Finite Group DatabasebetaFinite Group DatabasebetaOrder 8064.
802 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 802 conjugacy classes of subgroups — too many to draw. Only its 28 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 | #72277107 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #72276697 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #72276513 | — | 3 | 1 | no | maximal |
| 4 | #72214654 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×S4×AGL(1,𝔽7) | — | 4 | 1 | no | maximal |
| 6 | ΓL(1,𝔽8)×#5935 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | ΓL(1,𝔽8)×#997 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C7⋊#5934 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C7×#5935 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #90408⋊C3 | — | 3 | 1 | no | — |
| 11 | ΓL(1,𝔽8)×#453 | — | 3 | 1 | no | — |
| 12 | #13052183 | — | 2 | 1 | no | — |
| 13 | #13052163 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #13052144 | — | 3 | 1 | no | — |
| 15 | #13052144 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | #89340⋊C3 | — | 3 | 1 | no | — |
| 17 | #89270⋊C3 | — | 3 | 1 | no | — |
| 18 | #89270⋊C3 | — | 3 | 1 | no | — |
| 19 | C3×#73026 | — | 7 | 1 | no | maximal, Hall |
| 20 | C2×A4×AGL(1,𝔽7) | — | 4 | 1 | no | — |
| 21 | C2×A4⋊AGL(1,𝔽7) | — | 4 | 1 | no | — |
| 22 | C2×A4⋊AGL(1,𝔽7) | — | 8 | 1 | no | — |
| 23 | C2×ΓL(1,𝔽8)×S4 | — | 4 | 1 | no | — |
| 24 | S4×AGL(1,𝔽7) | — | 4 | 4 | no | — |
| 25 | ΓL(1,𝔽8)×GL(2,𝔽3) | — | 8 | 1 | no | — |
| 26 | #11728738 | — | 3 | 1 | no | Hall |
| 27 | C2×D7×S4 | — | 4 | 1 | no | — |
| 28 | C7⋊#997 | — | 2 | 1 | no | — |
| 29 | C7×#997 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 30 | ΓL(1,𝔽8)×H(5,𝔽2) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 31 | C2×D4×AGL(1,𝔽7) | — | 6 | 1 | no | — |
| 32 | C2×D4×AGL(1,𝔽7) | — | 6 | 1 | no | — |
| 33 | ΓL(1,𝔽8)×AGL(1,ℤ/8) | — | 6 | 1 | no | — |
| 34 | ΓL(1,𝔽8)×#127 | — | 3 | 1 | no | — |
| 35 | #6501⋊C3 | — | 6 | 1 | no | — |
| 36 | #6501⋊C3 | — | 6 | 1 | no | — |
| 37 | #6500⋊C3 | — | 3 | 1 | no | — |
| 38 | #6500⋊C3 | — | 3 | 1 | no | — |
| 39 | #6500⋊C3 | — | 6 | 1 | no | — |
| 40 | #6500⋊C3 | — | 6 | 1 | no | — |
| 41 | C4×C4⋊AGL(1,𝔽7) | — | 6 | 1 | no | — |
| 42 | ΓL(1,𝔽8)×#104 | — | 6 | 1 | no | — |
| 43 | ΓL(1,𝔽8)×#100 | — | 3 | 1 | no | — |
| 44 | #6408⋊C3 | — | 3 | 1 | no | — |
| 45 | #6408⋊C3 | — | 3 | 1 | no | — |
| 46 | #6408⋊C3 | — | 6 | 1 | no | — |
| 47 | C3×#5935 | — | 7 | 1 | no | — |
| 48 | C3×#5934 | — | 7 | 1 | no | — |
| 49 | C3×B(4,𝔽2)⋊C3 | — | 7 | 1 | no | — |
| 50 | C2×A4×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 51 | C2×A4×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 52 | C2×D3×AGL(1,𝔽7) | — | 16 | 1 | no | — |
| 53 | A4×AGL(1,𝔽7) | — | 4 | 2 | no | — |
| 54 | A4⋊AGL(1,𝔽7) | — | 4 | 2 | no | — |
| 55 | A4⋊AGL(1,𝔽7) | — | 8 | 2 | no | — |
| 56 | ΓL(1,𝔽8)×S4 | — | 4 | 2 | no | — |
| 57 | ΓL(1,𝔽8)×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 58 | #90408 | — | 3 | 1 | no | — |
| 59 | C7×#453 | — | 3 | 1 | no | — |
| 60 | #90016 | — | 3 | 1 | no | — |
| 61 | #90016 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 62 | #89340 | — | 3 | 1 | no | — |
| 63 | #89270 | — | 3 | 1 | no | — |
| 64 | #89270 | — | 3 | 1 | no | — |
| 65 | #73026 | — | 7 | 1 | no | — |
| 66 | C3×#2196 | — | 21 | 1 | no | — |
| 67 | C2×D7⋊A4 | — | 8 | 1 | no | — |
| 68 | C2×A4×D7 | — | 4 | 1 | no | — |
| 69 | C23×AGL(1,𝔽7) | — | 3 | 1 | no | — |
