Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
437 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 53 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 | 2 | yes | maximal, complemented |
| 2 | C2×A4×C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #73035 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C2×#2215 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C2×C4⋊S4 | — | 2 | 2 | no | — |
| 6 | C2×C8×A4 | C22 | 1 | 1 | yes | characteristic |
| 7 | #5402 | — | 2 | 2 | no | — |
| 8 | #5402 | C22 | 1 | 4 | yes | complemented |
| 9 | C2×D48 | — | 4 | 1 | no | maximal |
| 10 | A4×C16 | C22 | 1 | 2 | yes | complemented |
| 11 | C22×D16 | — | 3 | 1 | no | — |
| 12 | C23×C16 | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×#466 | — | 3 | 2 | no | — |
| 14 | #2215 | — | 3 | 4 | no | — |
| 15 | #2215 | — | 3 | 4 | no | — |
| 16 | C2×#366 | — | 3 | 1 | no | — |
| 17 | C2×#357 | — | 3 | 2 | no | — |
| 18 | C22×S4 | — | 4 | 2 | no | — |
| 19 | C2×C4×A4 | D4 | 1 | 1 | yes | characteristic |
| 20 | C4⋊S4 | — | 2 | 4 | no | — |
| 21 | C4⋊S4 | — | 4 | 2 | no | — |
| 22 | C2×D24 | — | 4 | 2 | no | — |
| 23 | C8×A4 | D4 | 1 | 1 | yes | characteristic |
| 24 | C8×A4 | C23 | 1 | 1 | yes | characteristic |
| 25 | C2×C3×C16 | — | 4 | 1 | no | — |
| 26 | D48 | — | 4 | 4 | no | — |
| 27 | C22×D8 | — | 3 | 2 | no | — |
| 28 | C23×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | C2×#121 | — | 6 | 2 | no | — |
| 30 | C2×D16 | — | 3 | 4 | no | — |
| 31 | C2×D16 | — | 6 | 1 | no | — |
| 32 | C2×D16 | — | 6 | 2 | no | — |
| 33 | C2×D16 | — | 6 | 1 | no | — |
| 34 | C22×C16 | — | 3 | 1 | no | — |
| 35 | C22×C16 | — | 3 | 1 | no | — |
| 36 | C22×C16 | — | 3 | 2 | no | — |
| 37 | C22×C16 | C2×D3 | 1 | 2 | yes | complemented |
| 38 | #466 | — | 3 | 4 | no | — |
| 39 | #466 | — | 3 | 4 | no | — |
| 40 | #466 | — | 6 | 2 | no | — |
| 41 | #466 | — | 6 | 2 | no | — |
| 42 | C2×#107 | — | 3 | 2 | no | — |
| 43 | C2×#102 | — | 6 | 2 | no | — |
| 44 | #366 | — | 3 | 4 | no | — |
| 45 | #357 | — | 3 | 8 | no | — |
| 46 | C22×A4 | D8 | 1 | 1 | yes | characteristic |
| 47 | C2×S4 | — | 4 | 4 | no | — |
| 48 | C2×S4 | — | 8 | 4 | no | — |
| 49 | C2×D12 | — | 8 | 2 | no | — |
| 50 | C4×A4 | D8 | 1 | 1 | yes | characteristic |
| 51 | C4×A4 | C2×D4 | 1 | 1 | yes | characteristic |
| 52 | C2×C3×C8 | — | 4 | 1 | no | — |
| 53 | D24 | — | 4 | 4 | no | — |
| 54 | D24 | — | 8 | 2 | no | — |
| 55 | C3×C16 | — | 4 | 2 | no | — |
| 56 | C22×D4 | — | 6 | 2 | no | — |
| 57 | C22×D4 | — | 12 | 2 | no | — |
| 58 | C23×C4 | D12 | 1 | 1 | yes | characteristic |
| 59 | C2×D8 | — | 3 | 4 | no | — |
| 60 | C2×D8 | — | 6 | 2 | no | — |
| 61 | C2×D8 | — | 6 | 2 | no | — |
| 62 | C2×D8 | — | 6 | 2 | no | — |
| 63 | C2×D8 | — | 6 | 2 | no | — |
| 64 | C2×D8 | — | 6 | 2 | no | — |
| 65 | C22×C8 | — | 3 | 1 | no | — |
| 66 | C22×C8 | — | 3 | 1 | no | — |
| 67 | C22×C8 | — | 3 | 1 | no | — |
| 68 | C22×C8 | — | 3 | 1 | no | — |
