Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3342 conjugacy classes of subgroups; only the 698 normal subgroups are shown.
This group has 3342 conjugacy classes of subgroups — too many to draw. Only its 698 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×#459 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | C2×#2328 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #28259 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 4 | C2×#2054 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×#1843 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#1804 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C23×D8 | C22 | 1 | 4 | yes | — |
| 8 | C22×#127 | C22 | 1 | 2 | yes | — |
| 9 | C22×C4×D4 | C22 | 1 | 4 | yes | — |
| 10 | C2×#459 | — | 2 | 32 | no | — |
| 11 | C2×#459 | C22 | 1 | 32 | yes | complemented |
| 12 | C22×C4⋊C8 | C22 | 1 | 4 | yes | — |
| 13 | C22×C4⋊C8 | C22 | 1 | 2 | yes | — |
| 14 | C22×#102 | C22 | 1 | 8 | yes | — |
| 15 | #2328 | — | 2 | 8 | no | — |
| 16 | #2328 | C22 | 1 | 16 | yes | complemented |
| 17 | C2×#390 | C22 | 1 | 4 | yes | — |
| 18 | C2×#386 | — | 2 | 8 | no | — |
| 19 | C2×#382 | C22 | 1 | 2 | yes | — |
| 20 | C2×#376 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2054 | C22 | 1 | 32 | yes | complemented |
| 22 | #1843 | C22 | 1 | 8 | yes | complemented |
| 23 | #1804 | C22 | 1 | 32 | yes | complemented |
| 24 | C2×#340 | C22 | 1 | 4 | yes | — |
| 25 | C23×D4 | — | 2 | 4 | no | — |
| 26 | C23×D4 | C23 | 1 | 4 | yes | — |
| 27 | C24×C4 | — | 2 | 4 | no | — |
| 28 | C22×D8 | — | 2 | 32 | no | — |
| 29 | C22×D8 | — | 4 | 8 | no | — |
| 30 | C22×D8 | C23 | 1 | 16 | yes | — |
| 31 | C23×C8 | C23 | 1 | 4 | yes | — |
| 32 | C2×#127 | — | 2 | 4 | no | — |
| 33 | C2×#127 | — | 2 | 8 | no | — |
| 34 | C2×#127 | D4 | 1 | 16 | yes | complemented |
| 35 | C2×#127 | C23 | 1 | 8 | yes | — |
| 36 | C2×C4×D4 | — | 2 | 8 | no | — |
| 37 | C2×C4×D4 | — | 2 | 16 | no | — |
| 38 | C2×C4×D4 | — | 2 | 16 | no | — |
| 39 | C2×C4×D4 | C23 | 1 | 16 | yes | — |
| 40 | C22×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 41 | C22×C22⋊C4 | — | 2 | 4 | no | — |
| 42 | C22×C22⋊C4 | — | 2 | 8 | no | — |
| 43 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 44 | C22×C42 | C23 | 1 | 2 | yes | — |
| 45 | #459 | — | 2 | 128 | no | — |
| 46 | C2×C4⋊C8 | — | 2 | 16 | no | — |
| 47 | C2×C4⋊C8 | — | 4 | 4 | no | — |
| 48 | C2×C4⋊C8 | C23 | 1 | 16 | yes | — |
| 49 | C2×C4⋊C8 | C23 | 1 | 8 | yes | — |
| 50 | C2×#102 | — | 2 | 32 | no | — |
| 51 | C2×#102 | C23 | 1 | 32 | yes | — |
| 52 | #390 | — | 2 | 8 | no | — |
| 53 | #390 | C23 | 1 | 16 | yes | — |
