Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
272 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 35 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 | C2×#5402 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | A4×M32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #73105 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13067 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C2×C4⋊S4 | — | 2 | 1 | no | — |
| 6 | C2×C8×A4 | C22 | 1 | 1 | yes | characteristic |
| 7 | #5409 | — | 2 | 1 | no | — |
| 8 | #5402 | — | 2 | 1 | no | — |
| 9 | #5402 | C4 | 1 | 2 | yes | — |
| 10 | A4×C16 | — | 2 | 1 | no | — |
| 11 | #4516 | — | 4 | 1 | no | maximal |
| 12 | C22×M32 | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×#466 | — | 3 | 1 | no | — |
| 14 | C2×#361 | — | 3 | 1 | no | — |
| 15 | C2×#361 | — | 3 | 1 | no | — |
| 16 | #1810 | — | 3 | 1 | no | — |
| 17 | #1387 | — | 3 | 1 | no | — |
| 18 | #1387 | — | 3 | 1 | no | — |
| 19 | C22×S4 | — | 4 | 1 | no | — |
| 20 | C2×C4×A4 | D4 | 1 | 1 | yes | characteristic |
| 21 | C4⋊S4 | — | 2 | 2 | no | — |
| 22 | C4⋊S4 | — | 4 | 1 | no | — |
| 23 | C2×D24 | — | 4 | 1 | no | — |
| 24 | C8×A4 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C8×A4 | D4 | 1 | 1 | yes | characteristic |
| 26 | A4⋊C8 | — | 4 | 1 | no | — |
| 27 | C3×M32 | — | 4 | 1 | no | — |
| 28 | #819 | — | 4 | 1 | no | — |
| 29 | C22×D8 | — | 3 | 1 | no | — |
| 30 | C23×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 31 | C2×#121 | — | 6 | 1 | no | — |
| 32 | C2×M32 | — | 3 | 1 | no | — |
| 33 | C2×M32 | — | 3 | 1 | no | — |
| 34 | C2×M32 | — | 6 | 1 | no | — |
| 35 | C22×C16 | — | 2 | 1 | no | — |
| 36 | #466 | — | 3 | 2 | no | — |
| 37 | #466 | — | 3 | 2 | no | — |
| 38 | #466 | — | 6 | 1 | no | — |
| 39 | #466 | — | 6 | 1 | no | — |
| 40 | C2×#108 | — | 3 | 1 | no | — |
| 41 | C2×#108 | — | 3 | 1 | no | — |
| 42 | C2×#107 | — | 3 | 1 | no | — |
| 43 | C2×#102 | — | 6 | 1 | no | — |
| 44 | #407 | — | 6 | 1 | no | — |
| 45 | #361 | — | 6 | 2 | no | — |
| 46 | #361 | — | 6 | 1 | no | — |
| 47 | #361 | — | 6 | 1 | no | — |
| 48 | #341 | — | 6 | 1 | no | — |
| 49 | C22×A4 | SD16 | 1 | 1 | yes | characteristic |
| 50 | C2×S4 | — | 4 | 2 | no | — |
| 51 | C2×S4 | — | 8 | 2 | no | — |
| 52 | C2×D12 | — | 8 | 1 | no | — |
| 53 | C4×A4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 54 | C4×A4 | D8 | 1 | 1 | yes | characteristic |
| 55 | C2×C3×C8 | — | 4 | 1 | no | — |
| 56 | #208 | — | 8 | 1 | no | — |
| 57 | D24 | — | 4 | 2 | no | — |
| 58 | D24 | — | 8 | 1 | no | — |
| 59 | C3×C16 | — | 8 | 1 | no | — |
| 60 | C22×D4 | — | 6 | 1 | no | — |
| 61 | C22×D4 | — | 12 | 1 | no | — |
| 62 | C23×C4 | D12 | 1 | 1 | yes | characteristic |
| 63 | C2×D8 | — | 3 | 2 | no | — |
| 64 | C2×D8 | — | 6 | 1 | no | — |
| 65 | C2×D8 | — | 6 | 1 | no | — |
| 66 | C2×D8 | — | 6 | 1 | no | — |
| 67 | C2×D8 | — | 6 | 1 | no | — |
| 68 | C2×D8 | — | 6 | 1 | no | — |
| 69 | C2×M16 | — | 6 | 1 | no | — |
| 70 | C2×M16 | — | 6 | 1 | no | — |
| 71 | C22×C8 | — | 3 | 1 | no | — |
| 72 | C22×C8 | — | 3 | 1 | no | — |
