Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
3876 conjugacy classes of subgroups; only the 387 normal subgroups are shown.
This group has 3876 conjugacy classes of subgroups — too many to draw. Only its 387 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 | C24×C3×Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#5227 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #84032 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #72132 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #72126 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #32879 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C23×C3×Q8 | — | 2 | 24 | no | — |
| 8 | C23×C3×Q8 | — | 2 | 3 | no | — |
| 9 | C23×C3×Q8 | C22 | 1 | 6 | yes | — |
| 10 | C24×C3×C4 | — | 2 | 1 | no | — |
| 11 | C24×C3×C4 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#953 | — | 2 | 1 | no | — |
| 13 | C22×#215 | C22 | 1 | 4 | yes | complemented |
| 14 | C2×#930 | C22 | 1 | 3 | yes | — |
| 15 | #5227 | — | 2 | 24 | no | — |
| 16 | C2×#835 | C22 | 1 | 12 | yes | complemented |
| 17 | #5213 | — | 2 | 3 | no | — |
| 18 | #5212 | C22 | 1 | 3 | yes | — |
| 19 | C2×#830 | C22 | 1 | 6 | yes | — |
| 20 | C24×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | C2×#448 | — | 3 | 6 | no | — |
| 22 | #2847 | — | 3 | 1 | no | — |
| 23 | #1787 | — | 3 | 6 | no | — |
| 24 | #1779 | — | 3 | 1 | no | — |
| 25 | C25×C3 | D4 | 1 | 1 | yes | characteristic |
| 26 | C22×C3×Q8 | — | 2 | 128 | no | — |
| 27 | C22×C3×Q8 | — | 2 | 48 | no | — |
| 28 | C22×C3×Q8 | — | 2 | 24 | no | — |
| 29 | C22×C3×Q8 | — | 2 | 48 | no | — |
| 30 | C22×C3×Q8 | — | 2 | 12 | no | — |
| 31 | C22×C3×Q8 | — | 2 | 6 | no | — |
| 32 | C22×C3×Q8 | D4 | 1 | 24 | yes | complemented |
| 33 | C22×C3×Q8 | C23 | 1 | 4 | yes | — |
| 34 | C23×C3×C4 | — | 2 | 24 | no | — |
| 35 | C23×C3×C4 | — | 2 | 6 | no | — |
| 36 | C23×C3×C4 | — | 2 | 12 | no | — |
| 37 | C23×C3×C4 | D4 | 1 | 3 | yes | — |
| 38 | C23×C3×C4 | C23 | 1 | 3 | yes | — |
| 39 | C22×#67 | — | 2 | 3 | no | — |
| 40 | C22×D12 | C23 | 1 | 1 | yes | characteristic |
| 41 | #953 | — | 4 | 4 | no | — |
| 42 | C2×#217 | — | 2 | 3 | no | — |
| 43 | C2×#215 | — | 2 | 24 | no | — |
| 44 | #930 | — | 2 | 12 | no | — |
| 45 | C2×(C2×D3)⋊C4 | — | 2 | 3 | no | — |
| 46 | C2×C4⋊Dic3 | C23 | 1 | 3 | yes | — |
| 47 | C22×C3⋊C8 | C23 | 1 | 4 | yes | — |
| 48 | #835 | — | 2 | 24 | no | — |
| 49 | C3⋊#95 | — | 2 | 3 | no | — |
| 50 | #830 | — | 2 | 12 | no | — |
| 51 | C23×Q8 | — | 2 | 24 | no | — |
| 52 | C23×Q8 | — | 2 | 3 | no | — |
| 53 | C23×Q8 | C2×D3 | 1 | 6 | yes | — |
| 54 | C24×C4 | — | 2 | 1 | no | — |
| 55 | C24×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 56 | C22×SD16 | — | 3 | 4 | no | — |
| 57 | C2×#121 | — | 3 | 3 | no | — |
| 58 | C2×#120 | — | 6 | 1 | no | — |
| 59 | #448 | — | 6 | 24 | no | — |
