Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1494 conjugacy classes of subgroups; only the 259 normal subgroups are shown.
This group has 1494 conjugacy classes of subgroups — too many to draw. Only its 259 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 | C22×C3×#131 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #84098 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#5115 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #71990 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #71971 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #32933 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C22×C3×#41 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×C3×#131 | — | 2 | 12 | no | — |
| 9 | C22×C3×M16 | C22 | 1 | 3 | yes | — |
| 10 | C23×C3×C8 | C22 | 1 | 3 | yes | — |
| 11 | C2×#930 | C22 | 1 | 6 | yes | — |
| 12 | C22×D24 | C22 | 1 | 1 | yes | characteristic |
| 13 | #5231 | — | 2 | 2 | no | — |
| 14 | #5223 | — | 2 | 6 | no | — |
| 15 | #5212 | C22 | 1 | 6 | yes | — |
| 16 | #5115 | — | 2 | 12 | no | — |
| 17 | C2×#821 | C22 | 1 | 6 | yes | — |
| 18 | C2×#818 | C22 | 1 | 3 | yes | — |
| 19 | #4551 | C22 | 1 | 6 | yes | — |
| 20 | C22×#131 | D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | C2×#466 | — | 3 | 3 | no | — |
| 22 | #2615 | — | 3 | 2 | no | — |
| 23 | #1895 | — | 3 | 6 | no | — |
| 24 | #1808 | — | 3 | 3 | no | — |
| 25 | C2×C3×#41 | — | 2 | 6 | no | — |
| 26 | C2×C3×#41 | — | 2 | 6 | no | — |
| 27 | C22×C3×Q8 | D4 | 1 | 1 | yes | characteristic |
| 28 | C22×C3×D4 | D4 | 1 | 3 | yes | — |
| 29 | C23×C3×C4 | C23 | 1 | 3 | yes | — |
| 30 | C22×#67 | — | 2 | 6 | no | — |
| 31 | C22×D12 | C23 | 1 | 2 | yes | — |
| 32 | C3×#131 | — | 2 | 32 | no | — |
| 33 | C2×C3×M16 | — | 2 | 6 | no | — |
| 34 | C2×C3×M16 | — | 2 | 12 | no | — |
| 35 | C22×C3×C8 | — | 2 | 12 | no | — |
| 36 | C22×C3×C8 | D4 | 1 | 12 | yes | complemented |
| 37 | C22×C3×C8 | C23 | 1 | 3 | yes | — |
| 38 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×#217 | — | 2 | 6 | no | — |
| 40 | #930 | — | 2 | 12 | no | — |
| 41 | #930 | — | 2 | 12 | no | — |
| 42 | C2×(C2×D3)⋊C4 | — | 2 | 6 | no | — |
| 43 | C2×C4⋊Dic3 | C23 | 1 | 6 | yes | — |
| 44 | C2×D24 | — | 2 | 6 | no | — |
| 45 | C3⋊#95 | — | 2 | 6 | no | — |
| 46 | #821 | — | 2 | 12 | no | — |
| 47 | #818 | — | 2 | 6 | no | — |
| 48 | C22×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 49 | C22×D8 | — | 3 | 1 | no | — |
| 50 | C2×#131 | — | 2 | 12 | no | — |
| 51 | C22×M16 | C2×D3 | 1 | 3 | yes | — |