| 54 | #386 | — | 2 | 64 | no | — |
| 55 | #382 | — | 2 | 4 | no | — |
| 56 | #382 | C23 | 1 | 8 | yes | — |
| 57 | #376 | D4 | 1 | 4 | yes | — |
| 58 | #376 | C23 | 1 | 4 | yes | — |
| 59 | C2×#95 | — | 2 | 4 | no | — |
| 60 | C2×#95 | D4 | 1 | 4 | yes | — |
| 61 | #340 | C23 | 1 | 16 | yes | — |
| 62 | C25 | — | 2 | 4 | no | — |
| 63 | C22×D4 | — | 2 | 32 | no | — |
| 64 | C22×D4 | — | 2 | 8 | no | — |
| 65 | C22×D4 | — | 2 | 4 | no | — |
| 66 | C22×D4 | — | 2 | 16 | no | — |
| 67 | C22×D4 | — | 2 | 8 | no | — |
| 68 | C22×D4 | — | 2 | 8 | no | — |
| 69 | C22×D4 | — | 2 | 8 | no | — |
| 70 | C22×D4 | — | 2 | 8 | no | — |
| 71 | C22×D4 | — | 8 | 16 | no | — |
| 72 | C22×D4 | C2×D4 | 1 | 16 | yes | — |
| 73 | C22×D4 | C24 | 1 | 8 | yes | — |
| 74 | C23×C4 | — | 2 | 4 | no | — |
| 75 | C23×C4 | — | 2 | 16 | no | — |
| 76 | C23×C4 | — | 4 | 16 | no | — |
| 77 | C23×C4 | — | 8 | 16 | no | — |
| 78 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 79 | C23×C4 | C2×D4 | 1 | 4 | yes | — |
| 80 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 81 | C2×D8 | — | 2 | 64 | no | — |
| 82 | C2×D8 | — | 4 | 16 | no | — |
| 83 | C2×D8 | — | 4 | 64 | no | — |
| 84 | C22×C8 | — | 2 | 16 | no | — |
| 85 | C22×C8 | — | 2 | 8 | no | — |
| 86 | C22×C8 | C24 | 1 | 8 | yes | — |
| 87 | #127 | — | 2 | 32 | no | — |
| 88 | #127 | — | 2 | 16 | no | — |
| 89 | #127 | C2×D4 | 1 | 32 | yes | complemented |
| 90 | C4×D4 | — | 2 | 32 | no | — |
| 91 | C4×D4 | — | 2 | 32 | no | — |
| 92 | C4×D4 | — | 4 | 32 | no | — |
| 93 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 94 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 95 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 96 | C2×C4⋊C4 | C24 | 1 | 8 | yes | — |
| 97 | C2×C22⋊C4 | — | 2 | 16 | no | — |
| 98 | C2×C22⋊C4 | — | 2 | 32 | no | — |
| 99 | C2×C22⋊C4 | — | 4 | 16 | no | — |
| 100 | C2×C22⋊C4 | — | 8 | 16 | no | — |
| 101 | C2×C42 | — | 2 | 2 | no | — |
| 102 | C2×C42 | — | 2 | 2 | no | — |
| 103 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 104 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 105 | C2×C42 | C2×D4 | 1 | 8 | yes | — |
| 106 | C2×C42 | #41 | 1 | 8 | yes | — |
| 107 | C2×C42 | C24 | 1 | 2 | yes | — |
| 108 | C2×C42 | C24 | 1 | 4 | yes | — |
| 109 | C4⋊C8 | — | 2 | 32 | no | — |
| 110 | C4⋊C8 | — | 4 | 8 | no | — |
| 111 | #102 | — | 2 | 64 | no | — |
| 112 | #95 | — | 2 | 16 | no | — |
| 113 | #95 | C2×D4 | 1 | 16 | yes | — |
| 114 | C24 | — | 2 | 16 | no | — |
| 115 | C24 | — | 2 | 8 | no | — |
| 116 | C24 | — | 4 | 8 | no | — |
| 117 | C24 | — | 4 | 8 | no | — |
| 118 | C24 | — | 4 | 16 | no | — |