| 69 | C22×C8 | D12 | 1 | 1 | yes | characteristic |
| 70 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 71 | #121 | — | 6 | 4 | no | — |
| 72 | #121 | — | 6 | 4 | no | — |
| 73 | #121 | — | 12 | 2 | no | — |
| 74 | #121 | — | 12 | 2 | no | — |
| 75 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 76 | C2×C22⋊C4 | — | 12 | 2 | no | — |
| 77 | D16 | — | 6 | 4 | no | — |
| 78 | D16 | — | 6 | 4 | no | — |
| 79 | C2×C16 | — | 3 | 1 | no | — |
| 80 | C2×C16 | — | 3 | 2 | no | — |
| 81 | C2×C16 | — | 3 | 2 | no | — |
| 82 | C2×C16 | — | 3 | 2 | no | — |
| 83 | C2×C16 | — | 6 | 1 | no | — |
| 84 | C2×C16 | S4 | 1 | 1 | yes | characteristic, complemented |
| 85 | #107 | — | 3 | 4 | no | — |
| 86 | #107 | — | 6 | 2 | no | — |
| 87 | #102 | — | 6 | 4 | no | — |
| 88 | #102 | — | 6 | 4 | no | — |
| 89 | C22×D3 | — | 16 | 2 | no | — |
| 90 | C2×A4 | D16 | 1 | 2 | yes | complemented |
| 91 | C2×A4 | C2×D8 | 1 | 1 | yes | characteristic |
| 92 | S4 | — | 8 | 4 | no | — |
| 93 | C2×C3×C4 | — | 4 | 1 | no | — |
| 94 | D12 | — | 8 | 4 | no | — |
| 95 | D12 | — | 16 | 2 | no | — |
| 96 | C3×C8 | — | 4 | 1 | no | — |
| 97 | C3×C8 | — | 4 | 1 | no | — |
| 98 | C24 | — | 12 | 2 | no | — |
| 99 | C24 | D24 | 1 | 1 | yes | characteristic |
| 100 | C2×D4 | — | 6 | 4 | no | — |
| 101 | C2×D4 | — | 12 | 2 | no | — |
| 102 | C2×D4 | — | 12 | 2 | no | — |
| 103 | C2×D4 | — | 12 | 2 | no | — |
| 104 | C2×D4 | — | 12 | 2 | no | — |
| 105 | C2×D4 | — | 12 | 4 | no | — |
| 106 | C2×D4 | — | 12 | 4 | no | — |
| 107 | C2×D4 | — | 12 | 2 | no | — |
| 108 | C2×D4 | — | 24 | 4 | no | — |
| 109 | C2×D4 | — | 24 | 4 | no | — |
| 110 | C22×C4 | — | 3 | 1 | no | — |
| 111 | C22×C4 | — | 3 | 1 | no | — |
| 112 | C22×C4 | — | 3 | 1 | no | — |
| 113 | C22×C4 | — | 3 | 1 | no | — |
| 114 | C22×C4 | — | 12 | 2 | no | — |
| 115 | C22×C4 | D24 | 1 | 1 | yes | characteristic |
| 116 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 117 | D8 | — | 6 | 4 | no | — |
| 118 | D8 | — | 6 | 4 | no | — |
| 119 | D8 | — | 12 | 2 | no | — |
| 120 | D8 | — | 12 | 2 | no | — |
| 121 | C2×C8 | — | 3 | 1 | no | — |
| 122 | C2×C8 | — | 3 | 1 | no | — |
| 123 | C2×C8 | — | 3 | 1 | no | — |
| 124 | C2×C8 | — | 3 | 1 | no | — |
| 125 | C2×C8 | — | 3 | 1 | no | — |
| 126 | C2×C8 | — | 3 | 1 | no | — |
| 127 | C2×C8 | — | 3 | 1 | no | — |
| 128 | C2×C8 | — | 6 | 1 | no | — |
| 129 | C2×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 130 | C4⋊C4 | — | 6 | 4 | no | — |
| 131 | C4⋊C4 | — | 12 | 2 | no | — |
| 132 | C22⋊C4 | — | 12 | 4 | no | — |
| 133 | C22⋊C4 | — | 12 | 4 | no | — |
| 134 | C16 | — | 3 | 2 | no | — |
| 135 | C16 | C2×S4 | 1 | 2 | yes | complemented |
| 136 | C22×C3 | — | 4 | 1 | no | — |
| 137 | C2×D3 | — | 16 | 4 | no | — |
| 138 | C2×D3 | — | 32 | 4 | no | — |
| 139 | A4 | C2×D16 | 1 | 1 | yes | characteristic, complemented |
| 140 | C3×C4 | — | 4 | 1 | no | — |
| 141 | C3×C4 | — | 4 | 1 | no | — |
| 142 | C23 | — | 3 | 1 | no | — |
