Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4486 conjugacy classes of subgroups; only the 618 normal subgroups are shown.
This group has 4486 conjugacy classes of subgroups — too many to draw. Only its 618 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×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×#496 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×#1655 | C2 | 1 | 24 | yes | maximal, direct product factor |
| 4 | C22×#329 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C23×SD16 | — | 2 | 3 | no | — |
| 6 | C23×D8 | — | 2 | 3 | no | — |
| 7 | C2×H(3,𝔽4) | — | 2 | 18 | no | — |
| 8 | C2×H(3,𝔽4) | C22 | 1 | 12 | yes | complemented |
| 9 | C22×#127 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22×#124 | — | 2 | 3 | no | — |
| 11 | C22×#124 | C22 | 1 | 3 | yes | — |
| 12 | C22×#120 | — | 2 | 3 | no | — |
| 13 | C2×#496 | — | 2 | 18 | no | — |
| 14 | C2×#496 | C22 | 1 | 36 | yes | complemented |
| 15 | C22×#97 | C22 | 1 | 3 | yes | complemented |
| 16 | #1655 | C22 | 1 | 64 | yes | direct product factor |
| 17 | C2×#329 | C22 | 1 | 36 | yes | complemented |
| 18 | C26 | — | 2 | 1 | no | — |
| 19 | C23×Q8 | D4 | 1 | 3 | yes | — |
| 20 | C23×D4 | — | 2 | 3 | no | — |
| 21 | C23×D4 | — | 2 | 3 | no | — |
| 22 | C23×D4 | D4 | 1 | 3 | yes | — |
| 23 | C22×SD16 | — | 2 | 36 | no | — |
| 24 | C22×SD16 | — | 4 | 24 | no | — |
| 25 | C22×D8 | — | 2 | 36 | no | — |
| 26 | C22×D8 | — | 4 | 24 | no | — |
| 27 | C23×C8 | — | 2 | 3 | no | — |
| 28 | H(3,𝔽4) | — | 2 | 48 | no | — |
| 29 | H(3,𝔽4) | — | 2 | 72 | no | — |
| 30 | H(3,𝔽4) | C23 | 1 | 16 | yes | complemented |
| 31 | C2×#127 | — | 2 | 9 | no | — |
| 32 | C2×#127 | C23 | 1 | 6 | yes | complemented |
| 33 | C2×#124 | — | 2 | 36 | no | — |
| 34 | C2×#124 | — | 2 | 36 | no | — |
| 35 | C2×#124 | — | 2 | 18 | no | — |
| 36 | C2×#124 | D4 | 1 | 18 | yes | — |
| 37 | C2×#124 | C23 | 1 | 18 | yes | — |
| 38 | C2×#120 | — | 2 | 36 | no | — |
| 39 | C2×#120 | — | 2 | 36 | no | — |
| 40 | C22×C22⋊C4 | — | 2 | 3 | no | — |
| 41 | C22×C22⋊C4 | — | 2 | 6 | no | — |
| 42 | C22×C42 | — | 2 | 1 | no | — |
| 43 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 44 | #496 | — | 2 | 72 | no | — |
| 45 | #496 | C23 | 1 | 48 | yes | complemented |
| 46 | C2×#97 | — | 2 | 9 | no | — |
| 47 | C2×#97 | C23 | 1 | 18 | yes | complemented |
| 48 | #329 | C23 | 1 | 48 | yes | complemented |
| 49 | C25 | — | 2 | 12 | no | — |
| 50 | C25 | — | 2 | 3 | no | — |
| 51 | C25 | — | 4 | 1 | no | — |
| 52 | C25 | — | 8 | 12 | no | — |
| 53 | C22×Q8 | — | 2 | 9 | no | — |
| 54 | C22×Q8 | — | 4 | 12 | no | — |
| 55 | C22×Q8 | C2×D4 | 1 | 18 | yes | — |
| 56 | C22×D4 | — | 2 | 36 | no | — |