| 73 | C22×C8 | — | 3 | 1 | no | — |
| 74 | C22×C8 | — | 3 | 1 | no | — |
| 75 | C22×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 76 | C22×C8 | #67 | 1 | 1 | yes | characteristic |
| 77 | #121 | — | 6 | 2 | no | — |
| 78 | #121 | — | 6 | 2 | no | — |
| 79 | #121 | — | 12 | 1 | no | — |
| 80 | #121 | — | 12 | 1 | no | — |
| 81 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 82 | C2×C22⋊C4 | — | 12 | 1 | no | — |
| 83 | M32 | — | 3 | 2 | no | — |
| 84 | M32 | — | 3 | 1 | no | — |
| 85 | M32 | — | 6 | 1 | no | — |
| 86 | M32 | S4 | 1 | 1 | yes | characteristic, complemented |
| 87 | C2×C16 | — | 6 | 1 | no | — |
| 88 | C2×C16 | — | 6 | 1 | no | — |
| 89 | #108 | — | 6 | 1 | no | — |
| 90 | #108 | — | 6 | 1 | no | — |
| 91 | #108 | — | 6 | 1 | no | — |
| 92 | #108 | — | 6 | 1 | no | — |
| 93 | #107 | — | 3 | 2 | no | — |
| 94 | #107 | — | 6 | 1 | no | — |
| 95 | #102 | — | 6 | 2 | no | — |
| 96 | #102 | — | 6 | 2 | no | — |
| 97 | C22⋊C8 | — | 12 | 1 | no | — |
| 98 | C22⋊C8 | — | 12 | 1 | no | — |
| 99 | C22×D3 | — | 16 | 1 | no | — |
| 100 | C2×A4 | — | 2 | 1 | no | — |
| 101 | C2×A4 | #102 | 1 | 1 | yes | characteristic |
| 102 | S4 | — | 8 | 2 | no | — |
| 103 | C2×C3×C4 | — | 4 | 1 | no | — |
| 104 | D12 | — | 8 | 2 | no | — |
| 105 | D12 | — | 16 | 1 | no | — |
| 106 | C3×C8 | — | 4 | 1 | no | — |
| 107 | C3×C8 | — | 4 | 1 | no | — |
| 108 | C3⋊C8 | — | 16 | 1 | no | — |
| 109 | C24 | — | 12 | 1 | no | — |
| 110 | C24 | #204 | 1 | 1 | yes | characteristic |
| 111 | C2×D4 | — | 6 | 2 | no | — |
| 112 | C2×D4 | — | 12 | 1 | no | — |
| 113 | C2×D4 | — | 12 | 1 | no | — |
| 114 | C2×D4 | — | 12 | 1 | no | — |
| 115 | C2×D4 | — | 12 | 1 | no | — |
| 116 | C2×D4 | — | 12 | 2 | no | — |
| 117 | C2×D4 | — | 12 | 2 | no | — |
| 118 | C2×D4 | — | 12 | 1 | no | — |
| 119 | C2×D4 | — | 24 | 2 | no | — |
| 120 | C2×D4 | — | 24 | 2 | no | — |
| 121 | C22×C4 | — | 3 | 1 | no | — |
| 122 | C22×C4 | — | 3 | 1 | no | — |
| 123 | C22×C4 | — | 3 | 1 | no | — |
| 124 | C22×C4 | — | 3 | 1 | no | — |
| 125 | C22×C4 | — | 12 | 1 | no | — |
| 126 | C22×C4 | D24 | 1 | 1 | yes | characteristic |
| 127 | C22×C4 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 128 | D8 | — | 6 | 2 | no | — |
| 129 | D8 | — | 6 | 2 | no | — |
| 130 | D8 | — | 12 | 1 | no | — |
| 131 | D8 | — | 12 | 1 | no | — |
| 132 | M16 | — | 12 | 1 | no | — |
| 133 | M16 | — | 12 | 1 | no | — |
| 134 | M16 | — | 12 | 1 | no | — |
| 135 | M16 | — | 12 | 1 | no | — |
| 136 | C2×C8 | — | 3 | 1 | no | — |
| 137 | C2×C8 | — | 3 | 1 | no | — |
| 138 | C2×C8 | — | 3 | 1 | no | — |
| 139 | C2×C8 | — | 3 | 1 | no | — |
| 140 | C2×C8 | — | 3 | 1 | no | — |
| 141 | C2×C8 | — | 3 | 1 | no | — |
| 142 | C2×C8 | — | 3 | 1 | no | — |
| 143 | C2×C8 | — | 6 | 1 | no | — |
| 144 | C2×C8 | — | 12 | 1 | no | — |
| 145 | C2×C8 | — | 12 | 1 | no | — |
| 146 | C2×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 147 | C4⋊C4 | — | 6 | 2 | no | — |
| 148 | C4⋊C4 | — | 12 | 1 | no | — |
| 149 | C22⋊C4 | — | 12 | 2 | no | — |
| 150 | C22⋊C4 | — | 12 | 2 | no | — |
