Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1776 conjugacy classes of subgroups; only the 490 normal subgroups are shown.
This group has 1776 conjugacy classes of subgroups — too many to draw. Only its 490 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 | #32877 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 2 | C2×#2091 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | C2×#2070 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C2×#2069 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×#1785 | C2 | 1 | 3 | yes | maximal |
| 6 | C23×#41 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#430 | C22 | 1 | 12 | yes | — |
| 8 | C22×#101 | C22 | 1 | 6 | yes | — |
| 9 | C22×#100 | C22 | 1 | 6 | yes | — |
| 10 | C2×#407 | C22 | 1 | 6 | yes | — |
| 11 | #2091 | C22 | 1 | 32 | yes | complemented |
| 12 | #2070 | C22 | 1 | 32 | yes | complemented |
| 13 | #2069 | C22 | 1 | 32 | yes | complemented |
| 14 | #1785 | C22 | 1 | 24 | yes | — |
| 15 | C2×#338 | C22 | 1 | 4 | yes | — |
| 16 | C22×#41 | — | 4 | 6 | no | — |
| 17 | C22×#41 | — | 4 | 6 | no | — |
| 18 | C22×#41 | Q8 | 1 | 4 | yes | — |
| 19 | C22×#41 | C23 | 1 | 4 | yes | — |
| 20 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 21 | C23×D4 | Q8 | 1 | 2 | yes | — |
| 22 | C23×D4 | C23 | 1 | 1 | yes | characteristic |
| 23 | C24×C4 | Q8 | 1 | 2 | yes | — |
| 24 | C24×C4 | C23 | 1 | 1 | yes | characteristic |
| 25 | C22×M16 | C23 | 1 | 12 | yes | — |
| 26 | #430 | — | 4 | 12 | no | — |
| 27 | #430 | C23 | 1 | 48 | yes | — |
| 28 | C2×#108 | — | 4 | 12 | no | — |
| 29 | C2×#101 | — | 4 | 6 | no | — |
| 30 | C2×#101 | — | 4 | 6 | no | — |
| 31 | C2×#101 | C23 | 1 | 24 | yes | — |
| 32 | C2×#100 | — | 4 | 6 | no | — |
| 33 | C2×#100 | — | 4 | 6 | no | — |
| 34 | C2×#100 | C23 | 1 | 24 | yes | — |
| 35 | #407 | — | 4 | 6 | no | — |
| 36 | #407 | C23 | 1 | 24 | yes | — |
| 37 | C2×C22⋊C8 | — | 4 | 6 | no | — |
| 38 | #338 | C23 | 1 | 16 | yes | — |
| 39 | C25 | C2×Q8 | 1 | 2 | yes | — |
| 40 | C25 | C2×Q8 | 1 | 1 | yes | characteristic |
| 41 | C2×#41 | — | 4 | 24 | no | — |
| 42 | C2×#41 | — | 4 | 24 | no | — |
| 43 | C2×#41 | — | 4 | 48 | no | — |
| 44 | C2×#41 | — | 4 | 16 | no | — |
| 45 | C22×Q8 | — | 2 | 6 | no | — |
| 46 | C22×Q8 | — | 2 | 6 | no | — |
| 47 | C22×Q8 | C2×Q8 | 1 | 2 | yes | — |
| 48 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 49 | C22×D4 | — | 2 | 6 | no | — |
| 50 | C22×D4 | — | 2 | 6 | no | — |
| 51 | C22×D4 | — | 4 | 6 | no | — |
| 52 | C22×D4 | — | 4 | 6 | no | — |
| 53 | C22×D4 | C2×Q8 | 1 | 2 | yes | — |
| 54 | C22×D4 | C2×Q8 | 1 | 8 | yes | — |
| 55 | C22×D4 | C24 | 1 | 2 | yes | — |