| 60 | C2×#103 | — | 3 | 12 | no | — |
| 61 | C2×C22⋊C8 | — | 3 | 6 | no | — |
| 62 | #386 | — | 6 | 3 | no | — |
| 63 | #380 | — | 3 | 3 | no | — |
| 64 | C24×C3 | — | 2 | 6 | no | — |
| 65 | C24×C3 | — | 2 | 6 | no | — |
| 66 | C24×C3 | SD16 | 1 | 4 | yes | complemented |
| 67 | C24×C3 | C2×D4 | 1 | 3 | yes | — |
| 68 | C23×D3 | — | 2 | 1 | no | — |
| 69 | C2×C3×Q8 | — | 2 | 192 | no | — |
| 70 | C2×C3×Q8 | — | 2 | 96 | no | — |
| 71 | C2×C3×Q8 | — | 2 | 48 | no | — |
| 72 | C2×C3×Q8 | — | 2 | 96 | no | — |
| 73 | C2×C3×Q8 | — | 2 | 24 | no | — |
| 74 | C2×C3×Q8 | — | 2 | 12 | no | — |
| 75 | C2×C3×Q8 | C2×D4 | 1 | 24 | yes | — |
| 76 | C22×C3×C4 | — | 2 | 64 | no | — |
| 77 | C22×C3×C4 | — | 2 | 48 | no | — |
| 78 | C22×C3×C4 | — | 2 | 32 | no | — |
| 79 | C22×C3×C4 | — | 2 | 24 | no | — |
| 80 | C22×C3×C4 | — | 2 | 6 | no | — |
| 81 | C22×C3×C4 | — | 2 | 4 | no | — |
| 82 | C22×C3×C4 | — | 2 | 12 | no | — |
| 83 | C22×C3×C4 | — | 2 | 12 | no | — |
| 84 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 85 | C22×C3×C4 | C2×D4 | 1 | 12 | yes | — |
| 86 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 87 | C2×#67 | — | 4 | 6 | no | — |
| 88 | C2×#67 | — | 4 | 6 | no | — |
| 89 | C2×#67 | — | 4 | 6 | no | — |
| 90 | C22×Dic3 | — | 2 | 3 | no | — |
| 91 | C2×D12 | — | 2 | 6 | no | — |
| 92 | #217 | — | 4 | 6 | no | — |
| 93 | #215 | — | 4 | 16 | no | — |
| 94 | (C2×D3)⋊C4 | — | 4 | 6 | no | — |
| 95 | C4⋊Dic3 | — | 2 | 6 | no | — |
| 96 | C2×C3⋊C8 | — | 2 | 12 | no | — |
| 97 | C25 | #67 | 1 | 1 | yes | characteristic |
| 98 | C22×Q8 | — | 2 | 128 | no | — |
| 99 | C22×Q8 | — | 2 | 48 | no | — |
| 100 | C22×Q8 | — | 2 | 24 | no | — |
| 101 | C22×Q8 | — | 2 | 48 | no | — |
| 102 | C22×Q8 | — | 2 | 12 | no | — |
| 103 | C22×Q8 | — | 2 | 6 | no | — |
| 104 | C22×Q8 | #67 | 1 | 24 | yes | complemented |
| 105 | C22×Q8 | C22×D3 | 1 | 4 | yes | — |
| 106 | C22×D4 | — | 3 | 1 | no | — |
| 107 | C22×D4 | — | 6 | 3 | no | — |
| 108 | C23×C4 | — | 2 | 24 | no | — |
| 109 | C23×C4 | — | 2 | 6 | no | — |
| 110 | C23×C4 | — | 2 | 12 | no | — |
| 111 | C23×C4 | #67 | 1 | 3 | yes | — |
| 112 | C23×C4 | C22×D3 | 1 | 3 | yes | — |
| 113 | C2×SD16 | — | 6 | 24 | no | — |
| 114 | C22×C8 | — | 3 | 4 | no | — |
| 115 | #121 | — | 6 | 12 | no | — |
| 116 | #120 | — | 12 | 4 | no | — |
| 117 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 118 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 119 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 120 | #103 | — | 6 | 24 | no | — |
| 121 | C22⋊C8 | — | 6 | 12 | no | — |
| 122 | #95 | — | 6 | 3 | no | — |
| 123 | C23×C3 | — | 2 | 8 | no | — |
| 124 | C23×C3 | — | 2 | 24 | no | — |
| 125 | C23×C3 | — | 2 | 6 | no | — |
| 126 | C23×C3 | — | 2 | 6 | no | — |
| 127 | C23×C3 | — | 2 | 24 | no | — |
| 128 | C23×C3 | #120 | 1 | 6 | yes | — |
| 129 | C23×C3 | C2×SD16 | 1 | 6 | yes | — |