| 52 | C23×C8 | C2×D3 | 1 | 3 | yes | — |
| 53 | C2×#121 | — | 3 | 6 | no | — |
| 54 | #466 | — | 6 | 12 | no | — |
| 55 | C2×#107 | — | 3 | 3 | no | — |
| 56 | C2×#102 | — | 3 | 6 | no | — |
| 57 | #393 | — | 6 | 2 | no | — |
| 58 | #392 | — | 6 | 6 | no | — |
| 59 | #380 | — | 3 | 6 | no | — |
| 60 | #340 | — | 3 | 6 | no | — |
| 61 | C24×C3 | C2×D4 | 1 | 3 | yes | — |
| 62 | C23×D3 | — | 2 | 2 | no | — |
| 63 | C3×#41 | — | 2 | 8 | no | — |
| 64 | C3×#41 | — | 2 | 24 | no | — |
| 65 | C2×C3×Q8 | — | 2 | 6 | no | — |
| 66 | C2×C3×D4 | — | 2 | 12 | no | — |
| 67 | C2×C3×D4 | — | 2 | 6 | no | — |
| 68 | C22×C3×C4 | — | 2 | 6 | no | — |
| 69 | C22×C3×C4 | — | 2 | 6 | no | — |
| 70 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 71 | C22×C3×C4 | C2×D4 | 1 | 6 | yes | — |
| 72 | C22×C3×C4 | #41 | 1 | 6 | yes | — |
| 73 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 74 | C2×#67 | — | 4 | 12 | no | — |
| 75 | C2×#67 | — | 8 | 12 | no | — |
| 76 | C22×Dic3 | — | 2 | 6 | no | — |
| 77 | C2×D12 | — | 2 | 6 | no | — |
| 78 | C2×D12 | — | 2 | 6 | no | — |
| 79 | C3×M16 | — | 2 | 24 | no | — |
| 80 | C2×C3×C8 | — | 2 | 6 | no | — |
| 81 | C2×C3×C8 | — | 2 | 24 | no | — |
| 82 | C2×C3×C8 | C2×D4 | 1 | 6 | yes | — |
| 83 | C2×C3×C8 | C2×D4 | 1 | 6 | yes | — |
| 84 | #217 | — | 8 | 6 | no | — |
| 85 | (C2×D3)⋊C4 | — | 4 | 12 | no | — |
| 86 | C4⋊Dic3 | — | 2 | 6 | no | — |
| 87 | C4⋊Dic3 | — | 2 | 6 | no | — |
| 88 | D24 | — | 4 | 4 | no | — |
| 89 | C2×#41 | — | 2 | 6 | no | — |
| 90 | C2×#41 | — | 2 | 6 | no | — |
| 91 | C22×Q8 | D12 | 1 | 1 | yes | characteristic |
| 92 | C22×D4 | — | 3 | 2 | no | — |
| 93 | C22×D4 | — | 6 | 6 | no | — |
| 94 | C22×D4 | D12 | 1 | 3 | yes | — |
| 95 | C23×C4 | C22×D3 | 1 | 3 | yes | — |
| 96 | C2×D8 | — | 6 | 6 | no | — |
| 97 | #131 | — | 2 | 32 | no | — |
| 98 | C2×M16 | — | 2 | 6 | no | — |
| 99 | C2×M16 | — | 2 | 12 | no | — |
| 100 | C22×C8 | — | 2 | 12 | no | — |
| 101 | C22×C8 | #67 | 1 | 12 | yes | complemented |
| 102 | C22×C8 | C22×D3 | 1 | 3 | yes | — |
| 103 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 104 | #121 | — | 6 | 12 | no | — |
| 105 | #121 | — | 6 | 12 | no | — |
| 106 | C2×C4⋊C4 | — | 3 | 6 | no | — |
| 107 | C2×C22⋊C4 | — | 6 | 6 | no | — |
| 108 | C2×C22⋊C4 | — | 6 | 6 | no | — |
| 109 | #107 | — | 6 | 6 | no | — |
| 110 | #102 | — | 6 | 12 | no | — |
| 111 | #95 | — | 6 | 6 | no | — |
| 112 | C23×C3 | — | 2 | 6 | no | — |
| 113 | C23×C3 | — | 2 | 3 | no | — |
| 114 | C23×C3 | — | 2 | 6 | no | — |