| 70 | C23×AGL(1,𝔽7) | — | 6 | 1 | no | — |
| 71 | C23×AGL(1,𝔽7) | S4 | 1 | 1 | yes | characteristic, complemented |
| 72 | C2×C7⋊S4 | — | 4 | 1 | no | — |
| 73 | C2×C7⋊S4 | — | 8 | 1 | no | — |
| 74 | C2×C7×S4 | — | 4 | 1 | no | — |
| 75 | D7×S4 | — | 4 | 4 | no | — |
| 76 | #41×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 77 | C2×D4×ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 78 | C2×D4×ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 79 | C2×D4×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 80 | C2×D4×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 81 | C2×D4×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 82 | C2×(C22⋊D7)⋊C3 | — | 6 | 1 | no | — |
| 83 | C2×(C22⋊D7)⋊C3 | — | 6 | 1 | no | — |
| 84 | C2×(C22⋊D7)⋊C3 | — | 6 | 1 | no | — |
| 85 | C2×(C22⋊D7)⋊C3 | — | 6 | 1 | no | — |
| 86 | C2×(C22⋊D7)⋊C3 | — | 12 | 1 | no | — |
| 87 | D4×AGL(1,𝔽7) | — | 12 | 4 | no | — |
| 88 | D4×AGL(1,𝔽7) | — | 12 | 4 | no | — |
| 89 | C2×C4⋊AGL(1,𝔽7) | — | 6 | 1 | no | — |
| 90 | C2×C4×AGL(1,𝔽7) | — | 6 | 1 | no | — |
| 91 | C2×C4×AGL(1,𝔽7) | — | 6 | 1 | no | — |
| 92 | C7×GL(2,𝔽3) | — | 8 | 1 | no | — |
| 93 | SD16×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 94 | D8×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 95 | M16×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 96 | C42×ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 97 | (C22⋊Dic7)⋊C3 | — | 3 | 1 | no | — |
| 98 | (C22⋊Dic7)⋊C3 | — | 3 | 1 | no | — |
| 99 | (C22⋊Dic7)⋊C3 | — | 6 | 1 | no | — |
| 100 | (C22⋊Dic7)⋊C3 | — | 6 | 1 | no | — |
| 101 | (C2×AGL(1,𝔽7))⋊C4 | — | 6 | 1 | no | — |
| 102 | (C2×AGL(1,𝔽7))⋊C4 | — | 6 | 1 | no | — |
| 103 | (C2×AGL(1,𝔽7))⋊C4 | — | 12 | 1 | no | — |
| 104 | C4⋊(ΓL(1,𝔽8)⋊C4) | — | 6 | 1 | no | — |
| 105 | C22×C3×S4 | — | 28 | 1 | no | — |
| 106 | C3×#997 | — | 7 | 1 | no | — |
| 107 | C2×C3⋊AGL(1,𝔽7) | — | 16 | 1 | no | — |
| 108 | C2×D3×ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 109 | C2×C3×AGL(1,𝔽7) | — | 16 | 1 | no | — |
| 110 | A4×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 111 | A4×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 112 | D3×AGL(1,𝔽7) | — | 16 | 4 | no | — |
| 113 | C7×H(5,𝔽2) | C2×C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 114 | C2×D4×D7 | — | 6 | 1 | no | — |
| 115 | C2×D4×D7 | — | 6 | 1 | no | — |
| 116 | C7×AGL(1,ℤ/8) | — | 6 | 1 | no | — |
| 117 | C7×#127 | — | 3 | 1 | no | — |
| 118 | #6501 | — | 6 | 1 | no | — |
| 119 | #6501 | — | 6 | 1 | no | — |
| 120 | #6500 | — | 3 | 1 | no | — |
| 121 | #6500 | — | 3 | 1 | no | — |
| 122 | #6500 | — | 6 | 1 | no | — |
| 123 | #6500 | — | 6 | 1 | no | — |
| 124 | C4×D28 | — | 6 | 1 | no | — |
| 125 | C7×#104 | — | 6 | 1 | no | — |
| 126 | C7×#100 | — | 3 | 1 | no | — |
| 127 | #6408 | — | 3 | 1 | no | — |
| 128 | #6408 | — | 3 | 1 | no | — |
| 129 | #6408 | — | 6 | 1 | no | — |
| 130 | #5935 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 131 | #5934 | — | 7 | 1 | no | — |
| 132 | C3×D42 | — | 21 | 1 | no | — |
| 133 | C3×B(4,𝔽2) | — | 7 | 1 | no | — |
| 134 | C3×B(4,𝔽2) | — | 21 | 1 | no | — |
| 135 | C3×#453 | — | 21 | 1 | no | — |
| 136 | B(4,𝔽2)⋊C3 | — | 7 | 1 | no | — |
| 137 | B(4,𝔽2)⋊C3 | — | 14 | 1 | no | — |
| 138 | C3×#353 | — | 21 | 1 | no | — |
| 139 | C3×AΣL(1,𝔽16) | — | 21 | 1 | no | — |
| 140 | C3×AΣL(1,𝔽16) | — | 21 | 1 | no | — |
| 141 | C2×C22⋊ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 142 | C2×C22⋊ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 143 | C2×C7×A4 | — | 4 | 1 | no | — |