| 143 | C23 | — | 3 | 1 | no | — |
| 144 | C23 | — | 3 | 2 | no | — |
| 145 | C23 | — | 12 | 4 | no | — |
| 146 | C23 | — | 24 | 2 | no | — |
| 147 | C23 | — | 24 | 4 | no | — |
| 148 | C23 | — | 24 | 4 | no | — |
| 149 | C23 | — | 24 | 2 | no | — |
| 150 | C23 | D48 | 1 | 2 | yes | complemented |
| 151 | C23 | C2×D24 | 1 | 1 | yes | characteristic |
| 152 | D4 | — | 12 | 4 | no | — |
| 153 | D4 | — | 12 | 4 | no | — |
| 154 | D4 | — | 24 | 2 | no | — |
| 155 | D4 | — | 24 | 8 | no | — |
| 156 | D4 | — | 24 | 4 | no | — |
| 157 | D4 | — | 24 | 4 | no | — |
| 158 | D4 | — | 24 | 2 | no | — |
| 159 | C2×C4 | — | 3 | 1 | no | — |
| 160 | C2×C4 | — | 3 | 1 | no | — |
| 161 | C2×C4 | — | 3 | 1 | no | — |
| 162 | C2×C4 | — | 3 | 1 | no | — |
| 163 | C2×C4 | — | 3 | 1 | no | — |
| 164 | C2×C4 | — | 3 | 1 | no | — |
| 165 | C2×C4 | — | 3 | 1 | no | — |
| 166 | C2×C4 | — | 6 | 1 | no | — |
| 167 | C2×C4 | — | 12 | 4 | no | — |
| 168 | C2×C4 | — | 24 | 4 | no | — |
| 169 | C2×C4 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 170 | C8 | — | 3 | 1 | no | — |
| 171 | C8 | — | 3 | 1 | no | — |
| 172 | C8 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 173 | C8 | C22×S4 | 1 | 1 | yes | characteristic |
| 174 | C2×C3 | — | 4 | 2 | no | — |
| 175 | C2×C3 | — | 4 | 1 | no | — |
| 176 | D3 | — | 32 | 4 | no | — |
| 177 | C22 | — | 3 | 2 | no | — |
| 178 | C22 | — | 3 | 1 | no | — |
| 179 | C22 | — | 3 | 2 | no | — |
| 180 | C22 | — | 3 | 2 | no | — |
| 181 | C22 | — | 3 | 1 | no | — |
| 182 | C22 | — | 3 | 1 | no | — |
| 183 | C22 | — | 6 | 1 | no | — |
| 184 | C22 | — | 24 | 4 | no | — |
| 185 | C22 | — | 24 | 4 | no | — |
| 186 | C22 | — | 24 | 4 | no | — |
| 187 | C22 | — | 48 | 4 | no | — |
| 188 | C22 | — | 48 | 4 | no | — |
| 189 | C22 | C2×D48 | 1 | 1 | yes | characteristic, complemented |
| 190 | C22 | #5402 | 1 | 1 | yes | characteristic |
| 191 | C4 | — | 3 | 1 | no | — |
| 192 | C4 | — | 3 | 1 | no | — |
| 193 | C4 | — | 24 | 4 | no | — |
| 194 | C4 | #5402 | 1 | 1 | yes | characteristic |
| 195 | C4 | C2×C4⋊S4 | 1 | 1 | yes | characteristic |
| 196 | C3 | — | 4 | 1 | no | Sylow |
| 197 | C2 | — | 3 | 1 | no | — |
| 198 | C2 | — | 3 | 1 | no | — |
| 199 | C2 | — | 3 | 2 | no | — |
| 200 | C2 | — | 48 | 4 | no | — |
| 201 | C2 | #73035 | 1 | 2 | yes | direct product factor |
| 202 | C2 | C2×#5402 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#5402 | C2 | semidirect |
| 2. C2×A4×C16 | C2 | semidirect |
| 3. #73035 | C2 | direct (N × H) |
| 8. #5402 | C22 | semidirect |
| 10. A4×C16 | C22 | semidirect |
| 12. C23×C16 | D3 | semidirect |
| 37. C22×C16 | C2×D3 | semidirect |
| 84. C2×C16 | S4 | semidirect |
| 90. C2×A4 | D16 | semidirect |
| 135. C16 | C2×S4 | semidirect |
| 139. A4 | C2×D16 | semidirect |
| 150. C23 | D48 | semidirect |
| 189. C22 | C2×D48 | semidirect |
| 201. C2 | #73035 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5402 | 2 |
| C2×A4×C16 | 2 |
| #73035 | 2 |
| C2×#2215 | 3 |
| C2×D48 | 4 |