| 115 | C23×C3 | — | 2 | 3 | no | — |
| 116 | C23×C3 | #121 | 1 | 6 | yes | — |
| 117 | C23×C3 | C22×D4 | 1 | 1 | yes | characteristic |
| 118 | C22×D3 | — | 4 | 6 | no | — |
| 119 | C22×D3 | — | 4 | 6 | no | — |
| 120 | C22×D3 | — | 4 | 2 | no | — |
| 121 | C3×Q8 | — | 2 | 8 | no | — |
| 122 | C3×D4 | — | 2 | 24 | no | — |
| 123 | C2×C3×C4 | — | 2 | 6 | no | — |
| 124 | C2×C3×C4 | — | 2 | 6 | no | — |
| 125 | C2×C3×C4 | — | 2 | 6 | no | — |
| 126 | C2×C3×C4 | — | 2 | 12 | no | — |
| 127 | C2×C3×C4 | #121 | 1 | 6 | yes | — |
| 128 | C2×C3×C4 | C22×D4 | 1 | 3 | yes | — |
| 129 | C2×C3×C4 | C2×#41 | 1 | 3 | yes | — |
| 130 | #67 | — | 8 | 24 | no | — |
| 131 | C2×Dic3 | — | 4 | 6 | no | — |
| 132 | C2×Dic3 | — | 4 | 12 | no | — |
| 133 | D12 | — | 4 | 2 | no | — |
| 134 | D12 | — | 4 | 6 | no | — |
| 135 | C3×C8 | — | 2 | 6 | no | — |
| 136 | C3×C8 | #120 | 1 | 4 | yes | — |
| 137 | C24 | — | 6 | 2 | no | — |
| 138 | C24 | C2×D12 | 1 | 3 | yes | — |
| 139 | #41 | — | 2 | 8 | no | — |
| 140 | #41 | — | 2 | 24 | no | — |
| 141 | C2×Q8 | — | 2 | 6 | no | — |
| 142 | C2×D4 | — | 2 | 12 | no | — |
| 143 | C2×D4 | — | 2 | 6 | no | — |
| 144 | C2×D4 | — | 6 | 6 | no | — |
| 145 | C2×D4 | — | 6 | 6 | no | — |
| 146 | C2×D4 | — | 12 | 12 | no | — |
| 147 | C2×D4 | — | 24 | 12 | no | — |
| 148 | C22×C4 | — | 2 | 6 | no | — |
| 149 | C22×C4 | — | 2 | 6 | no | — |
| 150 | C22×C4 | — | 6 | 6 | no | — |
| 151 | C22×C4 | C2×D12 | 1 | 3 | yes | — |
| 152 | C22×C4 | #235 | 1 | 6 | yes | — |
| 153 | C22×C4 | C2×#67 | 1 | 6 | yes | — |
| 154 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 155 | D8 | — | 12 | 4 | no | — |
| 156 | M16 | — | 2 | 24 | no | — |
| 157 | C2×C8 | — | 2 | 6 | no | — |
| 158 | C2×C8 | — | 2 | 24 | no | — |
| 159 | C2×C8 | C2×#67 | 1 | 6 | yes | — |
| 160 | C2×C8 | C2×#67 | 1 | 6 | yes | — |
| 161 | C4⋊C4 | — | 6 | 6 | no | — |
| 162 | C4⋊C4 | — | 6 | 6 | no | — |
| 163 | C22⋊C4 | — | 12 | 12 | no | — |
| 164 | C22⋊C4 | — | 24 | 6 | no | — |
| 165 | C22×C3 | — | 2 | 6 | no | — |
| 166 | C22×C3 | — | 2 | 6 | no | — |
| 167 | C22×C3 | — | 2 | 6 | no | — |
| 168 | C22×C3 | — | 4 | 6 | no | — |
| 169 | C22×C3 | — | 4 | 6 | no | — |
| 170 | C22×C3 | C2×#121 | 1 | 3 | yes | — |
| 171 | C22×C3 | #576 | 1 | 3 | yes | — |
| 172 | C22×C3 | #578 | 1 | 1 | yes | characteristic |
| 173 | C2×D3 | — | 8 | 2 | no | — |
| 174 | C2×D3 | — | 8 | 6 | no | — |
| 175 | C2×D3 | — | 8 | 6 | no | — |
| 176 | C3×C4 | — | 2 | 6 | no | — |
| 177 | C3×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 178 | C3×C4 | #525 | 1 | 3 | yes | — |