| 144 | C2×C7×A4 | — | 8 | 1 | no | — |
| 145 | C23×ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 146 | C23×ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 147 | C23×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 148 | C23×ΓL(1,𝔽8) | C2×S4 | 1 | 1 | yes | characteristic |
| 149 | C2×D3×D7 | — | 16 | 1 | no | — |
| 150 | D7⋊A4 | — | 8 | 2 | no | — |
| 151 | A4×D7 | — | 4 | 2 | no | — |
| 152 | C22×AGL(1,𝔽7) | — | 4 | 2 | no | — |
| 153 | C22×AGL(1,𝔽7) | — | 6 | 1 | no | — |
| 154 | C22×AGL(1,𝔽7) | — | 6 | 1 | no | — |
| 155 | C22×AGL(1,𝔽7) | — | 12 | 2 | no | — |
| 156 | C22×AGL(1,𝔽7) | — | 12 | 2 | no | — |
| 157 | C22×AGL(1,𝔽7) | — | 12 | 2 | no | — |
| 158 | C22×AGL(1,𝔽7) | — | 12 | 1 | no | — |
| 159 | C22×AGL(1,𝔽7) | — | 24 | 1 | no | — |
| 160 | C7⋊S4 | — | 4 | 2 | no | — |
| 161 | C7⋊S4 | — | 8 | 2 | no | — |
| 162 | C7×S4 | — | 4 | 2 | no | — |
| 163 | C7⋊SL(2,𝔽3) | — | 16 | 1 | no | — |
| 164 | C7×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 165 | Q8×ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 166 | D4×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 167 | D4×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 168 | D4×ΓL(1,𝔽8) | — | 12 | 2 | no | — |
| 169 | D4×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 170 | D4×ΓL(1,𝔽8) | — | 12 | 2 | no | — |
| 171 | C2×C4×ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 172 | C2×C4×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 173 | C2×C4×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 174 | (C22⋊D7)⋊C3 | — | 12 | 2 | no | — |
| 175 | (C22⋊D7)⋊C3 | — | 12 | 2 | no | — |
| 176 | (C22⋊D7)⋊C3 | — | 12 | 2 | no | — |
| 177 | (C22⋊D7)⋊C3 | — | 12 | 2 | no | — |
| 178 | (C22⋊D7)⋊C3 | — | 24 | 2 | no | — |
| 179 | C2×ΓL(1,𝔽8)⋊C4 | — | 6 | 1 | no | — |
| 180 | C2×ΓL(1,𝔽8)⋊C4 | — | 6 | 1 | no | — |
| 181 | C2×ΓL(1,𝔽8)⋊C4 | — | 12 | 1 | no | — |
| 182 | C4⋊AGL(1,𝔽7) | — | 6 | 4 | no | — |
| 183 | C4×AGL(1,𝔽7) | — | 12 | 2 | no | — |
| 184 | C4×AGL(1,𝔽7) | — | 12 | 2 | no | — |
| 185 | C8×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 186 | C22×C3×A4 | — | 28 | 1 | no | — |
| 187 | C2×C3×S4 | — | 28 | 1 | no | — |
| 188 | C2×C3×S4 | — | 28 | 1 | no | — |
| 189 | C2×C3×S4 | — | 28 | 4 | no | — |
| 190 | C2×C3×S4 | — | 56 | 1 | no | — |
| 191 | C3×GL(2,𝔽3) | — | 56 | 1 | no | — |
| 192 | #2196 | — | 21 | 1 | no | Sylow |
| 193 | C2×C3×ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 194 | C3⋊AGL(1,𝔽7) | — | 16 | 2 | no | — |
| 195 | D3×ΓL(1,𝔽8) | — | 16 | 2 | no | — |
| 196 | C3×AGL(1,𝔽7) | — | 16 | 2 | no | — |
| 197 | C23×D7 | — | 3 | 1 | no | — |
| 198 | C23×D7 | — | 6 | 1 | no | — |
| 199 | C23×D7 | C3×S4 | 1 | 1 | yes | characteristic, complemented |
| 200 | C7×#41 | — | 6 | 1 | no | — |
| 201 | C2×C7×D4 | — | 3 | 1 | no | — |
| 202 | C2×C7×D4 | — | 3 | 1 | no | — |
| 203 | C2×C7×D4 | — | 6 | 1 | no | — |
| 204 | C2×C7×D4 | — | 6 | 1 | no | — |
| 205 | C2×C7×D4 | — | 6 | 1 | no | — |
| 206 | C2×C22⋊D7 | — | 6 | 1 | no | — |
| 207 | C2×C22⋊D7 | — | 6 | 1 | no | — |
| 208 | C2×C22⋊D7 | — | 6 | 1 | no | — |
| 209 | C2×C22⋊D7 | — | 6 | 1 | no | — |
| 210 | C2×C22⋊D7 | — | 12 | 1 | no | — |
| 211 | D4×D7 | — | 12 | 4 | no | — |
| 212 | D4×D7 | — | 12 | 4 | no | — |
| 213 | C2×D28 | — | 6 | 1 | no | — |
| 214 | C2×C4×D7 | — | 6 | 1 | no | — |
| 215 | C2×C4×D7 | — | 6 | 1 | no | — |
| 216 | C7×SD16 | — | 12 | 1 | no | — |
