Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5704 conjugacy classes of subgroups; only the 2898 normal subgroups are shown.
This group has 5704 conjugacy classes of subgroups — too many to draw. Only its 2898 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 | #52925 | C2 | 1 | 24 | yes | maximal, complemented |
| 2 | #52914 | C2 | 1 | 24 | yes | maximal, complemented |
| 3 | #35455 | C2 | 1 | 9 | yes | maximal, complemented |
| 4 | #35446 | C2 | 1 | 3 | yes | maximal |
| 5 | C2×#2614 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C22×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 7 | C24×Q8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×#543 | C22 | 1 | 3 | yes | complemented |
| 9 | C22×#123 | C22 | 1 | 9 | yes | — |
| 10 | #2844 | C22 | 1 | 24 | yes | complemented |
| 11 | #2840 | C22 | 1 | 8 | yes | complemented |
| 12 | #2835 | C22 | 1 | 24 | yes | complemented |
| 13 | #2832 | C22 | 1 | 72 | yes | complemented |
| 14 | #2811 | C22 | 1 | 8 | yes | complemented |
| 15 | #2785 | C22 | 1 | 24 | yes | complemented |
| 16 | #2763 | C22 | 1 | 72 | yes | complemented |
| 17 | #2736 | C22 | 1 | 24 | yes | complemented |
| 18 | #2614 | C22 | 1 | 48 | yes | complemented |
| 19 | #2561 | C22 | 1 | 36 | yes | complemented |
| 20 | #2503 | C22 | 1 | 36 | yes | complemented |
| 21 | #2488 | C22 | 1 | 144 | yes | complemented |
| 22 | #2406 | C22 | 1 | 36 | yes | — |
| 23 | #2405 | C22 | 1 | 36 | yes | — |
| 24 | C2×#395 | C22 | 1 | 12 | yes | complemented |
| 25 | C2×#393 | C22 | 1 | 12 | yes | complemented |
| 26 | #2308 | C22 | 1 | 9 | yes | — |
| 27 | C2×#381 | C22 | 1 | 9 | yes | complemented |
| 28 | #2273 | C22 | 1 | 3 | yes | complemented |
| 29 | C2×H(5,𝔽2) | — | 4 | 12 | no | — |
| 30 | C22×#41 | C23 | 1 | 6 | yes | — |
| 31 | C23×Q8 | — | 4 | 12 | no | — |
| 32 | C23×Q8 | C23 | 1 | 12 | yes | — |
| 33 | C23×D4 | C23 | 1 | 3 | yes | — |
| 34 | C23×D4 | C23 | 1 | 6 | yes | — |
| 35 | C24×C4 | C23 | 1 | 3 | yes | — |
| 36 | #543 | — | 4 | 12 | no | — |
| 37 | C2×#123 | — | 4 | 36 | no | — |
| 38 | C2×#123 | C23 | 1 | 72 | yes | — |
| 39 | C2×C22⋊Q8 | C23 | 1 | 36 | yes | — |
| 40 | C22×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 41 | C22×C22⋊C4 | C23 | 1 | 9 | yes | — |
| 42 | C22×C22⋊C4 | C23 | 1 | 18 | yes | — |
| 43 | #400 | C23 | 1 | 144 | yes | — |
| 44 | #397 | C23 | 1 | 72 | yes | — |
| 45 | #395 | C23 | 1 | 72 | yes | — |
| 46 | #395 | C23 | 1 | 24 | yes | — |
| 47 | #395 | C23 | 1 | 24 | yes | — |
| 48 | #394 | C23 | 1 | 72 | yes | — |
| 49 | #393 | C23 | 1 | 72 | yes | — |
| 50 | #393 | C23 | 1 | 72 | yes | — |
| 51 | #393 | C23 | 1 | 24 | yes | — |
| 52 | #391 | C23 | 1 | 72 | yes | — |
| 53 | #388 | C23 | 1 | 144 | yes | — |
| 54 | #386 | C23 | 1 | 72 | yes | — |
| 55 | #385 | C23 | 1 | 72 | yes | — |
| 56 | #382 | C23 | 1 | 72 | yes | — |
| 57 | #381 | C23 | 1 | 72 | yes | — |
| 58 | #381 | C23 | 1 | 18 | yes | — |
| 59 | #379 | C23 | 1 | 18 | yes | — |
| 60 | #379 | C23 | 1 | 24 | yes | — |
| 61 | C4×C22⋊C4 | C23 | 1 | 36 | yes | — |
| 62 | C2×#95 | C23 | 1 | 36 | yes | — |
| 63 | C25 | C24 | 1 | 3 | yes | — |
| 64 | C25 | C24 | 1 | 2 | yes | — |
| 65 | C25 | C24 | 1 | 1 | yes | characteristic |
| 66 | H(5,𝔽2) | — | 4 | 64 | no | — |
| 67 | C2×#41 | — | 4 | 36 | no | — |
| 68 | C22×Q8 | — | 4 | 72 | no | — |
| 69 | C22×Q8 | — | 4 | 64 | no | — |
| 70 | C22×Q8 | C24 | 1 | 12 | yes | — |
| 71 | C22×Q8 | C24 | 1 | 12 | yes | — |