| 130 | C23×C3 | #137 | 1 | 6 | yes | — |
| 131 | C23×C3 | C22×D4 | 1 | 1 | yes | characteristic |
| 132 | C22×D3 | — | 4 | 3 | no | — |
| 133 | C22×D3 | — | 4 | 1 | no | — |
| 134 | C22×D3 | — | 4 | 3 | no | — |
| 135 | C3×Q8 | — | 2 | 96 | no | — |
| 136 | C3×Q8 | — | 2 | 24 | no | — |
| 137 | C3×Q8 | #120 | 1 | 16 | yes | complemented |
| 138 | C2×C3×C4 | — | 2 | 96 | no | — |
| 139 | C2×C3×C4 | — | 2 | 24 | no | — |
| 140 | C2×C3×C4 | — | 2 | 24 | no | — |
| 141 | C2×C3×C4 | — | 2 | 6 | no | — |
| 142 | C2×C3×C4 | — | 2 | 24 | no | — |
| 143 | C2×C3×C4 | #120 | 1 | 6 | yes | — |
| 144 | C2×C3×C4 | C22×D4 | 1 | 6 | yes | — |
| 145 | #67 | — | 8 | 12 | no | — |
| 146 | C2×Dic3 | — | 4 | 3 | no | — |
| 147 | C2×Dic3 | — | 4 | 3 | no | — |
| 148 | C2×Dic3 | — | 4 | 3 | no | — |
| 149 | D12 | — | 4 | 4 | no | — |
| 150 | C3⋊C8 | — | 4 | 4 | no | — |
| 151 | C24 | — | 2 | 6 | no | — |
| 152 | C24 | — | 2 | 6 | no | — |
| 153 | C24 | — | 6 | 1 | no | — |
| 154 | C24 | #215 | 1 | 4 | yes | complemented |
| 155 | C24 | C2×#67 | 1 | 3 | yes | — |
| 156 | C2×Q8 | — | 2 | 192 | no | — |
| 157 | C2×Q8 | — | 2 | 96 | no | — |
| 158 | C2×Q8 | — | 2 | 48 | no | — |
| 159 | C2×Q8 | — | 2 | 96 | no | — |
| 160 | C2×Q8 | — | 2 | 24 | no | — |
| 161 | C2×Q8 | — | 2 | 12 | no | — |
| 162 | C2×Q8 | C2×#67 | 1 | 24 | yes | — |
| 163 | C2×D4 | — | 6 | 6 | no | — |
| 164 | C2×D4 | — | 12 | 6 | no | — |
| 165 | C2×D4 | — | 12 | 6 | no | — |
| 166 | C2×D4 | — | 12 | 6 | no | — |
| 167 | C22×C4 | — | 2 | 64 | no | — |
| 168 | C22×C4 | — | 2 | 48 | no | — |
| 169 | C22×C4 | — | 2 | 32 | no | — |
| 170 | C22×C4 | — | 2 | 24 | no | — |
| 171 | C22×C4 | — | 2 | 6 | no | — |
| 172 | C22×C4 | — | 2 | 4 | no | — |
| 173 | C22×C4 | — | 2 | 12 | no | — |
| 174 | C22×C4 | — | 2 | 12 | no | — |
| 175 | C22×C4 | — | 6 | 3 | no | — |
| 176 | C22×C4 | C2×#67 | 1 | 3 | yes | — |
| 177 | C22×C4 | C2×#67 | 1 | 12 | yes | — |
| 178 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 179 | SD16 | — | 12 | 16 | no | — |
| 180 | C2×C8 | — | 6 | 12 | no | — |
| 181 | C4⋊C4 | — | 6 | 6 | no | — |
| 182 | C22⋊C4 | — | 12 | 6 | no | — |
| 183 | C22⋊C4 | — | 12 | 6 | no | — |
| 184 | C22×C3 | — | 2 | 32 | no | — |
| 185 | C22×C3 | — | 2 | 6 | no | — |
| 186 | C22×C3 | — | 2 | 12 | no | — |
| 187 | C22×C3 | — | 2 | 6 | no | — |
| 188 | C22×C3 | — | 2 | 12 | no | — |
| 189 | C22×C3 | #448 | 1 | 12 | yes | complemented |
| 190 | C22×C3 | C2×#120 | 1 | 3 | yes | — |
| 191 | C22×C3 | C22×SD16 | 1 | 1 | yes | characteristic |
| 192 | C22×C3 | C2×#137 | 1 | 3 | yes | — |
| 193 | C2×D3 | — | 8 | 1 | no | — |
| 194 | C2×D3 | — | 8 | 3 | no | — |
| 195 | C2×D3 | — | 8 | 3 | no | — |
| 196 | C3×C4 | — | 2 | 16 | no | — |
| 197 | C3×C4 | — | 2 | 6 | no | — |
| 198 | C3×C4 | C2×#120 | 1 | 4 | yes | — |
| 199 | Dic3 | — | 8 | 3 | no | — |