| 217 | C7×D8 | — | 12 | 1 | no | — |
| 218 | C7×M16 | — | 6 | 1 | no | — |
| 219 | C42×C7 | — | 3 | 1 | no | — |
| 220 | C22⋊Dic7 | — | 3 | 1 | no | — |
| 221 | C22⋊Dic7 | — | 3 | 1 | no | — |
| 222 | C22⋊Dic7 | — | 6 | 1 | no | — |
| 223 | C22⋊Dic7 | — | 6 | 1 | no | — |
| 224 | (C2×D7)⋊C4 | — | 6 | 1 | no | — |
| 225 | (C2×D7)⋊C4 | — | 6 | 1 | no | — |
| 226 | (C2×D7)⋊C4 | — | 12 | 1 | no | — |
| 227 | C4⋊Dic7 | — | 6 | 1 | no | — |
| 228 | C22×S4 | — | 28 | 1 | no | — |
| 229 | C3×H(5,𝔽2) | — | 7 | 1 | no | — |
| 230 | C22×C3×D4 | — | 42 | 1 | no | — |
| 231 | C22×C3×D4 | — | 42 | 1 | no | — |
| 232 | #997 | — | 14 | 1 | no | — |
| 233 | #997 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 234 | C3×AGL(1,ℤ/8) | — | 42 | 1 | no | — |
| 235 | C3×#127 | — | 21 | 1 | no | — |
| 236 | C3×#121 | — | 42 | 1 | no | — |
| 237 | C3×#121 | — | 42 | 1 | no | — |
| 238 | C3×#120 | — | 21 | 1 | no | — |
| 239 | C3×#120 | — | 21 | 1 | no | — |
| 240 | C3×#120 | — | 42 | 1 | no | — |
| 241 | C3×#120 | — | 42 | 1 | no | — |
| 242 | C3×C4×D4 | — | 42 | 1 | no | — |
| 243 | C3×#104 | — | 42 | 1 | no | — |
| 244 | C3×#100 | — | 21 | 1 | no | — |
| 245 | C3×#99 | — | 21 | 1 | no | — |
| 246 | C3×#99 | — | 21 | 1 | no | — |
| 247 | C3×#99 | — | 42 | 1 | no | — |
| 248 | C2×D21 | — | 16 | 1 | no | — |
| 249 | C2×D3×C7 | — | 16 | 1 | no | — |
| 250 | C2×C3×D7 | — | 16 | 1 | no | — |
| 251 | C22⋊ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 252 | C22⋊ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 253 | C7×A4 | — | 4 | 1 | no | — |
| 254 | C7×A4 | — | 8 | 1 | no | — |
| 255 | C22×ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 256 | C22×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 257 | C22×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 258 | C22×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 259 | C22×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 260 | C22×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 261 | C22×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 262 | C22×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 263 | D3×D7 | — | 16 | 4 | no | — |
| 264 | C2×AGL(1,𝔽7) | — | 4 | 1 | no | — |
| 265 | C2×AGL(1,𝔽7) | — | 6 | 2 | no | — |
| 266 | C2×AGL(1,𝔽7) | — | 6 | 2 | no | — |
| 267 | C2×AGL(1,𝔽7) | — | 12 | 1 | no | — |
| 268 | C2×AGL(1,𝔽7) | — | 12 | 1 | no | — |
| 269 | C2×AGL(1,𝔽7) | — | 12 | 1 | no | — |
| 270 | C2×AGL(1,𝔽7) | — | 12 | 2 | no | — |
| 271 | C2×AGL(1,𝔽7) | — | 24 | 4 | no | — |
| 272 | C2×AGL(1,𝔽7) | — | 24 | 1 | no | — |
| 273 | C2×AGL(1,𝔽7) | — | 32 | 1 | no | — |
| 274 | C4×ΓL(1,𝔽8) | — | 6 | 2 | no | — |
| 275 | C4×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 276 | ΓL(1,𝔽8)⋊C4 | — | 12 | 1 | no | — |
| 277 | ΓL(1,𝔽8)⋊C4 | — | 12 | 1 | no | — |
| 278 | ΓL(1,𝔽8)⋊C4 | — | 24 | 1 | no | — |
| 279 | C22×C3×D3 | — | 112 | 1 | no | — |
| 280 | C2×C3×A4 | — | 28 | 1 | no | — |
| 281 | C2×C3×A4 | — | 28 | 2 | no | — |
| 282 | C2×C3×A4 | — | 56 | 1 | no | — |
| 283 | C3×S4 | — | 28 | 2 | no | — |
| 284 | C3×S4 | — | 28 | 2 | no | — |
| 285 | C3×S4 | — | 56 | 2 | no | — |
| 286 | C3×SL(2,𝔽3) | — | 56 | 1 | no | — |
| 287 | D42 | — | 21 | 1 | no | — |
| 288 | B(4,𝔽2) | — | 7 | 1 | no | — |
| 289 | B(4,𝔽2) | — | 21 | 1 | no | — |
| 290 | #453 | — | 3 | 1 | no | — |
| 291 | #353 | — | 21 | 1 | no | — |
| 292 | AΣL(1,𝔽16) | — | 21 | 1 | no | — |
| 293 | AΣL(1,𝔽16) | — | 21 | 1 | no | — |
| 294 | C3×ΓL(1,𝔽8) | — | 16 | 1 | no | Hall |
