Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
766 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 133 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 | #19128536 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #19128209 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #19127890 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #19127889 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #19127888 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19127673 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #458×C83 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #14581839 | — | 2 | 3 | no | — |
| 9 | #14581837 | — | 2 | 3 | no | — |
| 10 | #14581837 | C22 | 1 | 3 | yes | — |
| 11 | C2×#13050589 | — | 2 | 3 | no | — |
| 12 | C2×#13050589 | C22 | 1 | 3 | yes | — |
| 13 | #41×D83 | — | 2 | 3 | no | — |
| 14 | C22×C4×D83 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #14581775 | C22 | 1 | 3 | yes | complemented |
| 16 | #14581770 | — | 2 | 3 | no | — |
| 17 | #14581765 | — | 2 | 3 | no | — |
| 18 | #14581762 | C22 | 1 | 3 | yes | complemented |
| 19 | #14581760 | C22 | 1 | 3 | yes | complemented |
| 20 | #14581754 | — | 2 | 3 | no | — |
| 21 | (C22⋊D83)⋊C4 | — | 2 | 3 | no | — |
| 22 | #14581748 | C22 | 1 | 3 | yes | complemented |
| 23 | #14581707 | C22 | 1 | 3 | yes | complemented |
| 24 | #14581706 | C22 | 1 | 6 | yes | complemented |
| 25 | H(5,𝔽2)×C83 | C22 | 1 | 1 | yes | characteristic |
| 26 | #123×C83 | C22 | 1 | 3 | yes | complemented |
| 27 | #99×C83 | C22 | 1 | 3 | yes | complemented |
| 28 | C23×D83 | C23 | 1 | 1 | yes | characteristic |
| 29 | #13050591 | — | 2 | 1 | no | — |
| 30 | #13050590 | — | 2 | 6 | no | — |
| 31 | #13050590 | — | 2 | 3 | no | — |
| 32 | #13050589 | — | 2 | 3 | no | — |
| 33 | #13050589 | — | 2 | 6 | no | — |
| 34 | #13050589 | — | 2 | 6 | no | — |
| 35 | #13050589 | — | 2 | 6 | no | — |
| 36 | #13050589 | — | 2 | 6 | no | — |
| 37 | C2×Dic166 | — | 2 | 3 | no | — |
| 38 | C2×Dic166 | D4 | 1 | 3 | yes | complemented |
| 39 | Q8×D83 | — | 2 | 3 | no | — |
| 40 | C2×D332 | — | 2 | 3 | no | — |
| 41 | C2×C22⋊D83 | — | 2 | 3 | no | — |
| 42 | C2×C22⋊D83 | — | 2 | 3 | no | — |
| 43 | C2×C22⋊D83 | — | 2 | 3 | no | — |
| 44 | C2×C22⋊D83 | D4 | 1 | 3 | yes | complemented |
| 45 | C2×C22⋊D83 | C23 | 1 | 3 | yes | — |
| 46 | D4×D83 | — | 2 | 3 | no | — |
| 47 | D4×D83 | — | 2 | 6 | no | — |
| 48 | C22×Dic83 | — | 2 | 1 | no | — |
| 49 | C22×Dic83 | D4 | 1 | 3 | yes | complemented |
| 50 | C22×Dic83 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×C4×D83 | — | 2 | 3 | no | — |
| 52 | C2×C4×D83 | — | 2 | 3 | no | — |
| 53 | C2×C4×D83 | — | 2 | 3 | no | — |
| 54 | C2×C4×D83 | D4 | 1 | 3 | yes | complemented |
| 55 | Dic83⋊C4 | — | 2 | 3 | no | — |
| 56 | Dic83⋊C4 | — | 2 | 3 | no | — |
| 57 | C22⋊Dic83 | C23 | 1 | 3 | yes | — |
| 58 | (C2×D83)⋊C4 | — | 2 | 3 | no | — |
| 59 | (C2×D83)⋊C4 | — | 2 | 3 | no | — |
| 60 | (C2×D83)⋊C4 | — | 2 | 3 | no | — |
