Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1103 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1103 conjugacy classes of subgroups — too many to draw. Only its 192 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 | #60646 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33360 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #13585 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #13583 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#577 | — | 2 | 2 | no | — |
| 6 | #3499 | C22 | 1 | 1 | yes | characteristic |
| 7 | #3498 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | #3154 | — | 2 | 6 | no | — |
| 9 | #3152 | C4 | 1 | 3 | yes | complemented |
| 10 | #3151 | C22 | 1 | 3 | yes | — |
| 11 | #3147 | C22 | 1 | 3 | yes | — |
| 12 | C2×#492 | C4 | 1 | 3 | yes | complemented |
| 13 | #2884 | — | 2 | 2 | no | — |
| 14 | C4×#131 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #2129 | — | 2 | 6 | no | — |
| 16 | #2126 | C22 | 1 | 12 | yes | complemented |
| 17 | #1409 | C22 | 1 | 4 | yes | complemented |
| 18 | #1408 | C22 | 1 | 12 | yes | complemented |
| 19 | C2×2−1+4 | D4 | 1 | 2 | yes | — |
| 20 | C2×H(5,𝔽2) | — | 2 | 1 | no | — |
| 21 | #577 | — | 4 | 16 | no | — |
| 22 | C2×#137 | — | 2 | 6 | no | — |
| 23 | C2×AGL(1,ℤ/8) | — | 2 | 6 | no | — |
| 24 | C2×#135 | — | 2 | 6 | no | — |
| 25 | C22×SD16 | — | 2 | 6 | no | — |
| 26 | C2×#131 | — | 2 | 1 | no | — |
| 27 | #550 | — | 4 | 1 | no | — |
| 28 | D4×Q8 | — | 4 | 1 | no | — |
| 29 | #547 | — | 4 | 3 | no | — |
| 30 | D42 | — | 4 | 3 | no | — |
| 31 | #536 | D4 | 1 | 6 | yes | — |
| 32 | #535 | — | 2 | 3 | no | — |
| 33 | #532 | C2×C4 | 1 | 3 | yes | — |
| 34 | #532 | C23 | 1 | 3 | yes | — |
| 35 | C2×C4⋊Q8 | C2×C4 | 1 | 3 | yes | — |
| 36 | C2×#127 | C2×C4 | 1 | 3 | yes | — |
| 37 | #519 | — | 4 | 2 | no | — |
| 38 | C4×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | #497 | — | 2 | 6 | no | — |
| 40 | #496 | — | 2 | 6 | no | — |
| 41 | #495 | — | 2 | 6 | no | — |
| 42 | #494 | C2×C4 | 1 | 3 | yes | — |
| 43 | #493 | C23 | 1 | 3 | yes | — |
| 44 | #492 | — | 2 | 6 | no | — |
| 45 | #492 | C2×C4 | 1 | 3 | yes | — |
| 46 | #492 | C2×C4 | 1 | 6 | yes | — |
| 47 | #492 | C23 | 1 | 1 | yes | characteristic |
| 48 | #410 | — | 2 | 12 | no | — |
| 49 | #410 | — | 4 | 6 | no | — |
| 50 | #405 | — | 2 | 3 | no | — |
| 51 | C4×M16 | C2×C4 | 1 | 3 | yes | — |
| 52 | C2×C4×C8 | C2×C4 | 1 | 3 | yes | — |
| 53 | #354 | — | 2 | 12 | no | — |
| 54 | #353 | C23 | 1 | 8 | yes | — |
| 55 | 2−1+4 | — | 4 | 8 | no | — |
| 56 | H(5,𝔽2) | — | 4 | 8 | no | — |
| 57 | C2×#41 | — | 2 | 6 | no | — |
| 58 | C2×#41 | — | 2 | 3 | no | — |
| 59 | C2×#41 | — | 4 | 1 | no | — |
| 60 | C2×#41 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C2×#41 | C2×D4 | 1 | 6 | yes | — |
| 62 | C22×Q8 | — | 2 | 2 | no | — |
| 63 | C22×Q8 | C22⋊C4 | 1 | 6 | yes | — |
| 64 | C22×D4 | — | 2 | 3 | no | — |
| 65 | C22×D4 | — | 4 | 3 | no | — |
| 66 | C22×D4 | — | 4 | 3 | no | — |
| 67 | #137 | — | 4 | 24 | no | — |
| 68 | AGL(1,ℤ/8) | — | 4 | 24 | no | — |