| 295 | C23×C7 | — | 2 | 1 | no | — |
| 296 | C23×C7 | — | 3 | 1 | no | — |
| 297 | C23×C7 | — | 6 | 1 | no | — |
| 298 | C23×C7 | C2×C3×S4 | 1 | 1 | yes | characteristic |
| 299 | C22×D7 | — | 4 | 2 | no | — |
| 300 | C22×D7 | — | 6 | 1 | no | — |
| 301 | C22×D7 | — | 6 | 1 | no | — |
| 302 | C22×D7 | — | 12 | 2 | no | — |
| 303 | C22×D7 | — | 12 | 2 | no | — |
| 304 | C22×D7 | — | 12 | 2 | no | — |
| 305 | C22×D7 | — | 12 | 1 | no | — |
| 306 | C22×D7 | — | 24 | 1 | no | — |
| 307 | C7×Q8 | — | 2 | 1 | no | — |
| 308 | C7×D4 | — | 6 | 1 | no | — |
| 309 | C7×D4 | — | 12 | 1 | no | — |
| 310 | C7×D4 | — | 12 | 2 | no | — |
| 311 | C7×D4 | — | 12 | 1 | no | — |
| 312 | C7×D4 | — | 12 | 2 | no | — |
| 313 | C2×C4×C7 | — | 3 | 1 | no | — |
| 314 | C2×C4×C7 | — | 6 | 1 | no | — |
| 315 | C2×C4×C7 | — | 6 | 1 | no | — |
| 316 | C22⋊D7 | — | 12 | 2 | no | — |
| 317 | C22⋊D7 | — | 12 | 2 | no | — |
| 318 | C22⋊D7 | — | 12 | 2 | no | — |
| 319 | C22⋊D7 | — | 12 | 2 | no | — |
| 320 | C22⋊D7 | — | 24 | 2 | no | — |
| 321 | C2×Dic7 | — | 6 | 1 | no | — |
| 322 | C2×Dic7 | — | 6 | 1 | no | — |
| 323 | C2×Dic7 | — | 12 | 1 | no | — |
| 324 | D28 | — | 6 | 4 | no | — |
| 325 | C4×D7 | — | 12 | 2 | no | — |
| 326 | C4×D7 | — | 12 | 2 | no | — |
| 327 | C7×C8 | — | 12 | 1 | no | — |
| 328 | C24×C3 | — | 7 | 1 | no | — |
| 329 | C24×C3 | — | 21 | 1 | no | — |
| 330 | C24×C3 | — | 42 | 1 | no | — |
| 331 | C22×A4 | — | 28 | 1 | no | — |
| 332 | C22×A4 | — | 56 | 1 | no | — |
| 333 | C2×S4 | — | 4 | 1 | no | — |
| 334 | C2×S4 | — | 28 | 1 | no | — |
| 335 | C2×S4 | — | 28 | 4 | no | — |
| 336 | C2×S4 | — | 56 | 1 | no | — |
| 337 | C3×#41 | — | 42 | 1 | no | — |
| 338 | C2×C3×D4 | — | 21 | 1 | no | — |
| 339 | C2×C3×D4 | — | 21 | 1 | no | — |
| 340 | C2×C3×D4 | — | 42 | 1 | no | — |
| 341 | C2×C3×D4 | — | 42 | 1 | no | — |
| 342 | C2×C3×D4 | — | 42 | 1 | no | — |
| 343 | C2×C3×D4 | — | 42 | 1 | no | — |
| 344 | C2×C3×D4 | — | 42 | 1 | no | — |
| 345 | C2×C3×D4 | — | 42 | 1 | no | — |
| 346 | C2×C3×D4 | — | 42 | 1 | no | — |
| 347 | C2×C3×D4 | — | 42 | 1 | no | — |
| 348 | C2×C3×D4 | — | 84 | 4 | no | — |
| 349 | C2×C3×D4 | — | 84 | 4 | no | — |
| 350 | C2×C3×D4 | — | 84 | 1 | no | — |
| 351 | C22×C3×C4 | — | 42 | 1 | no | — |
| 352 | C22×C3×C4 | — | 42 | 1 | no | — |
| 353 | GL(2,𝔽3) | — | 8 | 1 | no | — |
| 354 | C3×SD16 | — | 84 | 1 | no | — |
| 355 | C3×D8 | — | 84 | 1 | no | — |
| 356 | C3×M16 | — | 42 | 1 | no | — |
| 357 | C3×C4⋊C4 | — | 42 | 1 | no | — |
| 358 | C3×C22⋊C4 | — | 21 | 1 | no | — |
| 359 | C3×C22⋊C4 | — | 21 | 1 | no | — |
| 360 | C3×C22⋊C4 | — | 42 | 1 | no | — |
| 361 | C3×C22⋊C4 | — | 42 | 1 | no | — |
| 362 | C3×C22⋊C4 | — | 42 | 1 | no | — |
| 363 | C3×C22⋊C4 | — | 42 | 1 | no | — |
| 364 | C3×C22⋊C4 | — | 84 | 1 | no | — |
| 365 | C3×C42 | — | 21 | 1 | no | — |
| 366 | C2×C3×C7 | — | 16 | 1 | no | — |
| 367 | D21 | — | 16 | 2 | no | — |
| 368 | C3×D7 | — | 16 | 2 | no | — |
| 369 | D3×C7 | — | 16 | 2 | no | — |
| 370 | C2×ΓL(1,𝔽8) | — | 6 | 1 | no | — |
| 371 | C2×ΓL(1,𝔽8) | — | 12 | 1 | no | — |
| 372 | C2×ΓL(1,𝔽8) | — | 24 | 1 | no | — |
| 373 | C2×ΓL(1,𝔽8) | — | 32 | 1 | no | — |
| 374 | C2×ΓL(1,𝔽8) | #1020⋊C2 | 1 | 1 | yes | characteristic |
| 375 | AGL(1,𝔽7) | — | 4 | 2 | no | — |
| 376 | AGL(1,𝔽7) | — | 12 | 2 | no | — |
| 377 | AGL(1,𝔽7) | — | 32 | 2 | no | — |
| 378 | C22×C32 | — | 112 | 1 | no | — |
| 379 | C2×C3×D3 | — | 112 | 1 | no | — |
| 380 | C2×C3×D3 | — | 112 | 1 | no | — |