| 61 | C4×Dic83 | — | 2 | 3 | no | — |
| 62 | #41×C83 | — | 2 | 3 | no | — |
| 63 | C2×D4×C83 | — | 2 | 3 | no | — |
| 64 | C2×D4×C83 | C23 | 1 | 3 | yes | — |
| 65 | C22×C4×C83 | C23 | 1 | 1 | yes | characteristic |
| 66 | C4⋊C4×C83 | — | 2 | 3 | no | — |
| 67 | C22⋊C4×C83 | — | 2 | 3 | no | — |
| 68 | C22⋊C4×C83 | C23 | 1 | 3 | yes | — |
| 69 | C23×C83 | — | 2 | 1 | no | — |
| 70 | C23×C83 | C2×D4 | 1 | 3 | yes | — |
| 71 | C23×C83 | C24 | 1 | 1 | yes | characteristic |
| 72 | C22×D83 | — | 2 | 3 | no | — |
| 73 | C22×D83 | — | 2 | 3 | no | — |
| 74 | C22×D83 | — | 4 | 2 | no | — |
| 75 | C22×D83 | C2×D4 | 1 | 3 | yes | — |
| 76 | Q8×C83 | — | 2 | 1 | no | — |
| 77 | D4×C83 | — | 2 | 3 | no | — |
| 78 | D4×C83 | — | 2 | 6 | no | — |
| 79 | C2×C4×C83 | — | 2 | 3 | no | — |
| 80 | C2×C4×C83 | — | 2 | 3 | no | — |
| 81 | C2×C4×C83 | — | 2 | 3 | no | — |
| 82 | C2×C4×C83 | C2×D4 | 1 | 3 | yes | — |
| 83 | C22⋊D83 | — | 2 | 6 | no | — |
| 84 | C22⋊D83 | — | 2 | 6 | no | — |
| 85 | C22⋊D83 | — | 2 | 6 | no | — |
| 86 | C22⋊D83 | — | 4 | 6 | no | — |
| 87 | C2×Dic83 | — | 2 | 3 | no | — |
| 88 | C2×Dic83 | — | 2 | 3 | no | — |
| 89 | C2×Dic83 | — | 2 | 3 | no | — |
| 90 | C2×Dic83 | — | 2 | 3 | no | — |
| 91 | C2×Dic83 | C2×D4 | 1 | 3 | yes | — |
| 92 | C2×Dic83 | C2×D4 | 1 | 3 | yes | — |
| 93 | C2×Dic83 | C2×D4 | 1 | 3 | yes | — |
| 94 | D332 | — | 2 | 3 | no | — |
| 95 | D332 | — | 4 | 6 | no | — |
| 96 | C4×D83 | — | 2 | 6 | no | — |
| 97 | C4×D83 | — | 2 | 3 | no | — |
| 98 | C4×D83 | — | 2 | 3 | no | — |
| 99 | C4×D83 | — | 4 | 3 | no | — |
| 100 | C4×D83 | — | 4 | 1 | no | — |
| 101 | Dic166 | — | 2 | 3 | no | — |
| 102 | Dic166 | — | 2 | 6 | no | — |
| 103 | C22×C83 | — | 2 | 3 | no | — |
| 104 | C22×C83 | — | 4 | 1 | no | — |
| 105 | C22×C83 | — | 4 | 3 | no | — |
| 106 | C22×C83 | — | 8 | 1 | no | — |
| 107 | C22×C83 | C22×D4 | 1 | 3 | yes | — |
| 108 | C2×D83 | — | 2 | 6 | no | — |
| 109 | C2×D83 | — | 2 | 3 | no | — |
| 110 | C2×D83 | — | 4 | 3 | no | — |
| 111 | C2×D83 | — | 4 | 1 | no | — |
| 112 | C2×D83 | — | 4 | 3 | no | — |
| 113 | C2×D83 | — | 8 | 3 | no | — |
| 114 | C4×C83 | — | 2 | 3 | no | — |
| 115 | C4×C83 | — | 4 | 3 | no | — |
| 116 | C4×C83 | — | 4 | 1 | no | — |
| 117 | Dic83 | — | 2 | 3 | no | — |
| 118 | Dic83 | — | 4 | 3 | no | — |
| 119 | Dic83 | #120 | 1 | 3 | yes | — |
| 120 | Dic83 | #120 | 1 | 1 | yes | characteristic |
| 121 | C2×C83 | — | 2 | 3 | no | — |
| 122 | C2×C83 | — | 4 | 3 | no | — |
| 123 | C2×C83 | C2×#120 | 1 | 1 | yes | characteristic |
| 124 | D83 | — | 4 | 2 | no | — |
| 125 | D83 | — | 4 | 3 | no | — |
| 126 | #3025 | — | 83 | 1 | no | maximal, Sylow |
| 127 | C83 | #3025 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 128 | #585 | — | 83 | 1 | no | — |
| 129 | #525 | — | 83 | 3 | no | — |
| 130 | #458 | — | 83 | 3 | no | — |
| 131 | #458 | D83 | 1 | 1 | yes | characteristic, complemented |
| 132 | B(4,𝔽2) | — | 83 | 1 | no | — |
| 133 | B(4,𝔽2) | — | 83 | 3 | no | — |
