Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
16913 conjugacy classes of subgroups; only the 1808 normal subgroups are shown.
This group has 16913 conjugacy classes of subgroups — too many to draw. Only its 1808 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 | #63103 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #60076 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #60065 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #20612 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | C2×#1791 | C2 | 1 | 12 | yes | maximal, complemented |
| 6 | 2−1+6 | — | 2 | 12 | no | — |
| 7 | 2−1+6 | C4 | 1 | 4 | yes | — |
| 8 | H(7,𝔽2) | — | 2 | 12 | no | — |
| 9 | H(7,𝔽2) | C4 | 1 | 12 | yes | — |
| 10 | C2×#585 | — | 2 | 24 | no | — |
| 11 | C2×#585 | C22 | 1 | 12 | yes | — |
| 12 | C2×#585 | C22 | 1 | 3 | yes | — |
| 13 | C2×#568 | C22 | 1 | 3 | yes | complemented |
| 14 | C22×#131 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #2889 | — | 2 | 12 | no | — |
| 16 | #2888 | — | 2 | 12 | no | — |
| 17 | C2×#413 | C22 | 1 | 36 | yes | complemented |
| 18 | #2880 | — | 2 | 24 | no | — |
| 19 | #2879 | C22 | 1 | 18 | yes | complemented |
| 20 | C2×#408 | C22 | 1 | 18 | yes | complemented |
| 21 | #1791 | C22 | 1 | 16 | yes | complemented |
| 22 | #1791 | C22 | 1 | 48 | yes | complemented |
| 23 | #585 | — | 2 | 64 | no | — |
| 24 | #585 | — | 2 | 192 | no | — |
| 25 | #585 | — | 2 | 48 | no | — |
| 26 | #585 | — | 2 | 144 | no | — |
| 27 | #585 | — | 2 | 72 | no | — |
| 28 | #585 | — | 2 | 24 | no | — |
| 29 | #585 | — | 2 | 72 | no | — |
| 30 | #585 | — | 2 | 24 | no | — |
| 31 | #585 | C2×C4 | 1 | 48 | yes | — |
| 32 | #585 | C23 | 1 | 16 | yes | — |
| 33 | C2×2−1+4 | — | 2 | 96 | no | — |
| 34 | C2×2−1+4 | — | 2 | 48 | no | — |
| 35 | C2×2−1+4 | — | 2 | 36 | no | — |
| 36 | C2×2−1+4 | C2×C4 | 1 | 6 | yes | — |
| 37 | C2×2−1+4 | D4 | 1 | 12 | yes | — |
| 38 | C2×H(5,𝔽2) | — | 2 | 96 | no | — |
| 39 | C2×H(5,𝔽2) | — | 2 | 144 | no | — |
| 40 | C2×H(5,𝔽2) | — | 2 | 36 | no | — |
| 41 | C2×H(5,𝔽2) | C2×C4 | 1 | 48 | yes | — |
| 42 | C2×H(5,𝔽2) | C2×C4 | 1 | 18 | yes | — |
| 43 | C2×H(5,𝔽2) | D4 | 1 | 12 | yes | — |
| 44 | C22×#41 | — | 2 | 64 | no | — |
| 45 | C22×#41 | — | 2 | 72 | no | — |
| 46 | C22×#41 | D4 | 1 | 24 | yes | — |
| 47 | C22×#41 | C23 | 1 | 18 | yes | — |
| 48 | C22×#41 | C23 | 1 | 1 | yes | characteristic |
| 49 | #568 | — | 2 | 24 | no | — |
| 50 | C2×#131 | — | 2 | 12 | no | — |
| 51 | C2×#131 | — | 2 | 12 | no | — |
| 52 | C22×M16 | C23 | 1 | 18 | yes | complemented |
| 53 | C22×M16 | C23 | 1 | 3 | yes | complemented |
| 54 | C23×C8 | C23 | 1 | 3 | yes | complemented |
| 55 | #413 | — | 2 | 72 | no | — |
| 56 | #413 | C23 | 1 | 48 | yes | complemented |
