect |
| 12. C3×#72214179 | C2 | semidirect |
| 18. C3×AGL(1,𝔽8)2 | C2×C3 | semidirect |
| 25. #72214179 | D3 | direct (N × H) |
| 26. AGL(1,𝔽8)2 | C3×D3 | semidirect |
| 29. C3×#90632 | C2×ΓL(1,𝔽8) | semidirect |
| 30. C3×#90631 | C2×ΓL(1,𝔽8) | semidirect |
| 31. C23×C3×AGL(1,𝔽8) | C2×ΓL(1,𝔽8) | semidirect |
| 76. #90632 | D3×ΓL(1,𝔽8) | semidirect |
| 77. #90631 | D3×ΓL(1,𝔽8) | semidirect |
| 80. C23×AGL(1,𝔽8) | D3×ΓL(1,𝔽8) | semidirect |
| 83. C26×D3 | #3861 | semidirect |
| 92. D3×AGL(1,𝔽8) | AΓL(1,𝔽8) | semidirect |
| 114. C26×C3 | C2×#3861 | semidirect |
| 135. C3×AGL(1,𝔽8) | C2×AΓL(1,𝔽8) | semidirect |
| 221. C26 | D3×#3861 | semidirect |
| 231. AGL(1,𝔽8) | D3×AΓL(1,𝔽8) | semidirect |
| 267. C23×D3 | #11924306 | semidirect |
| 328. C23×C3 | C2×#11924306 | semidirect |
| 487. C23 | D3×#11924306 | semidirect |
| 502. D3 | #72214179 | direct (N × H) |
| 520. C3 | C2×#72214179 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×#11924306 | 8 |
| D3×#90632⋊C3 | 7 |
| D3×#13062401 | 7 |
| C2×D3×C22⋊AΓL(1,𝔽8) | 7 |
| D3×AGL(1,𝔽8)2 | 3 |
| C3×#72214179 | 2 |
| C2×#72214179 | 3 |