| 151 | C16 | — | 2 | 1 | no | — |
| 152 | C16 | — | 6 | 1 | no | — |
| 153 | C22×C3 | — | 4 | 1 | no | — |
| 154 | C2×D3 | — | 16 | 2 | no | — |
| 155 | C2×D3 | — | 32 | 2 | no | — |
| 156 | A4 | #361 | 1 | 1 | yes | characteristic, complemented |
| 157 | C3×C4 | — | 4 | 1 | no | — |
| 158 | C3×C4 | — | 4 | 1 | no | — |
| 159 | C23 | — | 2 | 1 | no | — |
| 160 | C23 | — | 3 | 1 | no | — |
| 161 | C23 | — | 3 | 1 | no | — |
| 162 | C23 | — | 6 | 1 | no | — |
| 163 | C23 | — | 12 | 2 | no | — |
| 164 | C23 | — | 24 | 1 | no | — |
| 165 | C23 | — | 24 | 2 | no | — |
| 166 | C23 | — | 24 | 2 | no | — |
| 167 | C23 | — | 24 | 1 | no | — |
| 168 | C23 | #821 | 1 | 1 | yes | characteristic |
| 169 | D4 | — | 12 | 2 | no | — |
| 170 | D4 | — | 12 | 2 | no | — |
| 171 | D4 | — | 24 | 1 | no | — |
| 172 | D4 | — | 24 | 2 | no | — |
| 173 | D4 | — | 24 | 4 | no | — |
| 174 | D4 | — | 24 | 2 | no | — |
| 175 | D4 | — | 24 | 1 | no | — |
| 176 | C2×C4 | — | 3 | 1 | no | — |
| 177 | C2×C4 | — | 3 | 1 | no | — |
| 178 | C2×C4 | — | 3 | 1 | no | — |
| 179 | C2×C4 | — | 3 | 1 | no | — |
| 180 | C2×C4 | — | 3 | 1 | no | — |
| 181 | C2×C4 | — | 3 | 1 | no | — |
| 182 | C2×C4 | — | 3 | 1 | no | — |
| 183 | C2×C4 | — | 6 | 1 | no | — |
| 184 | C2×C4 | — | 12 | 2 | no | — |
| 185 | C2×C4 | — | 24 | 2 | no | — |
| 186 | C2×C4 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 187 | C8 | — | 3 | 1 | no | — |
| 188 | C8 | — | 3 | 1 | no | — |
| 189 | C8 | — | 24 | 1 | no | — |
| 190 | C8 | — | 24 | 1 | no | — |
| 191 | C8 | C4×S4 | 1 | 1 | yes | characteristic |
| 192 | C8 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 193 | C2×C3 | — | 4 | 1 | no | — |
| 194 | C2×C3 | — | 8 | 1 | no | — |
| 195 | D3 | — | 32 | 2 | no | — |
| 196 | C22 | — | 3 | 1 | no | — |
| 197 | C22 | — | 3 | 1 | no | — |
| 198 | C22 | — | 3 | 1 | no | — |
| 199 | C22 | — | 6 | 1 | no | — |
| 200 | C22 | — | 6 | 1 | no | — |
| 201 | C22 | — | 6 | 1 | no | — |
| 202 | C22 | — | 6 | 1 | no | — |
| 203 | C22 | — | 24 | 2 | no | — |
| 204 | C22 | — | 24 | 2 | no | — |
| 205 | C22 | — | 24 | 2 | no | — |
| 206 | C22 | — | 48 | 2 | no | — |
| 207 | C22 | — | 48 | 2 | no | — |
| 208 | C22 | #4516 | 1 | 1 | yes | characteristic, complemented |
| 209 | C22 | #5401 | 1 | 1 | yes | characteristic |
| 210 | C4 | — | 3 | 1 | no | — |
| 211 | C4 | — | 3 | 1 | no | — |
| 212 | C4 | — | 24 | 2 | no | — |
| 213 | C4 | #5402 | 1 | 1 | yes | characteristic |
| 214 | C4 | (C2×S4)⋊C4 | 1 | 1 | yes | characteristic |
| 215 | C3 | — | 4 | 1 | no | Sylow |
| 216 | C2 | — | 2 | 1 | no | — |
| 217 | C2 | — | 3 | 1 | no | — |
| 218 | C2 | — | 3 | 1 | no | — |
| 219 | C2 | — | 6 | 1 | no | — |
| 220 | C2 | — | 48 | 2 | no | — |
| 221 | C2 | #73113 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×M32 | C2 | semidirect |
| 3. #73105 | C2 | semidirect |
| 12. C22×M32 | D3 | semidirect |
| 86. M32 | S4 | semidirect |
| 156. A4 | #361 | semidirect |
| 208. C22 | #4516 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5402 | 2 |
| A4×M32 | 2 |
| #73105 | 2 |
| #13067 | 3 |
| #4516 | 4 |