| 57 | C2×#101 | — | 2 | 36 | no | — |
| 58 | C2×#100 | — | 2 | 36 | no | — |
| 59 | #408 | — | 2 | 36 | no | — |
| 60 | #408 | — | 2 | 36 | no | — |
| 61 | #408 | C23 | 1 | 12 | yes | complemented |
| 62 | #408 | C23 | 1 | 36 | yes | complemented |
| 63 | #407 | — | 2 | 36 | no | — |
| 64 | C2×C22⋊C8 | — | 2 | 36 | no | — |
| 65 | 2−1+4 | — | 2 | 192 | no | — |
| 66 | 2−1+4 | — | 2 | 288 | no | — |
| 67 | 2−1+4 | — | 2 | 192 | no | — |
| 68 | 2−1+4 | — | 2 | 96 | no | — |
| 69 | 2−1+4 | — | 2 | 144 | no | — |
| 70 | 2−1+4 | — | 2 | 48 | no | — |
| 71 | 2−1+4 | — | 2 | 48 | no | — |
| 72 | H(5,𝔽2) | — | 2 | 64 | no | — |
| 73 | H(5,𝔽2) | — | 2 | 288 | no | — |
| 74 | H(5,𝔽2) | — | 2 | 576 | no | — |
| 75 | H(5,𝔽2) | — | 2 | 96 | no | — |
| 76 | H(5,𝔽2) | — | 2 | 144 | no | — |
| 77 | H(5,𝔽2) | — | 2 | 288 | no | — |
| 78 | H(5,𝔽2) | — | 2 | 144 | no | — |
| 79 | H(5,𝔽2) | — | 2 | 48 | no | — |
| 80 | H(5,𝔽2) | C22×C4 | 1 | 64 | yes | — |
| 81 | C2×#41 | — | 2 | 288 | no | — |
| 82 | C2×#41 | — | 2 | 48 | no | — |
| 83 | C2×#41 | — | 2 | 192 | no | — |
| 84 | C2×#41 | — | 2 | 576 | no | — |
| 85 | C2×#41 | — | 2 | 288 | no | — |
| 86 | C2×#41 | — | 2 | 144 | no | — |
| 87 | C2×#41 | — | 2 | 96 | no | — |
| 88 | C2×#41 | — | 2 | 96 | no | — |
| 89 | C2×#41 | — | 2 | 288 | no | — |
| 90 | C2×#41 | — | 2 | 144 | no | — |
| 91 | C2×#41 | — | 2 | 96 | no | — |
| 92 | C2×#41 | — | 2 | 72 | no | — |
| 93 | C2×#41 | — | 2 | 72 | no | — |
| 94 | C2×#41 | C22⋊C4 | 1 | 72 | yes | — |
| 95 | C2×#41 | C22⋊C4 | 1 | 24 | yes | — |
| 96 | C2×#41 | C22×C4 | 1 | 36 | yes | — |
| 97 | C2×#41 | C2×D4 | 1 | 36 | yes | — |
| 98 | C2×#41 | C2×D4 | 1 | 36 | yes | — |
| 99 | C2×#41 | C2×D4 | 1 | 12 | yes | — |
| 100 | C2×#41 | C2×D4 | 1 | 12 | yes | — |
| 101 | C2×#41 | C24 | 1 | 12 | yes | — |
| 102 | C22×Q8 | — | 2 | 64 | no | — |
| 103 | C22×Q8 | — | 2 | 72 | no | — |
| 104 | C22×Q8 | C22⋊C4 | 1 | 24 | yes | — |
| 105 | C22×Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 106 | C22×Q8 | C2×D4 | 1 | 18 | yes | — |
| 107 | C22×D4 | — | 2 | 192 | no | — |
| 108 | C22×D4 | — | 2 | 72 | no | — |
| 109 | C22×D4 | — | 2 | 144 | no | — |
| 110 | C22×D4 | C22⋊C4 | 1 | 72 | yes | — |
| 111 | C22×D4 | C22×C4 | 1 | 36 | yes | — |
| 112 | C22×D4 | C22×C4 | 1 | 3 | yes | — |
| 113 | C22×D4 | C2×D4 | 1 | 18 | yes | — |
| 114 | C23×C4 | — | 2 | 48 | no | — |
| 115 | C23×C4 | C2×D4 | 1 | 36 | yes | — |
| 116 | C23×C4 | C24 | 1 | 3 | yes | — |
| 117 | #131 | — | 4 | 16 | no | — |
| 118 | #131 | — | 4 | 48 | no | — |
| 119 | C2×M16 | — | 2 | 18 | no | — |
| 120 | C2×M16 | — | 2 | 18 | no | — |
| 121 | C2×M16 | — | 2 | 72 | no | — |
| 122 | C2×M16 | C24 | 1 | 12 | yes | — |
| 123 | C22×C8 | — | 2 | 18 | no | — |