| 69 | #135 | — | 4 | 24 | no | — |
| 70 | C2×Q16 | — | 2 | 6 | no | — |
| 71 | C2×SD16 | — | 2 | 6 | no | — |
| 72 | C2×SD16 | — | 2 | 12 | no | — |
| 73 | C2×SD16 | — | 2 | 2 | no | — |
| 74 | C2×SD16 | — | 4 | 24 | no | — |
| 75 | C2×D8 | — | 2 | 6 | no | — |
| 76 | #131 | — | 2 | 8 | no | — |
| 77 | C2×M16 | — | 2 | 3 | no | — |
| 78 | C22×C8 | — | 2 | 3 | no | — |
| 79 | C4⋊Q8 | — | 4 | 3 | no | — |
| 80 | C4⋊Q8 | C22⋊C4 | 1 | 6 | yes | — |
| 81 | C4⋊Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C4⋊Q8 | C22×C4 | 1 | 3 | yes | — |
| 83 | #127 | — | 2 | 3 | no | — |
| 84 | #127 | — | 4 | 3 | no | — |
| 85 | #127 | C22×C4 | 1 | 3 | yes | — |
| 86 | #127 | C24 | 1 | 1 | yes | characteristic |
| 87 | #124 | — | 2 | 3 | no | — |
| 88 | #124 | C2×D4 | 1 | 6 | yes | — |
| 89 | #123 | — | 4 | 6 | no | — |
| 90 | C22⋊Q8 | — | 4 | 6 | no | — |
| 91 | #121 | — | 4 | 6 | no | — |
| 92 | #121 | — | 4 | 3 | no | — |
| 93 | #121 | — | 4 | 3 | no | — |
| 94 | #120 | — | 4 | 6 | no | — |
| 95 | C4×Q8 | — | 2 | 1 | no | — |
| 96 | C4×Q8 | — | 8 | 8 | no | — |
| 97 | C4×D4 | — | 2 | 3 | no | — |
| 98 | C4×D4 | — | 4 | 3 | no | — |
| 99 | C4×D4 | — | 4 | 3 | no | — |
| 100 | #117 | — | 4 | 12 | no | — |
| 101 | #117 | C22⋊C4 | 1 | 3 | yes | — |
| 102 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 103 | C2×C42 | C22×C4 | 1 | 3 | yes | — |
| 104 | #99 | — | 2 | 8 | no | — |
| 105 | #99 | — | 4 | 12 | no | — |
| 106 | #97 | — | 2 | 3 | no | — |
| 107 | C4×C8 | — | 2 | 3 | no | — |
| 108 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 109 | C4×C8 | C22×C4 | 1 | 3 | yes | — |
| 110 | C24 | — | 4 | 3 | no | — |
| 111 | #41 | — | 2 | 4 | no | — |
| 112 | #41 | — | 4 | 12 | no | — |
| 113 | #41 | — | 4 | 12 | no | — |
| 114 | #41 | — | 8 | 12 | no | — |
| 115 | #41 | — | 8 | 4 | no | — |
| 116 | C2×Q8 | — | 2 | 6 | no | — |
| 117 | C2×Q8 | — | 2 | 1 | no | — |
| 118 | C2×Q8 | — | 4 | 12 | no | — |
| 119 | C2×Q8 | — | 4 | 8 | no | — |
| 120 | C2×Q8 | #99 | 1 | 1 | yes | characteristic |
| 121 | C2×Q8 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 122 | C2×D4 | — | 2 | 6 | no | — |
| 123 | C2×D4 | — | 2 | 3 | no | — |
| 124 | C2×D4 | — | 4 | 12 | no | — |
| 125 | C2×D4 | — | 4 | 1 | no | — |
| 126 | C2×D4 | — | 4 | 3 | no | — |
| 127 | C2×D4 | — | 4 | 6 | no | — |
| 128 | C2×D4 | — | 4 | 3 | no | — |
| 129 | C2×D4 | — | 4 | 3 | no | — |
| 130 | C2×D4 | — | 8 | 12 | no | — |
| 131 | C2×D4 | — | 16 | 6 | no | — |
| 132 | C2×D4 | #99 | 1 | 3 | yes | — |
| 133 | C2×D4 | C22×D4 | 1 | 2 | yes | — |
| 134 | C22×C4 | — | 2 | 3 | no | — |
| 135 | C22×C4 | — | 2 | 6 | no | — |
| 136 | C22×C4 | — | 2 | 6 | no | — |
| 137 | C22×C4 | — | 4 | 3 | no | — |
| 138 | C22×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 139 | Q16 | — | 4 | 12 | no | — |
| 140 | SD16 | — | 4 | 12 | no | — |
| 141 | SD16 | — | 4 | 24 | no | — |
| 142 | SD16 | — | 4 | 4 | no | — |
| 143 | D8 | — | 4 | 12 | no | — |
| 144 | M16 | — | 2 | 12 | no | — |
| 145 | C2×C8 | — | 2 | 12 | no | — |
| 146 | C2×C8 | — | 2 | 1 | no | — |
| 147 | C2×C8 | — | 2 | 3 | no | — |