| 134 | #410 | — | 83 | 3 | no | — |
| 135 | 2−1+4 | — | 166 | 3 | no | — |
| 136 | H(5,𝔽2) | — | 83 | 3 | no | — |
| 137 | H(5,𝔽2) | — | 166 | 3 | no | — |
| 138 | H(5,𝔽2) | C2×D83 | 1 | 1 | yes | characteristic |
| 139 | C2×#41 | — | 83 | 3 | no | — |
| 140 | C2×#41 | — | 166 | 3 | no | — |
| 141 | C2×#41 | — | 166 | 3 | no | — |
| 142 | C23×C4 | — | 83 | 1 | no | — |
| 143 | #123 | — | 83 | 3 | no | — |
| 144 | #123 | C2×D83 | 1 | 3 | yes | complemented |
| 145 | C22⋊Q8 | — | 166 | 3 | no | — |
| 146 | #121 | — | 166 | 3 | no | — |
| 147 | #120 | — | 83 | 3 | no | — |
| 148 | #120 | — | 83 | 3 | no | — |
| 149 | C4×D4 | — | 166 | 3 | no | — |
| 150 | C4×D4 | — | 166 | 3 | no | — |
| 151 | #117 | — | 83 | 3 | no | — |
| 152 | #99 | — | 83 | 3 | no | — |
| 153 | #99 | — | 83 | 6 | no | — |
| 154 | #99 | C2×D83 | 1 | 3 | yes | complemented |
| 155 | C24 | — | 83 | 1 | no | — |
| 156 | #41 | — | 2 | 3 | no | — |
| 157 | #41 | — | 166 | 6 | no | — |
| 158 | #41 | — | 166 | 6 | no | — |
| 159 | #41 | — | 166 | 6 | no | — |
| 160 | #41 | — | 166 | 6 | no | — |
| 161 | #41 | — | 166 | 3 | no | — |
| 162 | #41 | — | 166 | 6 | no | — |
| 163 | #41 | — | 166 | 1 | no | — |
| 164 | #41 | — | 166 | 3 | no | — |
| 165 | C2×Q8 | — | 83 | 3 | no | — |
| 166 | C2×Q8 | — | 166 | 3 | no | — |
| 167 | C2×Q8 | — | 166 | 3 | no | — |
| 168 | C2×D4 | — | 2 | 3 | no | — |
| 169 | C2×D4 | — | 83 | 3 | no | — |
| 170 | C2×D4 | — | 83 | 3 | no | — |
| 171 | C2×D4 | — | 166 | 3 | no | — |
| 172 | C2×D4 | — | 166 | 6 | no | — |
| 173 | C2×D4 | — | 166 | 3 | no | — |
| 174 | C2×D4 | — | 166 | 3 | no | — |
| 175 | C2×D4 | — | 166 | 3 | no | — |
| 176 | C2×D4 | — | 166 | 3 | no | — |
| 177 | C2×D4 | C22×D83 | 1 | 3 | yes | — |
| 178 | C22×C4 | — | 83 | 1 | no | — |
| 179 | C22×C4 | — | 83 | 3 | no | — |
| 180 | C22×C4 | — | 83 | 3 | no | — |
| 181 | C22×C4 | — | 166 | 3 | no | — |
| 182 | C22×C4 | — | 166 | 3 | no | — |
| 183 | C22×C4 | — | 166 | 3 | no | — |
| 184 | C22×C4 | — | 166 | 1 | no | — |
| 185 | C22×C4 | C22×D83 | 1 | 1 | yes | characteristic |
| 186 | C4⋊C4 | — | 2 | 3 | no | — |
| 187 | C4⋊C4 | — | 166 | 3 | no | — |
| 188 | C4⋊C4 | — | 166 | 3 | no | — |
| 189 | C22⋊C4 | — | 2 | 3 | no | — |
| 190 | C22⋊C4 | — | 83 | 3 | no | — |
| 191 | C22⋊C4 | — | 166 | 3 | no | — |
| 192 | C22⋊C4 | — | 166 | 3 | no | — |
| 193 | C22⋊C4 | — | 166 | 3 | no | — |
| 194 | C22⋊C4 | C22×D83 | 1 | 3 | yes | — |
| 195 | C42 | — | 166 | 3 | no | — |
| 196 | C23 | — | 2 | 1 | no | — |
| 197 | C23 | — | 332 | 2 | no | — |
| 198 | C23 | — | 83 | 3 | no | — |
| 199 | C23 | — | 166 | 3 | no | — |
| 200 | C23 | — | 166 | 3 | no | — |
| 201 | C23 | D4×D83 | 1 | 3 | yes | — |
| 202 | C23 | C23×D83 | 1 | 1 | yes | characteristic |
| 203 | Q8 | — | 2 | 1 | no | — |
| 204 | Q8 | — | 166 | 6 | no | — |
| 205 | Q8 | — | 166 | 3 | no | — |
| 206 | D4 | — | 2 | 3 | no | — |
| 207 | D4 | — | 2 | 6 | no | — |
| 208 | D4 | — | 332 | 6 | no | — |
| 209 | D4 | — | 332 | 6 | no | — |