| 124 | C22×C8 | — | 2 | 18 | no | — |
| 125 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 126 | C22×C8 | C24 | 1 | 3 | yes | — |
| 127 | #101 | — | 2 | 48 | no | — |
| 128 | #100 | — | 2 | 48 | no | — |
| 129 | C22⋊C8 | — | 2 | 24 | no | — |
| 130 | C22⋊C8 | — | 2 | 72 | no | — |
| 131 | C24 | — | 2 | 48 | no | — |
| 132 | C24 | C2×C22⋊C4 | 1 | 36 | yes | — |
| 133 | C24 | C23×C4 | 1 | 3 | yes | — |
| 134 | #41 | — | 2 | 64 | no | — |
| 135 | #41 | — | 2 | 576 | no | — |
| 136 | #41 | — | 2 | 192 | no | — |
| 137 | #41 | — | 2 | 576 | no | — |
| 138 | #41 | — | 2 | 192 | no | — |
| 139 | #41 | — | 2 | 96 | no | — |
| 140 | #41 | — | 2 | 288 | no | — |
| 141 | #41 | — | 2 | 192 | no | — |
| 142 | #41 | — | 2 | 144 | no | — |
| 143 | #41 | — | 2 | 72 | no | — |
| 144 | #41 | — | 2 | 72 | no | — |
| 145 | #41 | — | 2 | 144 | no | — |
| 146 | #41 | — | 2 | 24 | no | — |
| 147 | #41 | — | 2 | 24 | no | — |
| 148 | #41 | #120 | 1 | 16 | yes | — |
| 149 | #41 | #120 | 1 | 48 | yes | — |
| 150 | C2×Q8 | — | 2 | 96 | no | — |
| 151 | C2×Q8 | — | 2 | 288 | no | — |
| 152 | C2×Q8 | — | 2 | 48 | no | — |
| 153 | C2×Q8 | — | 2 | 96 | no | — |
| 154 | C2×Q8 | — | 2 | 72 | no | — |
| 155 | C2×Q8 | C2×C22⋊C4 | 1 | 36 | yes | — |
| 156 | C2×Q8 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 157 | C2×Q8 | C22×D4 | 1 | 12 | yes | — |
| 158 | C2×D4 | — | 2 | 576 | no | — |
| 159 | C2×D4 | — | 2 | 144 | no | — |
| 160 | C2×D4 | — | 2 | 96 | no | — |
| 161 | C2×D4 | — | 2 | 288 | no | — |
| 162 | C2×D4 | — | 2 | 192 | no | — |
| 163 | C2×D4 | — | 2 | 144 | no | — |
| 164 | C2×D4 | — | 2 | 288 | no | — |
| 165 | C2×D4 | — | 2 | 72 | no | — |
| 166 | C2×D4 | C2×C22⋊C4 | 1 | 72 | yes | — |
| 167 | C2×D4 | C2×C22⋊C4 | 1 | 36 | yes | — |
| 168 | C2×D4 | C2×C22⋊C4 | 1 | 36 | yes | — |
| 169 | C2×D4 | C23×C4 | 1 | 24 | yes | — |
| 170 | C2×D4 | C22×D4 | 1 | 12 | yes | — |
| 171 | C22×C4 | — | 2 | 64 | no | — |
| 172 | C22×C4 | — | 2 | 144 | no | — |
| 173 | C22×C4 | — | 2 | 192 | no | — |
| 174 | C22×C4 | — | 2 | 72 | no | — |
| 175 | C22×C4 | — | 2 | 72 | no | — |
| 176 | C22×C4 | C2×C22⋊C4 | 1 | 72 | yes | — |
| 177 | C22×C4 | C2×C22⋊C4 | 1 | 36 | yes | — |
| 178 | C22×C4 | C23×C4 | 1 | 3 | yes | — |
| 179 | C22×C4 | C22×D4 | 1 | 24 | yes | — |
| 180 | C22×C4 | C22×D4 | 1 | 18 | yes | — |
| 181 | C22×C4 | C22×D4 | 1 | 18 | yes | — |
| 182 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 183 | M16 | — | 2 | 24 | no | — |
| 184 | M16 | — | 4 | 36 | no | — |
| 185 | M16 | — | 4 | 12 | no | — |
| 186 | C2×C8 | — | 2 | 3 | no | — |
| 187 | C2×C8 | — | 2 | 12 | no | — |
| 188 | C2×C8 | — | 2 | 9 | no | — |
| 189 | C2×C8 | — | 4 | 12 | no | — |
| 190 | C2×C8 | — | 4 | 36 | no | — |