| 56 | C23×C4 | — | 2 | 6 | no | — |
| 57 | C23×C4 | — | 2 | 6 | no | — |
| 58 | C23×C4 | — | 4 | 6 | no | — |
| 59 | C23×C4 | — | 4 | 6 | no | — |
| 60 | C23×C4 | C2×Q8 | 1 | 1 | yes | characteristic |
| 61 | C23×C4 | C2×Q8 | 1 | 2 | yes | — |
| 62 | C23×C4 | C2×Q8 | 1 | 2 | yes | — |
| 63 | C23×C4 | C2×Q8 | 1 | 8 | yes | — |
| 64 | C23×C4 | C24 | 1 | 2 | yes | — |
| 65 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 66 | C2×M16 | — | 2 | 12 | no | — |
| 67 | C2×M16 | — | 2 | 12 | no | — |
| 68 | C2×M16 | — | 2 | 12 | no | — |
| 69 | C2×M16 | — | 2 | 12 | no | — |
| 70 | C2×M16 | — | 2 | 12 | no | — |
| 71 | C2×M16 | C24 | 1 | 24 | yes | — |
| 72 | C22×C8 | — | 2 | 12 | no | — |
| 73 | #108 | — | 4 | 48 | no | — |
| 74 | #101 | — | 4 | 24 | no | — |
| 75 | #100 | — | 4 | 24 | no | — |
| 76 | C22⋊C8 | — | 4 | 24 | no | — |
| 77 | C24 | — | 2 | 3 | no | — |
| 78 | C24 | — | 2 | 3 | no | — |
| 79 | C24 | — | 2 | 6 | no | — |
| 80 | C24 | — | 2 | 6 | no | — |
| 81 | C24 | — | 8 | 2 | no | — |
| 82 | C24 | — | 8 | 4 | no | — |
| 83 | C24 | C22×Q8 | 1 | 4 | yes | — |
| 84 | C24 | C22×Q8 | 1 | 2 | yes | — |
| 85 | C24 | C22×Q8 | 1 | 1 | yes | characteristic |
| 86 | #41 | — | 4 | 64 | no | — |
| 87 | #41 | — | 4 | 64 | no | — |
| 88 | C2×Q8 | — | 2 | 12 | no | — |
| 89 | C2×Q8 | — | 2 | 12 | no | — |
| 90 | C2×Q8 | — | 4 | 12 | no | — |
| 91 | C2×Q8 | — | 4 | 4 | no | — |
| 92 | C2×D4 | — | 2 | 12 | no | — |
| 93 | C2×D4 | — | 2 | 12 | no | — |
| 94 | C2×D4 | — | 4 | 24 | no | — |
| 95 | C2×D4 | — | 4 | 24 | no | — |
| 96 | C2×D4 | — | 4 | 8 | no | — |
| 97 | C2×D4 | — | 4 | 12 | no | — |
| 98 | C2×D4 | — | 4 | 4 | no | — |
| 99 | C22×C4 | — | 2 | 3 | no | — |
| 100 | C22×C4 | — | 2 | 3 | no | — |
| 101 | C22×C4 | — | 2 | 6 | no | — |
| 102 | C22×C4 | — | 2 | 6 | no | — |
| 103 | C22×C4 | — | 2 | 12 | no | — |
| 104 | C22×C4 | — | 2 | 12 | no | — |
| 105 | C22×C4 | — | 4 | 24 | no | — |
| 106 | C22×C4 | — | 4 | 24 | no | — |
| 107 | C22×C4 | — | 4 | 8 | no | — |
| 108 | C22×C4 | — | 4 | 12 | no | — |
| 109 | C22×C4 | — | 4 | 4 | no | — |
| 110 | C22×C4 | C22×Q8 | 1 | 2 | yes | — |
| 111 | C22×C4 | C22×Q8 | 1 | 4 | yes | — |
| 112 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 113 | C22×C4 | H(5,𝔽2) | 1 | 6 | yes | — |
| 114 | C22×C4 | H(5,𝔽2) | 1 | 6 | yes | — |
| 115 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 116 | M16 | — | 2 | 24 | no | — |
| 117 | M16 | — | 2 | 24 | no | — |
| 118 | M16 | — | 4 | 12 | no | — |
| 119 | M16 | — | 4 | 12 | no | — |
| 120 | C2×C8 | — | 2 | 24 | no | — |
| 121 | C2×C8 | — | 4 | 12 | no | — |
| 122 | C2×C8 | — | 4 | 12 | no | — |