| 148 | C4⋊C4 | — | 2 | 3 | no | — |
| 149 | C4⋊C4 | — | 4 | 3 | no | — |
| 150 | C4⋊C4 | — | 4 | 6 | no | — |
| 151 | C4⋊C4 | — | 4 | 1 | no | — |
| 152 | C4⋊C4 | — | 8 | 12 | no | — |
| 153 | C22⋊C4 | — | 2 | 3 | no | — |
| 154 | C22⋊C4 | — | 4 | 6 | no | — |
| 155 | C22⋊C4 | — | 4 | 6 | no | — |
| 156 | C22⋊C4 | — | 4 | 8 | no | — |
| 157 | C42 | — | 2 | 3 | no | — |
| 158 | C42 | — | 8 | 12 | no | — |
| 159 | C42 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 160 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 161 | C23 | — | 2 | 2 | no | — |
| 162 | C23 | — | 4 | 3 | no | — |
| 163 | C23 | — | 8 | 6 | no | — |
| 164 | C23 | — | 16 | 6 | no | — |
| 165 | C23 | C2×#99 | 1 | 3 | yes | — |
| 166 | Q8 | — | 4 | 6 | no | — |
| 167 | Q8 | — | 4 | 2 | no | — |
| 168 | Q8 | — | 4 | 12 | no | — |
| 169 | Q8 | #353 | 1 | 4 | yes | — |
| 170 | D4 | — | 2 | 6 | no | — |
| 171 | D4 | — | 4 | 2 | no | — |
| 172 | D4 | — | 4 | 6 | no | — |
| 173 | D4 | — | 8 | 2 | no | — |
| 174 | D4 | — | 8 | 6 | no | — |
| 175 | D4 | — | 8 | 6 | no | — |
| 176 | D4 | — | 8 | 6 | no | — |
| 177 | D4 | — | 8 | 6 | no | — |
| 178 | D4 | — | 16 | 6 | no | — |
| 179 | C2×C4 | — | 2 | 1 | no | — |
| 180 | C2×C4 | — | 2 | 6 | no | — |
| 181 | C2×C4 | — | 2 | 3 | no | — |
| 182 | C2×C4 | — | 2 | 6 | no | — |
| 183 | C2×C4 | — | 4 | 6 | no | — |
| 184 | C2×C4 | — | 4 | 12 | no | — |
| 185 | C2×C4 | — | 4 | 1 | no | — |
| 186 | C2×C4 | — | 8 | 6 | no | — |
| 187 | C2×C4 | — | 8 | 6 | no | — |
| 188 | C2×C4 | — | 8 | 8 | no | — |
| 189 | C2×C4 | C2×#99 | 1 | 3 | yes | — |
| 190 | C2×C4 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 191 | C8 | — | 2 | 2 | no | — |
| 192 | C8 | — | 2 | 6 | no | — |
| 193 | C22 | — | 2 | 3 | no | — |
| 194 | C22 | — | 4 | 3 | no | — |
| 195 | C22 | — | 4 | 2 | no | — |
| 196 | C22 | — | 8 | 2 | no | — |
| 197 | C22 | — | 8 | 6 | no | — |
| 198 | C22 | — | 8 | 3 | no | — |
| 199 | C22 | — | 32 | 6 | no | — |
| 200 | C22 | C22×#99 | 1 | 1 | yes | characteristic |
| 201 | C4 | — | 2 | 2 | no | — |
| 202 | C4 | — | 2 | 1 | no | — |
| 203 | C4 | — | 4 | 3 | no | — |
| 204 | C4 | — | 4 | 6 | no | — |
| 205 | C4 | — | 8 | 1 | no | — |
| 206 | C4 | — | 16 | 8 | no | — |
| 207 | C4 | C2×#353 | 1 | 6 | yes | — |
| 208 | C2 | — | 2 | 1 | no | — |
| 209 | C2 | — | 4 | 3 | no | — |
| 210 | C2 | — | 8 | 2 | no | — |
| 211 | C2 | — | 16 | 3 | no | — |
| 212 | C2 | C22×#353 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#353 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| prime radical | #60646 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×#41 ◁ #3499 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33360 | C2 | semidirect |
| 3. #13585 | C2 | semidirect |
| 4. #13583 | C2 | semidirect |
| 7. #3498 | C4 | semidirect |
| 9. #3152 | C4 | semidirect |
| 12. C2×#492 | C4 | semidirect |
| 14. C4×#131 | C4 | semidirect |
| 16. #2126 | C22 | semidirect |
| 17. #1409 | C22 | semidirect |
| 18. #1408 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60646 | 2 |
| #33360 | 2 |
| #13585 | 2 |
| #13583 | 2 |