| 179 | Dic3 | — | 8 | 6 | no | — |
| 180 | C23 | — | 2 | 3 | no | — |
| 181 | C23 | — | 2 | 6 | no | — |
| 182 | C23 | — | 2 | 3 | no | — |
| 183 | C23 | — | 2 | 6 | no | — |
| 184 | C23 | — | 12 | 6 | no | — |
| 185 | C23 | — | 12 | 2 | no | — |
| 186 | C23 | — | 12 | 6 | no | — |
| 187 | C23 | #930 | 1 | 6 | yes | — |
| 188 | C23 | C22×D12 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 2 | 8 | no | — |
| 190 | D4 | — | 2 | 24 | no | — |
| 191 | D4 | — | 12 | 6 | no | — |
| 192 | D4 | — | 12 | 2 | no | — |
| 193 | D4 | — | 24 | 24 | no | — |
| 194 | C2×C4 | — | 2 | 6 | no | — |
| 195 | C2×C4 | — | 2 | 12 | no | — |
| 196 | C2×C4 | — | 2 | 6 | no | — |
| 197 | C2×C4 | — | 2 | 6 | no | — |
| 198 | C2×C4 | — | 12 | 12 | no | — |
| 199 | C2×C4 | — | 12 | 6 | no | — |
| 200 | C2×C4 | #930 | 1 | 6 | yes | — |
| 201 | C2×C4 | C2×#235 | 1 | 3 | yes | — |
| 202 | C2×C4 | C22×#67 | 1 | 3 | yes | — |
| 203 | C8 | — | 2 | 6 | no | — |
| 204 | C8 | #953 | 1 | 4 | yes | — |
| 205 | C2×C3 | — | 4 | 6 | no | — |
| 206 | C2×C3 | #2847 | 1 | 1 | yes | characteristic |
| 207 | C2×C3 | #3058 | 1 | 3 | yes | — |
| 208 | C2×C3 | #3061 | 1 | 3 | yes | — |
| 209 | D3 | — | 16 | 2 | no | — |
| 210 | C22 | — | 2 | 6 | no | — |
| 211 | C22 | — | 2 | 6 | no | — |
| 212 | C22 | — | 2 | 6 | no | — |
| 213 | C22 | — | 4 | 6 | no | — |
| 214 | C22 | — | 4 | 6 | no | — |
| 215 | C22 | — | 24 | 2 | no | — |
| 216 | C22 | — | 24 | 6 | no | — |
| 217 | C22 | — | 24 | 6 | no | — |
| 218 | C22 | #5752 | 1 | 3 | yes | — |
| 219 | C22 | #5753 | 1 | 1 | yes | characteristic |
| 220 | C22 | C2×#930 | 1 | 3 | yes | — |
| 221 | C4 | — | 2 | 6 | no | — |
| 222 | C4 | — | 24 | 6 | no | — |
| 223 | C4 | #5791 | 1 | 3 | yes | — |
| 224 | C4 | C2×#953 | 1 | 1 | yes | characteristic |
| 225 | C3 | #32933 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 226 | C2 | — | 4 | 6 | no | — |
| 227 | C2 | — | 48 | 2 | no | — |
| 228 | C2 | #83799 | 1 | 3 | yes | — |
| 229 | C2 | #83801 | 1 | 3 | yes | — |
| 230 | C2 | #84032 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×#131 | C2 | semidirect |
| 2. #84098 | C2 | semidirect |
| 3. C2×#5115 | C2 | semidirect |
| 4. #71990 | C2 | semidirect |
| 5. #71971 | C2 | semidirect |
| 20. C22×#131 | D3 | semidirect |
| 36. C22×C3×C8 | D4 | semidirect |
| 101. C22×C8 | #67 | semidirect |
| 225. C3 | #32933 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#131 | 2 |
| #84098 | 2 |
| C2×#5115 | 2 |
| #71990 | 2 |
| #71971 | 2 |
| #32933 | 3 |