| 191 | C23 | — | 2 | 64 | no | — |
| 192 | C23 | — | 2 | 72 | no | — |
| 193 | C23 | — | 8 | 3 | no | — |
| 194 | C23 | — | 8 | 36 | no | — |
| 195 | C23 | — | 16 | 48 | no | — |
| 196 | C23 | C22×C22⋊C4 | 1 | 24 | yes | — |
| 197 | C23 | C22×C22⋊C4 | 1 | 18 | yes | — |
| 198 | C23 | C24×C4 | 1 | 1 | yes | characteristic |
| 199 | Q8 | — | 2 | 64 | no | — |
| 200 | Q8 | — | 2 | 24 | no | — |
| 201 | Q8 | — | 2 | 72 | no | — |
| 202 | Q8 | #379 | 1 | 16 | yes | — |
| 203 | D4 | — | 2 | 192 | no | — |
| 204 | D4 | — | 2 | 48 | no | — |
| 205 | D4 | — | 2 | 144 | no | — |
| 206 | D4 | — | 2 | 24 | no | — |
| 207 | D4 | — | 2 | 72 | no | — |
| 208 | D4 | #379 | 1 | 48 | yes | — |
| 209 | C2×C4 | — | 2 | 192 | no | — |
| 210 | C2×C4 | — | 2 | 72 | no | — |
| 211 | C2×C4 | — | 2 | 24 | no | — |
| 212 | C2×C4 | — | 2 | 24 | no | — |
| 213 | C2×C4 | — | 2 | 144 | no | — |
| 214 | C2×C4 | C22×C22⋊C4 | 1 | 24 | yes | — |
| 215 | C2×C4 | C22×C22⋊C4 | 1 | 18 | yes | — |
| 216 | C2×C4 | C2×#120 | 1 | 36 | yes | — |
| 217 | C2×C4 | C2×#120 | 1 | 12 | yes | — |
| 218 | C2×C4 | C23×D4 | 1 | 3 | yes | — |
| 219 | C2×C4 | C23×D4 | 1 | 3 | yes | — |
| 220 | C8 | — | 4 | 12 | no | — |
| 221 | C8 | — | 4 | 1 | no | — |
| 222 | C8 | — | 4 | 3 | no | — |
| 223 | C22 | — | 2 | 24 | no | — |
| 224 | C22 | — | 4 | 18 | no | — |
| 225 | C22 | — | 4 | 1 | no | — |
| 226 | C22 | — | 4 | 24 | no | — |
| 227 | C22 | — | 8 | 72 | no | — |
| 228 | C22 | — | 8 | 64 | no | — |
| 229 | C22 | C2×#379 | 1 | 12 | yes | — |
| 230 | C22 | C23×C22⋊C4 | 1 | 3 | yes | — |
| 231 | C4 | — | 2 | 24 | no | — |
| 232 | C4 | C2×#379 | 1 | 12 | yes | — |
| 233 | C4 | C22×#120 | 1 | 3 | yes | — |
| 234 | C4 | C22×#120 | 1 | 1 | yes | characteristic |
| 235 | C2 | — | 2 | 3 | no | — |
| 236 | C2 | — | 2 | 12 | no | — |
| 237 | C2 | — | 4 | 24 | no | — |
| 238 | C2 | C22×#379 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#120 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C22×#379 |
| prime radical | #63103 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #60076 | C2 | semidirect |
| 3. #60065 | C2 | semidirect |
| 4. #20612 | C2 | semidirect |
| 5. C2×#1791 | C2 | semidirect |
| 13. C2×#568 | C22 | semidirect |
| 14. C22×#131 | C22 | semidirect |
| 17. C2×#413 | C22 | semidirect |
| 19. #2879 | C22 | semidirect |
| 20. C2×#408 | C22 | semidirect |
| 21. #1791 | C22 | semidirect |
| 22. #1791 | C22 | semidirect |
| 52. C22×M16 | C23 | semidirect |
| 53. C22×M16 | C23 | semidirect |
| 54. C23×C8 | C23 | semidirect |
| 56. #413 | C23 | semidirect |
| 61. #408 | C23 | semidirect |
| 62. #408 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63103 | 2 |
| #60076 | 2 |
| #60065 | 2 |
| #20612 | 2 |
| C2×#1791 | 2 |