| 210 | D4 | — | 166 | 3 | no | — |
| 211 | D4 | — | 166 | 6 | no | — |
| 212 | D4 | — | 166 | 6 | no | — |
| 213 | D4 | — | 166 | 6 | no | — |
| 214 | C2×C4 | — | 2 | 3 | no | — |
| 215 | C2×C4 | — | 2 | 3 | no | — |
| 216 | C2×C4 | — | 2 | 3 | no | — |
| 217 | C2×C4 | — | 332 | 1 | no | — |
| 218 | C2×C4 | — | 332 | 3 | no | — |
| 219 | C2×C4 | — | 83 | 3 | no | — |
| 220 | C2×C4 | — | 83 | 3 | no | — |
| 221 | C2×C4 | — | 83 | 3 | no | — |
| 222 | C2×C4 | — | 166 | 3 | no | — |
| 223 | C2×C4 | — | 166 | 3 | no | — |
| 224 | C2×C4 | — | 166 | 6 | no | — |
| 225 | C2×C4 | — | 166 | 3 | no | — |
| 226 | C2×C4 | — | 166 | 3 | no | — |
| 227 | C2×C4 | — | 166 | 3 | no | — |
| 228 | C2×C4 | — | 166 | 3 | no | — |
| 229 | C2×C4 | D4×D83 | 1 | 3 | yes | — |
| 230 | C22 | — | 2 | 3 | no | — |
| 231 | C22 | — | 4 | 1 | no | — |
| 232 | C22 | — | 4 | 3 | no | — |
| 233 | C22 | — | 8 | 1 | no | — |
| 234 | C22 | — | 332 | 3 | no | — |
| 235 | C22 | — | 332 | 3 | no | — |
| 236 | C22 | — | 332 | 1 | no | — |
| 237 | C22 | — | 664 | 3 | no | — |
| 238 | C22 | — | 166 | 6 | no | — |
| 239 | C22 | — | 166 | 3 | no | — |
| 240 | C22 | C2×D4×D83 | 1 | 3 | yes | — |
| 241 | C4 | — | 2 | 3 | no | — |
| 242 | C4 | — | 4 | 3 | no | — |
| 243 | C4 | — | 4 | 1 | no | — |
| 244 | C4 | — | 332 | 3 | no | — |
| 245 | C4 | — | 83 | 1 | no | — |
| 246 | C4 | — | 83 | 3 | no | — |
| 247 | C4 | — | 166 | 3 | no | — |
| 248 | C2 | — | 2 | 3 | no | — |
| 249 | C2 | — | 4 | 3 | no | — |
| 250 | C2 | — | 332 | 2 | no | — |
| 251 | C2 | — | 332 | 3 | no | — |
| 252 | C2 | #120×D83 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #120×D83 |
| commutator, cosocle | C23×C83 | C24 |
| Frattini | C23 | C23×D83 |
| Fitting | #458×C83 | C2 |
| socle | C2×C83 | C2×#120 |
| hypercenter | #458 | D83 |
| hypocenter | C83 | #3025 |
| LCS | G ▷ C23×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #458 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #19128536 | C2 | semidirect |
| 2. #19128209 | C2 | semidirect |
| 3. #19127890 | C2 | semidirect |
| 4. #19127889 | C2 | semidirect |
| 5. #19127888 | C2 | semidirect |
| 6. #19127673 | C2 | semidirect |
| 7. #458×C83 | C2 | semidirect |
| 14. C22×C4×D83 | C22 | semidirect |
| 15. #14581775 | C22 | semidirect |
| 18. #14581762 | C22 | semidirect |
| 19. #14581760 | C22 | semidirect |
| 22. #14581748 | C22 | semidirect |
| 23. #14581707 | C22 | semidirect |
| 24. #14581706 | C22 | semidirect |
| 26. #123×C83 | C22 | semidirect |
| 27. #99×C83 | C22 | semidirect |
| 38. C2×Dic166 | D4 | semidirect |
| 44. C2×C22⋊D83 | D4 | semidirect |
| 49. C22×Dic83 | D4 | semidirect |
| 54. C2×C4×D83 | D4 | semidirect |
| 127. C83 | #3025 | semidirect |
| 131. #458 | D83 | semidirect |
| 144. #123 | C2×D83 | semidirect |
| 154. #99 | C2×D83 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19128536 | 2 |
| #19128209 | 2 |
| #19127890 | 2 |
| #19127889 | 2 |
| #19127888 | 2 |
| #19127673 | 2 |
| #458×C83 | 2 |
| #3025 | 83 |