| 381 | C2×C3×D3 | — | 112 | 4 | no | — |
| 382 | C3×A4 | — | 28 | 1 | no | — |
| 383 | C3×A4 | — | 56 | 1 | no | — |
| 384 | H(5,𝔽2) | D3×AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 385 | C22×D4 | — | 42 | 1 | no | — |
| 386 | C22×D4 | — | 42 | 1 | no | — |
| 387 | AGL(1,ℤ/8) | — | 6 | 1 | no | — |
| 388 | #127 | — | 3 | 1 | no | — |
| 389 | #121 | — | 42 | 1 | no | — |
| 390 | #121 | — | 42 | 1 | no | — |
| 391 | #120 | — | 21 | 1 | no | — |
| 392 | #120 | — | 21 | 1 | no | — |
| 393 | #120 | — | 42 | 1 | no | — |
| 394 | #120 | — | 42 | 1 | no | — |
| 395 | C4×D4 | — | 42 | 1 | no | — |
| 396 | #104 | — | 6 | 1 | no | — |
| 397 | #100 | — | 3 | 1 | no | — |
| 398 | #99 | — | 21 | 1 | no | — |
| 399 | #99 | — | 21 | 1 | no | — |
| 400 | #99 | — | 42 | 1 | no | — |
| 401 | C22×C7 | — | 3 | 1 | no | — |
| 402 | C22×C7 | — | 4 | 1 | no | — |
| 403 | C22×C7 | — | 6 | 1 | no | — |
| 404 | C22×C7 | — | 8 | 1 | no | — |
| 405 | C22×C7 | — | 12 | 1 | no | — |
| 406 | C22×C7 | — | 12 | 1 | no | — |
| 407 | C22×C7 | — | 12 | 1 | no | — |
| 408 | C22×C7 | — | 12 | 1 | no | — |
| 409 | C2×D7 | — | 4 | 1 | no | — |
| 410 | C2×D7 | — | 6 | 2 | no | — |
| 411 | C2×D7 | — | 6 | 2 | no | — |
| 412 | C2×D7 | — | 12 | 1 | no | — |
| 413 | C2×D7 | — | 12 | 1 | no | — |
| 414 | C2×D7 | — | 12 | 1 | no | — |
| 415 | C2×D7 | — | 12 | 2 | no | — |
| 416 | C2×D7 | — | 24 | 4 | no | — |
| 417 | C2×D7 | — | 24 | 1 | no | — |
| 418 | C4×C7 | — | 6 | 2 | no | — |
| 419 | C4×C7 | — | 6 | 1 | no | — |
| 420 | Dic7 | — | 12 | 1 | no | — |
| 421 | Dic7 | — | 12 | 1 | no | — |
| 422 | Dic7 | — | 24 | 1 | no | — |
| 423 | C23×C3 | — | 7 | 1 | no | — |
| 424 | C23×C3 | — | 14 | 1 | no | — |
| 425 | C23×C3 | — | 21 | 1 | no | — |
| 426 | C23×C3 | — | 28 | 2 | no | — |
| 427 | C23×C3 | — | 42 | 1 | no | — |
| 428 | C23×C3 | — | 42 | 1 | no | — |
| 429 | C23×C3 | — | 42 | 1 | no | — |
| 430 | C23×C3 | — | 84 | 1 | no | — |
| 431 | C23×C3 | — | 84 | 2 | no | — |
| 432 | C23×C3 | — | 84 | 2 | no | — |
| 433 | C23×C3 | — | 84 | 2 | no | — |
| 434 | C23×C3 | — | 168 | 1 | no | — |
| 435 | C22×D3 | — | 112 | 1 | no | — |
| 436 | C2×A4 | — | 4 | 1 | no | — |
| 437 | C2×A4 | — | 8 | 1 | no | — |
| 438 | C2×A4 | — | 28 | 2 | no | — |
| 439 | C2×A4 | — | 56 | 2 | no | — |
| 440 | C2×A4 | — | 56 | 1 | no | — |
| 441 | C2×A4 | — | 112 | 1 | no | — |
| 442 | S4 | — | 4 | 2 | no | — |
| 443 | S4 | — | 28 | 2 | no | — |
| 444 | S4 | — | 56 | 2 | no | — |
| 445 | C3×Q8 | — | 14 | 1 | no | — |
| 446 | C3×D4 | — | 42 | 1 | no | — |
| 447 | C3×D4 | — | 42 | 4 | no | — |
| 448 | C3×D4 | — | 84 | 1 | no | — |
| 449 | C3×D4 | — | 84 | 2 | no | — |
| 450 | C3×D4 | — | 84 | 1 | no | — |
| 451 | C3×D4 | — | 84 | 2 | no | — |
| 452 | C3×D4 | — | 84 | 2 | no | — |
| 453 | C3×D4 | — | 84 | 2 | no | — |
| 454 | C3×D4 | — | 84 | 2 | no | — |
| 455 | C3×D4 | — | 84 | 2 | no | — |
| 456 | C3×D4 | — | 168 | 2 | no | — |
| 457 | C2×C3×C4 | — | 21 | 1 | no | — |
| 458 | C2×C3×C4 | — | 42 | 1 | no | — |
| 459 | C2×C3×C4 | — | 42 | 1 | no | — |
| 460 | C2×C3×C4 | — | 42 | 1 | no | — |
| 461 | C2×C3×C4 | — | 42 | 1 | no | — |
| 462 | C2×C3×C4 | — | 84 | 2 | no | — |
| 463 | C2×C3×C4 | — | 84 | 2 | no | — |
| 464 | C2×C3×C4 | — | 84 | 1 | no | — |
| 465 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 466 | SL(2,𝔽3) | — | 112 | 1 | no | — |
| 467 | C3×C8 | — | 84 | 1 | no | — |
| 468 | C3×C7 | — | 16 | 1 | no | — |
| 469 | ΓL(1,𝔽8) | — | 32 | 1 | no | — |