| 200 | C23 | — | 2 | 8 | no | — |
| 201 | C23 | — | 2 | 24 | no | — |
| 202 | C23 | — | 2 | 24 | no | — |
| 203 | C23 | — | 2 | 6 | no | — |
| 204 | C23 | — | 2 | 6 | no | — |
| 205 | C23 | — | 12 | 1 | no | — |
| 206 | C23 | — | 12 | 3 | no | — |
| 207 | C23 | — | 12 | 3 | no | — |
| 208 | C23 | C2×#215 | 1 | 6 | yes | — |
| 209 | C23 | #942 | 1 | 6 | yes | — |
| 210 | C23 | #953 | 1 | 6 | yes | — |
| 211 | C23 | C22×#67 | 1 | 1 | yes | characteristic |
| 212 | Q8 | — | 2 | 96 | no | — |
| 213 | Q8 | — | 2 | 24 | no | — |
| 214 | Q8 | #953 | 1 | 16 | yes | complemented |
| 215 | D4 | — | 12 | 4 | no | — |
| 216 | D4 | — | 24 | 12 | no | — |
| 217 | C2×C4 | — | 2 | 96 | no | — |
| 218 | C2×C4 | — | 2 | 24 | no | — |
| 219 | C2×C4 | — | 2 | 24 | no | — |
| 220 | C2×C4 | — | 2 | 6 | no | — |
| 221 | C2×C4 | — | 2 | 24 | no | — |
| 222 | C2×C4 | — | 12 | 3 | no | — |
| 223 | C2×C4 | — | 12 | 3 | no | — |
| 224 | C2×C4 | — | 12 | 3 | no | — |
| 225 | C2×C4 | #953 | 1 | 6 | yes | — |
| 226 | C2×C4 | C22×#67 | 1 | 6 | yes | — |
| 227 | C8 | — | 12 | 4 | no | — |
| 228 | C2×C3 | — | 2 | 12 | no | — |
| 229 | C2×C3 | #2846 | 1 | 1 | yes | characteristic |
| 230 | C2×C3 | C2×#448 | 1 | 3 | yes | — |
| 231 | C2×C3 | #3004 | 1 | 3 | yes | — |
| 232 | D3 | — | 16 | 1 | no | — |
| 233 | C22 | — | 2 | 32 | no | — |
| 234 | C22 | — | 2 | 12 | no | — |
| 235 | C22 | — | 2 | 12 | no | — |
| 236 | C22 | — | 2 | 6 | no | — |
| 237 | C22 | — | 2 | 6 | no | — |
| 238 | C22 | — | 24 | 3 | no | — |
| 239 | C22 | — | 24 | 3 | no | — |
| 240 | C22 | — | 24 | 1 | no | — |
| 241 | C22 | #5227 | 1 | 12 | yes | complemented |
| 242 | C22 | C22×#215 | 1 | 1 | yes | characteristic |
| 243 | C22 | C2×#942 | 1 | 3 | yes | — |
| 244 | C22 | C2×#953 | 1 | 3 | yes | — |
| 245 | C4 | — | 2 | 16 | no | — |
| 246 | C4 | — | 2 | 6 | no | — |
| 247 | C4 | — | 24 | 3 | no | — |
| 248 | C4 | C2×#953 | 1 | 4 | yes | — |
| 249 | C3 | #32879 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 250 | C2 | — | 2 | 12 | no | — |
| 251 | C2 | — | 48 | 1 | no | — |
| 252 | C2 | C2×#5227 | 1 | 3 | yes | — |
| 253 | C2 | #84059 | 1 | 3 | yes | — |
| 254 | C2 | #84128 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3×Q8 | C2 | semidirect |
| 2. C2×#5227 | C2 | semidirect |
| 4. #72132 | C2 | semidirect |
| 5. #72126 | C2 | semidirect |
| 13. C22×#215 | C22 | semidirect |
| 16. C2×#835 | C22 | semidirect |
| 20. C24×Q8 | D3 | semidirect |
| 32. C22×C3×Q8 | D4 | semidirect |
| 66. C24×C3 | SD16 | semidirect |
| 104. C22×Q8 | #67 | semidirect |
| 137. C3×Q8 | #120 | semidirect |
| 154. C24 | #215 | semidirect |
| 189. C22×C3 | #448 | semidirect |
| 214. Q8 | #953 | semidirect |
| 241. C22 | #5227 | semidirect |
| 249. C3 | #32879 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×Q8 | 2 |
| C2×#5227 | 2 |
| #84032 | 2 |
| #72132 | 2 |
| #72126 | 2 |
| #32879 | 3 |