| 123 | C23 | — | 2 | 6 | no | — |
| 124 | C23 | — | 2 | 12 | no | — |
| 125 | C23 | — | 4 | 2 | no | — |
| 126 | C23 | — | 4 | 12 | no | — |
| 127 | C23 | — | 4 | 3 | no | — |
| 128 | C23 | — | 4 | 6 | no | — |
| 129 | C23 | — | 4 | 1 | no | — |
| 130 | C23 | — | 4 | 2 | no | — |
| 131 | C23 | — | 8 | 12 | no | — |
| 132 | C23 | — | 8 | 2 | no | — |
| 133 | C23 | — | 8 | 6 | no | — |
| 134 | C23 | — | 8 | 4 | no | — |
| 135 | C23 | — | 8 | 12 | no | — |
| 136 | C23 | #543 | 1 | 3 | yes | — |
| 137 | C23 | #543 | 1 | 3 | yes | — |
| 138 | C23 | C23×Q8 | 1 | 1 | yes | characteristic |
| 139 | Q8 | — | 4 | 8 | no | — |
| 140 | Q8 | — | 4 | 8 | no | — |
| 141 | D4 | — | 4 | 32 | no | — |
| 142 | D4 | — | 4 | 8 | no | — |
| 143 | D4 | — | 4 | 8 | no | — |
| 144 | C2×C4 | — | 2 | 6 | no | — |
| 145 | C2×C4 | — | 2 | 12 | no | — |
| 146 | C2×C4 | — | 4 | 32 | no | — |
| 147 | C2×C4 | — | 4 | 3 | no | — |
| 148 | C2×C4 | — | 4 | 6 | no | — |
| 149 | C2×C4 | — | 4 | 1 | no | — |
| 150 | C2×C4 | — | 4 | 2 | no | — |
| 151 | C2×C4 | — | 4 | 8 | no | — |
| 152 | C2×C4 | — | 4 | 8 | no | — |
| 153 | C2×C4 | #543 | 1 | 6 | yes | — |
| 154 | C2×C4 | #560 | 1 | 12 | yes | — |
| 155 | C2×C4 | #560 | 1 | 4 | yes | — |
| 156 | C2×C4 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 157 | C8 | — | 4 | 24 | no | — |
| 158 | C22 | — | 2 | 6 | no | — |
| 159 | C22 | — | 4 | 3 | no | — |
| 160 | C22 | — | 4 | 1 | no | — |
| 161 | C22 | — | 4 | 12 | no | — |
| 162 | C22 | — | 4 | 2 | no | — |
| 163 | C22 | — | 4 | 4 | no | — |
| 164 | C22 | — | 8 | 12 | no | — |
| 165 | C22 | — | 8 | 6 | no | — |
| 166 | C22 | — | 8 | 12 | no | — |
| 167 | C22 | — | 8 | 2 | no | — |
| 168 | C22 | — | 8 | 4 | no | — |
| 169 | C22 | #2842 | 1 | 3 | yes | — |
| 170 | C22 | #2842 | 1 | 1 | yes | characteristic |
| 171 | C22 | C2×#543 | 1 | 3 | yes | — |
| 172 | C4 | — | 4 | 2 | no | — |
| 173 | C4 | — | 4 | 4 | no | — |
| 174 | C4 | #2842 | 1 | 4 | yes | — |
| 175 | C4 | C2×#560 | 1 | 4 | yes | — |
| 176 | C2 | — | 2 | 3 | no | — |
| 177 | C2 | — | 2 | 3 | no | — |
| 178 | C2 | — | 8 | 2 | no | — |
| 179 | C2 | — | 8 | 4 | no | — |
| 180 | C2 | #32877 | 1 | 2 | yes | direct product factor |
| 181 | C2 | C2×#2842 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #32877 | C2 | direct (N × H) |
| 2. C2×#2091 | C2 | semidirect |
| 3. C2×#2070 | C2 | semidirect |
| 4. C2×#2069 | C2 | semidirect |
| 11. #2091 | C22 | semidirect |
| 12. #2070 | C22 | semidirect |
| 13. #2069 | C22 | semidirect |
| 180. C2 | #32877 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #32877 | 2 |
| C2×#2091 | 2 |
| C2×#2070 | 2 |
| C2×#2069 | 2 |
| C2×#1785 | 2 |