| 470 | ΓL(1,𝔽8) | #73026 | 1 | 1 | yes | characteristic, complemented |
| 471 | C2×C32 | — | 112 | 1 | no | — |
| 472 | C2×C32 | — | 112 | 2 | no | — |
| 473 | C3×D3 | — | 112 | 2 | no | — |
| 474 | C3×D3 | — | 112 | 2 | no | — |
| 475 | C24 | — | 7 | 1 | no | — |
| 476 | C24 | — | 21 | 1 | no | — |
| 477 | C24 | — | 42 | 1 | no | — |
| 478 | #41 | — | 6 | 1 | no | — |
| 479 | C2×D4 | — | 3 | 1 | no | — |
| 480 | C2×D4 | — | 3 | 1 | no | — |
| 481 | C2×D4 | — | 6 | 1 | no | — |
| 482 | C2×D4 | — | 6 | 1 | no | — |
| 483 | C2×D4 | — | 6 | 1 | no | — |
| 484 | C2×D4 | — | 42 | 1 | no | — |
| 485 | C2×D4 | — | 42 | 1 | no | — |
| 486 | C2×D4 | — | 42 | 1 | no | — |
| 487 | C2×D4 | — | 42 | 1 | no | — |
| 488 | C2×D4 | — | 42 | 1 | no | — |
| 489 | C2×D4 | — | 84 | 4 | no | — |
| 490 | C2×D4 | — | 84 | 4 | no | — |
| 491 | C2×D4 | — | 84 | 1 | no | — |
| 492 | C22×C4 | — | 42 | 1 | no | — |
| 493 | C22×C4 | — | 42 | 1 | no | — |
| 494 | SD16 | — | 12 | 1 | no | — |
| 495 | D8 | — | 12 | 1 | no | — |
| 496 | M16 | — | 6 | 1 | no | — |
| 497 | C4⋊C4 | — | 42 | 1 | no | — |
| 498 | C22⋊C4 | — | 21 | 1 | no | — |
| 499 | C22⋊C4 | — | 21 | 1 | no | — |
| 500 | C22⋊C4 | — | 42 | 1 | no | — |
| 501 | C22⋊C4 | — | 42 | 1 | no | — |
| 502 | C22⋊C4 | — | 42 | 1 | no | — |
| 503 | C22⋊C4 | — | 42 | 1 | no | — |
| 504 | C22⋊C4 | — | 84 | 1 | no | — |
| 505 | C42 | — | 3 | 1 | no | — |
| 506 | C2×C7 | — | 6 | 1 | no | — |
| 507 | C2×C7 | — | 12 | 1 | no | — |
| 508 | C2×C7 | — | 24 | 1 | no | — |
| 509 | C2×C7 | C3×#1020⋊C2 | 1 | 1 | yes | characteristic |
| 510 | D7 | — | 4 | 2 | no | — |
| 511 | D7 | — | 12 | 2 | no | — |
| 512 | C22×C3 | — | 21 | 1 | no | — |
| 513 | C22×C3 | — | 28 | 1 | no | — |
| 514 | C22×C3 | — | 28 | 1 | no | — |
| 515 | C22×C3 | — | 42 | 1 | no | — |
| 516 | C22×C3 | — | 42 | 2 | no | — |
| 517 | C22×C3 | — | 42 | 2 | no | — |
| 518 | C22×C3 | — | 56 | 1 | no | — |
| 519 | C22×C3 | — | 84 | 1 | no | — |
| 520 | C22×C3 | — | 84 | 1 | no | — |
| 521 | C22×C3 | — | 84 | 1 | no | — |
| 522 | C22×C3 | — | 84 | 1 | no | — |
| 523 | C22×C3 | — | 84 | 1 | no | — |
| 524 | C22×C3 | — | 84 | 1 | no | — |
| 525 | C22×C3 | — | 84 | 1 | no | — |
| 526 | C22×C3 | — | 84 | 2 | no | — |
| 527 | C22×C3 | — | 112 | 1 | no | — |
| 528 | C22×C3 | — | 168 | 4 | no | — |
| 529 | C22×C3 | — | 168 | 1 | no | — |
| 530 | C22×C3 | — | 224 | 1 | no | — |
| 531 | C2×D3 | — | 16 | 1 | no | — |
| 532 | C2×D3 | — | 112 | 1 | no | — |
| 533 | C2×D3 | — | 112 | 4 | no | — |
| 534 | A4 | — | 4 | 1 | no | — |
| 535 | A4 | — | 8 | 1 | no | — |
| 536 | A4 | — | 56 | 1 | no | — |
| 537 | A4 | — | 112 | 1 | no | — |
| 538 | C3×C4 | — | 42 | 1 | no | — |
| 539 | C3×C4 | — | 42 | 2 | no | — |
| 540 | C3×C4 | — | 84 | 1 | no | — |
| 541 | C3×C4 | — | 84 | 1 | no | — |
| 542 | C3×C4 | — | 168 | 1 | no | — |
| 543 | C32 | — | 112 | 1 | no | Sylow |
| 544 | C23 | — | 2 | 1 | no | — |
| 545 | C23 | — | 3 | 1 | no | — |
| 546 | C23 | — | 6 | 1 | no | — |
| 547 | C23 | — | 28 | 2 | no | — |
| 548 | C23 | — | 42 | 1 | no | — |
| 549 | C23 | — | 42 | 1 | no | — |
| 550 | C23 | — | 84 | 1 | no | — |
| 551 | C23 | — | 84 | 2 | no | — |
| 552 | C23 | — | 84 | 2 | no | — |
| 553 | C23 | — | 84 | 2 | no | — |
| 554 | C23 | — | 168 | 1 | no | — |
| 555 | C23 | S4×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 556 | Q8 | — | 2 | 1 | no | — |
| 557 | D4 | — | 6 | 1 | no | — |
| 558 | D4 | — | 12 | 1 | no | — |
| 559 | D4 | — | 12 | 2 | no | — |