| 119 | C24 | — | 4 | 16 | no | — |
| 120 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 121 | C2×D4 | — | 2 | 16 | no | — |
| 122 | C2×D4 | — | 2 | 64 | no | — |
| 123 | C2×D4 | — | 2 | 32 | no | — |
| 124 | C2×D4 | — | 2 | 32 | no | — |
| 125 | C2×D4 | — | 2 | 8 | no | — |
| 126 | C2×D4 | — | 2 | 8 | no | — |
| 127 | C2×D4 | — | 2 | 8 | no | — |
| 128 | C2×D4 | — | 2 | 8 | no | — |
| 129 | C2×D4 | — | 2 | 8 | no | — |
| 130 | C2×D4 | — | 4 | 8 | no | — |
| 131 | C2×D4 | — | 4 | 8 | no | — |
| 132 | C2×D4 | — | 4 | 16 | no | — |
| 133 | C2×D4 | — | 4 | 16 | no | — |
| 134 | C2×D4 | — | 8 | 64 | no | — |
| 135 | C2×D4 | — | 8 | 32 | no | — |
| 136 | C2×D4 | C22×D4 | 1 | 16 | yes | — |
| 137 | C22×C4 | — | 2 | 16 | no | — |
| 138 | C22×C4 | — | 2 | 4 | no | — |
| 139 | C22×C4 | — | 2 | 8 | no | — |
| 140 | C22×C4 | — | 2 | 8 | no | — |
| 141 | C22×C4 | — | 4 | 8 | no | — |
| 142 | C22×C4 | — | 4 | 16 | no | — |
| 143 | C22×C4 | — | 4 | 32 | no | — |
| 144 | C22×C4 | — | 8 | 32 | no | — |
| 145 | C22×C4 | — | 8 | 64 | no | — |
| 146 | C22×C4 | #121 | 1 | 16 | yes | — |
| 147 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 148 | C22×C4 | C22×D4 | 1 | 8 | yes | — |
| 149 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 150 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 151 | C22×C4 | C2×#41 | 1 | 4 | yes | — |
| 152 | C22×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 153 | C22×C4 | 2−1+4 | 1 | 2 | yes | — |
| 154 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 155 | D8 | — | 4 | 64 | no | — |
| 156 | C2×C8 | — | 2 | 16 | no | — |
| 157 | C2×C8 | — | 2 | 8 | no | — |
| 158 | C2×C8 | — | 4 | 16 | no | — |
| 159 | C4⋊C4 | — | 2 | 8 | no | — |
| 160 | C4⋊C4 | — | 2 | 8 | no | — |
| 161 | C4⋊C4 | — | 4 | 8 | no | — |
| 162 | C22⋊C4 | — | 4 | 32 | no | — |
| 163 | C22⋊C4 | — | 8 | 32 | no | — |
| 164 | C42 | — | 2 | 4 | no | — |
| 165 | C42 | — | 2 | 2 | no | — |
| 166 | C42 | — | 2 | 2 | no | — |
| 167 | C42 | #121 | 1 | 16 | yes | — |
| 168 | C42 | C22×D4 | 1 | 8 | yes | — |
| 169 | C42 | C2×#41 | 1 | 8 | yes | — |
| 170 | C23 | — | 2 | 16 | no | — |
| 171 | C23 | — | 4 | 8 | no | — |
| 172 | C23 | — | 4 | 8 | no | — |
| 173 | C23 | — | 4 | 64 | no | — |
| 174 | C23 | — | 4 | 32 | no | — |
| 175 | C23 | — | 4 | 16 | no | — |
| 176 | C23 | — | 4 | 16 | no | — |
| 177 | C23 | — | 8 | 32 | no | — |
| 178 | C23 | — | 8 | 16 | no | — |
| 179 | C23 | — | 8 | 16 | no | — |
| 180 | C23 | C2×#121 | 1 | 4 | yes | — |
| 181 | C23 | #537 | 1 | 2 | yes | — |
| 182 | C23 | #576 | 1 | 8 | yes | — |