| 57 | C22×D4 | — | 2 | 9 | no | — |
| 58 | C22×D4 | — | 2 | 18 | no | — |
| 59 | C22×D4 | — | 4 | 12 | no | — |
| 60 | C22×D4 | — | 4 | 12 | no | — |
| 61 | C22×D4 | — | 4 | 12 | no | — |
| 62 | C22×D4 | — | 4 | 9 | no | — |
| 63 | C22×D4 | — | 8 | 12 | no | — |
| 64 | C22×D4 | C2×D4 | 1 | 18 | yes | — |
| 65 | C23×C4 | — | 2 | 3 | no | — |
| 66 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 67 | C2×SD16 | — | 2 | 48 | no | — |
| 68 | C2×SD16 | — | 4 | 144 | no | — |
| 69 | C2×D8 | — | 2 | 48 | no | — |
| 70 | C2×D8 | — | 4 | 144 | no | — |
| 71 | C22×C8 | — | 2 | 24 | no | — |
| 72 | C22×C8 | — | 2 | 18 | no | — |
| 73 | #127 | — | 2 | 18 | no | — |
| 74 | #127 | — | 4 | 6 | no | — |
| 75 | #127 | C24 | 1 | 4 | yes | complemented |
| 76 | #124 | — | 2 | 144 | no | — |
| 77 | #124 | — | 2 | 48 | no | — |
| 78 | #124 | — | 2 | 36 | no | — |
| 79 | #124 | — | 2 | 36 | no | — |
| 80 | #124 | C2×D4 | 1 | 36 | yes | — |
| 81 | #124 | C24 | 1 | 12 | yes | — |
| 82 | #120 | — | 2 | 144 | no | — |
| 83 | #120 | — | 2 | 48 | no | — |
| 84 | C2×C22⋊C4 | — | 2 | 36 | no | — |
| 85 | C2×C22⋊C4 | — | 2 | 36 | no | — |
| 86 | C2×C22⋊C4 | — | 2 | 36 | no | — |
| 87 | C2×C42 | — | 2 | 12 | no | — |
| 88 | C2×C42 | C2×D4 | 1 | 9 | yes | — |
| 89 | C2×C42 | C24 | 1 | 3 | yes | — |
| 90 | #97 | — | 2 | 18 | no | — |
| 91 | #97 | C24 | 1 | 12 | yes | complemented |
| 92 | C24 | — | 2 | 16 | no | — |
| 93 | C24 | — | 2 | 18 | no | — |
| 94 | C24 | — | 4 | 12 | no | — |
| 95 | C24 | — | 4 | 24 | no | — |
| 96 | C24 | — | 4 | 6 | no | — |
| 97 | C24 | — | 8 | 72 | no | — |
| 98 | C24 | — | 8 | 64 | no | — |
| 99 | C24 | — | 8 | 12 | no | — |
| 100 | C24 | #120 | 1 | 1 | yes | characteristic |
| 101 | C2×Q8 | — | 2 | 18 | no | — |
| 102 | C2×Q8 | — | 4 | 36 | no | — |
| 103 | C2×Q8 | — | 8 | 18 | no | — |
| 104 | C2×Q8 | C22×D4 | 1 | 12 | yes | — |
| 105 | C2×D4 | — | 2 | 48 | no | — |
| 106 | C2×D4 | — | 2 | 18 | no | — |
| 107 | C2×D4 | — | 2 | 12 | no | — |
| 108 | C2×D4 | — | 4 | 36 | no | — |
| 109 | C2×D4 | — | 4 | 36 | no | — |
| 110 | C2×D4 | — | 4 | 36 | no | — |
| 111 | C2×D4 | — | 4 | 18 | no | — |
| 112 | C2×D4 | — | 8 | 18 | no | — |
| 113 | C2×D4 | — | 8 | 36 | no | — |
| 114 | C2×D4 | — | 8 | 72 | no | — |
| 115 | C2×D4 | C22×D4 | 1 | 12 | yes | — |
| 116 | C22×C4 | — | 2 | 12 | no | — |
| 117 | C22×C4 | — | 2 | 18 | no | — |
| 118 | C22×C4 | — | 4 | 12 | no | — |
| 119 | C22×C4 | #120 | 1 | 9 | yes | — |
| 120 | C22×C4 | C22×D4 | 1 | 9 | yes | — |
| 121 | SD16 | — | 4 | 96 | no | — |
| 122 | D8 | — | 4 | 96 | no | — |
| 123 | C2×C8 | — | 2 | 72 | no | — |
| 124 | C2×C8 | — | 2 | 12 | no | — |