| 560 | D4 | — | 12 | 1 | no | — |
| 561 | D4 | — | 12 | 2 | no | — |
| 562 | D4 | — | 42 | 4 | no | — |
| 563 | D4 | — | 84 | 2 | no | — |
| 564 | D4 | — | 84 | 2 | no | — |
| 565 | D4 | — | 84 | 2 | no | — |
| 566 | D4 | — | 84 | 2 | no | — |
| 567 | D4 | — | 168 | 2 | no | — |
| 568 | C2×C4 | — | 3 | 1 | no | — |
| 569 | C2×C4 | — | 6 | 1 | no | — |
| 570 | C2×C4 | — | 6 | 1 | no | — |
| 571 | C2×C4 | — | 42 | 1 | no | — |
| 572 | C2×C4 | — | 42 | 1 | no | — |
| 573 | C2×C4 | — | 84 | 2 | no | — |
| 574 | C2×C4 | — | 84 | 2 | no | — |
| 575 | C2×C4 | — | 84 | 1 | no | — |
| 576 | C8 | — | 12 | 1 | no | — |
| 577 | C7 | C3×#73026 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 578 | C2×C3 | — | 7 | 1 | no | — |
| 579 | C2×C3 | — | 16 | 1 | no | — |
| 580 | C2×C3 | — | 28 | 2 | no | — |
| 581 | C2×C3 | — | 42 | 1 | no | — |
| 582 | C2×C3 | — | 84 | 1 | no | — |
| 583 | C2×C3 | — | 84 | 2 | no | — |
| 584 | C2×C3 | — | 112 | 2 | no | — |
| 585 | C2×C3 | — | 168 | 1 | no | — |
| 586 | C2×C3 | — | 224 | 2 | no | — |
| 587 | C2×C3 | — | 224 | 1 | no | — |
| 588 | D3 | — | 16 | 2 | no | — |
| 589 | D3 | — | 112 | 2 | no | — |
| 590 | C22 | — | 3 | 1 | no | — |
| 591 | C22 | — | 4 | 1 | no | — |
| 592 | C22 | — | 6 | 1 | no | — |
| 593 | C22 | — | 8 | 1 | no | — |
| 594 | C22 | — | 12 | 1 | no | — |
| 595 | C22 | — | 12 | 1 | no | — |
| 596 | C22 | — | 12 | 1 | no | — |
| 597 | C22 | — | 12 | 1 | no | — |
| 598 | C22 | — | 28 | 1 | no | — |
| 599 | C22 | — | 42 | 2 | no | — |
| 600 | C22 | — | 42 | 2 | no | — |
| 601 | C22 | — | 84 | 1 | no | — |
| 602 | C22 | — | 84 | 1 | no | — |
| 603 | C22 | — | 84 | 2 | no | — |
| 604 | C22 | — | 84 | 1 | no | — |
| 605 | C22 | — | 168 | 1 | no | — |
| 606 | C22 | — | 168 | 4 | no | — |
| 607 | C4 | — | 6 | 2 | no | — |
| 608 | C4 | — | 6 | 1 | no | — |
| 609 | C4 | — | 84 | 1 | no | — |
| 610 | C4 | — | 84 | 1 | no | — |
| 611 | C4 | — | 168 | 1 | no | — |
| 612 | C3 | — | 7 | 1 | no | — |
| 613 | C3 | — | 16 | 1 | no | — |
| 614 | C3 | — | 224 | 1 | no | — |
| 615 | C2 | — | 6 | 1 | no | — |
| 616 | C2 | — | 12 | 1 | no | — |
| 617 | C2 | — | 24 | 1 | no | — |
| 618 | C2 | — | 28 | 2 | no | — |
| 619 | C2 | — | 84 | 2 | no | — |
| 620 | C2 | #72214629 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72214629 |
| commutator, cosocle, hypocenter | C7×#997 | C22×C3 |
| Frattini | C23 | S4×AGL(1,𝔽7) |
| Fitting | C7×H(5,𝔽2) | C2×C3×D3 |
| socle | C2×C7 | C3×#1020⋊C2 |
| LCS | G ▷ C7×#997 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72277107 | C3 | semidirect |
| 2. #72276697 | C2 | semidirect |
| 4. #72214654 | C2 | semidirect |
| 6. ΓL(1,𝔽8)×#5935 | C2 | semidirect |
| 7. ΓL(1,𝔽8)×#997 | C22 | semidirect |
| 8. C7⋊#5934 | C2×C3 | semidirect |
| 9. C7×#5935 | C2×C3 | semidirect |
| 13. #13052163 | C2×C3 | semidirect |
| 15. #13052144 | D3 | semidirect |
| 29. C7×#997 | C22×C3 | semidirect |
| 30. ΓL(1,𝔽8)×H(5,𝔽2) | C2×D3 | semidirect |
| 61. #90016 | C3×D3 | semidirect |
| 71. C23×AGL(1,𝔽7) | S4 | semidirect |
| 113. C7×H(5,𝔽2) | C2×C3×D3 | semidirect |
| 130. #5935 | AGL(1,𝔽7) | semidirect |
| 199. C23×D7 | C3×S4 | semidirect |
| 233. #997 | C2×AGL(1,𝔽7) | semidirect |
| 384. H(5,𝔽2) | D3×AGL(1,𝔽7) | semidirect |
| 470. ΓL(1,𝔽8) | #73026 | semidirect |
| 577. C7 | C3×#73026 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72277107 | 3 |
| #72276697 | 2 |
| #72276513 | 3 |
| #72214654 | 2 |
| C2×S4×AGL(1,𝔽7) | 4 |
| ΓL(1,𝔽8)×#5935 | 2 |
| C3×#73026 | 7 |