| 183 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 184 | D4 | — | 2 | 16 | no | — |
| 185 | D4 | — | 2 | 16 | no | — |
| 186 | D4 | — | 4 | 8 | no | — |
| 187 | D4 | — | 4 | 8 | no | — |
| 188 | D4 | — | 4 | 32 | no | — |
| 189 | D4 | — | 8 | 64 | no | — |
| 190 | C2×C4 | — | 2 | 32 | no | — |
| 191 | C2×C4 | — | 2 | 16 | no | — |
| 192 | C2×C4 | — | 2 | 4 | no | — |
| 193 | C2×C4 | — | 2 | 8 | no | — |
| 194 | C2×C4 | — | 4 | 8 | no | — |
| 195 | C2×C4 | — | 4 | 8 | no | — |
| 196 | C2×C4 | — | 4 | 16 | no | — |
| 197 | C2×C4 | — | 8 | 64 | no | — |
| 198 | C2×C4 | — | 8 | 16 | no | — |
| 199 | C2×C4 | C2×#121 | 1 | 16 | yes | — |
| 200 | C2×C4 | C2×#121 | 1 | 4 | yes | — |
| 201 | C2×C4 | #537 | 1 | 2 | yes | — |
| 202 | C2×C4 | D42 | 1 | 4 | yes | — |
| 203 | C2×C4 | #547 | 1 | 8 | yes | — |
| 204 | C2×C4 | #555 | 1 | 4 | yes | — |
| 205 | C2×C4 | C23×D4 | 1 | 2 | yes | — |
| 206 | C2×C4 | C22×#41 | 1 | 2 | yes | — |
| 207 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 209 | C8 | — | 4 | 8 | no | — |
| 210 | C22 | — | 4 | 32 | no | — |
| 211 | C22 | — | 4 | 16 | no | — |
| 212 | C22 | — | 8 | 8 | no | — |
| 213 | C22 | — | 8 | 8 | no | — |
| 214 | C22 | — | 8 | 64 | no | — |
| 215 | C22 | — | 8 | 16 | no | — |
| 216 | C22 | #2537 | 1 | 4 | yes | — |
| 217 | C22 | #3041 | 1 | 16 | yes | — |
| 218 | C22 | #3259 | 1 | 8 | yes | — |
| 219 | C22 | C22×#121 | 1 | 2 | yes | — |
| 220 | C22 | C2×#537 | 1 | 1 | yes | characteristic |
| 221 | C22 | C2×#576 | 1 | 4 | yes | — |
| 222 | C4 | — | 2 | 16 | no | — |
| 223 | C4 | — | 4 | 4 | no | — |
| 224 | C4 | — | 8 | 8 | no | — |
| 225 | C4 | #2537 | 1 | 4 | yes | — |
| 226 | C4 | C2×D42 | 1 | 1 | yes | characteristic |
| 227 | C4 | C2×#547 | 1 | 2 | yes | — |
| 228 | C4 | C2×#555 | 1 | 1 | yes | characteristic |
| 229 | C2 | — | 8 | 16 | no | — |
| 230 | C2 | #28259 | 1 | 8 | yes | direct product factor |
| 231 | C2 | C2×#2537 | 1 | 1 | yes | characteristic |
| 232 | C2 | C2×#3041 | 1 | 4 | yes | — |
| 233 | C2 | C2×#3259 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#459 | C2 | semidirect |
| 2. C2×#2328 | C2 | semidirect |
| 3. #28259 | C2 | direct (N × H) |
| 4. C2×#2054 | C2 | semidirect |
| 5. C2×#1843 | C2 | semidirect |
| 6. C2×#1804 | C2 | semidirect |
| 11. C2×#459 | C22 | semidirect |
| 16. #2328 | C22 | semidirect |
| 21. #2054 | C22 | semidirect |
| 22. #1843 | C22 | semidirect |
| 23. #1804 | C22 | semidirect |
| 34. C2×#127 | D4 | semidirect |
| 89. #127 | C2×D4 | semidirect |
| 230. C2 | #28259 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#459 | 2 |
| C2×#2328 | 2 |
| #28259 | 2 |
| C2×#2054 | 2 |
| C2×#1843 | 2 |
| C2×#1804 | 2 |