| 125 | C22⋊C4 | — | 2 | 48 | no | — |
| 126 | C22⋊C4 | — | 2 | 72 | no | — |
| 127 | C22⋊C4 | — | 2 | 24 | no | — |
| 128 | C42 | — | 2 | 16 | no | — |
| 129 | C42 | #120 | 1 | 6 | yes | — |
| 130 | C42 | C22×D4 | 1 | 9 | yes | — |
| 131 | C42 | C25 | 1 | 1 | yes | characteristic |
| 132 | C23 | — | 2 | 12 | no | — |
| 133 | C23 | — | 4 | 36 | no | — |
| 134 | C23 | — | 4 | 72 | no | — |
| 135 | C23 | — | 4 | 4 | no | — |
| 136 | C23 | — | 8 | 48 | no | — |
| 137 | C23 | — | 8 | 192 | no | — |
| 138 | C23 | — | 8 | 48 | no | — |
| 139 | C23 | — | 8 | 36 | no | — |
| 140 | C23 | — | 16 | 16 | no | — |
| 141 | C23 | #453 | 1 | 12 | yes | — |
| 142 | C23 | C2×#120 | 1 | 3 | yes | — |
| 143 | Q8 | — | 4 | 12 | no | — |
| 144 | Q8 | — | 8 | 18 | no | — |
| 145 | D4 | — | 4 | 12 | no | — |
| 146 | D4 | — | 4 | 12 | no | — |
| 147 | D4 | — | 4 | 12 | no | — |
| 148 | D4 | — | 8 | 18 | no | — |
| 149 | D4 | — | 8 | 36 | no | — |
| 150 | D4 | — | 8 | 48 | no | — |
| 151 | C2×C4 | — | 2 | 18 | no | — |
| 152 | C2×C4 | — | 2 | 18 | no | — |
| 153 | C2×C4 | — | 2 | 12 | no | — |
| 154 | C2×C4 | — | 4 | 36 | no | — |
| 155 | C2×C4 | C2×#120 | 1 | 9 | yes | — |
| 156 | C2×C4 | C23×D4 | 1 | 3 | yes | — |
| 157 | C8 | — | 2 | 24 | no | — |
| 158 | C22 | — | 4 | 12 | no | — |
| 159 | C22 | — | 4 | 24 | no | — |
| 160 | C22 | — | 8 | 64 | no | — |
| 161 | C22 | — | 8 | 72 | no | — |
| 162 | C22 | — | 8 | 12 | no | — |
| 163 | C22 | — | 16 | 24 | no | — |
| 164 | C22 | #1655 | 1 | 16 | yes | direct product factor |
| 165 | C22 | C2×#453 | 1 | 18 | yes | — |
| 166 | C22 | C22×#120 | 1 | 1 | yes | characteristic |
| 167 | C4 | — | 2 | 3 | no | — |
| 168 | C4 | — | 2 | 9 | no | — |
| 169 | C4 | — | 4 | 12 | no | — |
| 170 | C2 | — | 8 | 12 | no | — |
| 171 | C2 | — | 16 | 4 | no | — |
| 172 | C2 | C2×#1655 | 1 | 12 | yes | direct product factor |
| 173 | C2 | C22×#453 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×H(3,𝔽4) | C2 | semidirect |
| 2. C22×#496 | C2 | semidirect |
| 3. C2×#1655 | C2 | direct (N × H) |
| 4. C22×#329 | C2 | semidirect |
| 8. C2×H(3,𝔽4) | C22 | semidirect |
| 9. C22×#127 | C22 | semidirect |
| 14. C2×#496 | C22 | semidirect |
| 15. C22×#97 | C22 | semidirect |
| 16. #1655 | C22 | direct (N × H) |
| 17. C2×#329 | C22 | semidirect |
| 30. H(3,𝔽4) | C23 | semidirect |
| 32. C2×#127 | C23 | semidirect |
| 45. #496 | C23 | semidirect |
| 47. C2×#97 | C23 | semidirect |
| 48. #329 | C23 | semidirect |
| 75. #127 | C24 | semidirect |
| 91. #97 | C24 | semidirect |
| 164. C22 | #1655 | direct (N × H) |
| 172. C2 | C2×#1655 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×H(3,𝔽4) | 2 |
| C22×#496 | 2 |
| C2×#1655 | 2 |
| C22×#329 | 2 |