| 72 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 73 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 74 | C22×D4 | — | 2 | 36 | no | — |
| 75 | C22×D4 | — | 4 | 36 | no | — |
| 76 | C22×D4 | C24 | 1 | 12 | yes | — |
| 77 | C22×D4 | C24 | 1 | 6 | yes | — |
| 78 | C23×C4 | — | 2 | 36 | no | — |
| 79 | C23×C4 | C24 | 1 | 18 | yes | — |
| 80 | C23×C4 | C24 | 1 | 3 | yes | — |
| 81 | C23×C4 | C24 | 1 | 18 | yes | — |
| 82 | C23×C4 | C24 | 1 | 6 | yes | — |
| 83 | #123 | — | 4 | 144 | no | — |
| 84 | C22⋊Q8 | — | 4 | 72 | no | — |
| 85 | C2×C4⋊C4 | — | 4 | 36 | no | — |
| 86 | C2×C4⋊C4 | C24 | 1 | 72 | yes | — |
| 87 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 88 | C2×C22⋊C4 | — | 2 | 36 | no | — |
| 89 | C2×C22⋊C4 | — | 4 | 36 | no | — |
| 90 | C2×C22⋊C4 | C24 | 1 | 36 | yes | — |
| 91 | C2×C22⋊C4 | C24 | 1 | 72 | yes | — |
| 92 | C2×C22⋊C4 | C24 | 1 | 72 | yes | — |
| 93 | C2×C42 | C24 | 1 | 24 | yes | — |
| 94 | #95 | C24 | 1 | 72 | yes | — |
| 95 | #95 | C24 | 1 | 72 | yes | — |
| 96 | #95 | C24 | 1 | 24 | yes | — |
| 97 | #95 | C24 | 1 | 72 | yes | — |
| 98 | C24 | — | 2 | 18 | no | — |
| 99 | C24 | — | 4 | 4 | no | — |
| 100 | C24 | — | 4 | 6 | no | — |
| 101 | C24 | — | 4 | 12 | no | — |
| 102 | C24 | — | 4 | 8 | no | — |
| 103 | C24 | H(5,𝔽2) | 1 | 12 | yes | — |
| 104 | C24 | C25 | 1 | 6 | yes | — |
| 105 | C24 | C25 | 1 | 3 | yes | — |
| 106 | #41 | — | 4 | 96 | no | — |
| 107 | C2×Q8 | — | 4 | 36 | no | — |
| 108 | C2×Q8 | — | 4 | 12 | no | — |
| 109 | C2×Q8 | — | 4 | 6 | no | — |
| 110 | C2×Q8 | — | 4 | 12 | no | — |
| 111 | C2×Q8 | — | 4 | 24 | no | — |
| 112 | C2×Q8 | — | 4 | 192 | no | — |
| 113 | C2×D4 | — | 2 | 72 | no | — |
| 114 | C2×D4 | — | 4 | 18 | no | — |
| 115 | C2×D4 | — | 4 | 96 | no | — |
| 116 | C2×D4 | — | 4 | 48 | no | — |
| 117 | C22×C4 | — | 2 | 36 | no | — |
| 118 | C22×C4 | — | 2 | 18 | no | — |
| 119 | C22×C4 | — | 2 | 72 | no | — |
| 120 | C22×C4 | — | 2 | 36 | no | — |
| 121 | C22×C4 | — | 4 | 18 | no | — |
| 122 | C22×C4 | — | 4 | 36 | no | — |
| 123 | C22×C4 | — | 4 | 48 | no | — |
| 124 | C22×C4 | C25 | 1 | 36 | yes | — |
| 125 | C22×C4 | C25 | 1 | 12 | yes | — |
| 126 | C22×C4 | C25 | 1 | 6 | yes | — |
| 127 | C4⋊C4 | — | 4 | 72 | no | — |
| 128 | C4⋊C4 | — | 4 | 36 | no | — |
| 129 | C22⋊C4 | — | 2 | 72 | no | — |
| 130 | C22⋊C4 | — | 4 | 72 | no | — |
| 131 | C22⋊C4 | — | 4 | 72 | no | — |
| 132 | C42 | — | 4 | 24 | no | — |
| 133 | C23 | — | 2 | 12 | no | — |
| 134 | C23 | — | 2 | 18 | no | — |
| 135 | C23 | — | 2 | 24 | no | — |
| 136 | C23 | — | 4 | 24 | no | — |
| 137 | C23 | — | 4 | 8 | no | — |
| 138 | C23 | — | 4 | 12 | no | — |
| 139 | C23 | — | 8 | 36 | no | — |
| 140 | C23 | — | 16 | 12 | no | — |
| 141 | C23 | H(3,𝔽4) | 1 | 16 | yes | — |
| 142 | C23 | C2×H(5,𝔽2) | 1 | 18 | yes | — |
| 143 | C23 | C26 | 1 | 1 | yes | characteristic |
| 144 | Q8 | — | 4 | 48 | no | — |
| 145 | Q8 | — | 4 | 16 | no | — |
| 146 | Q8 | — | 4 | 8 | no | — |
| 147 | Q8 | — | 16 | 12 | no | — |
| 148 | D4 | — | 4 | 24 | no | — |
| 149 | D4 | — | 4 | 48 | no | — |
| 150 | C2×C4 | — | 2 | 36 | no | — |
| 151 | C2×C4 | — | 2 | 36 | no | — |
| 152 | C2×C4 | — | 2 | 18 | no | — |
| 153 | C2×C4 | — | 2 | 72 | no | — |
| 154 | C2×C4 | — | 4 | 24 | no | — |
| 155 | C2×C4 | — | 4 | 72 | no | — |
| 156 | C2×C4 | — | 4 | 12 | no | — |
| 157 | C2×C4 | — | 8 | 36 | no | — |
| 158 | C22 | — | 2 | 36 | no | — |
| 159 | C22 | — | 4 | 16 | no | — |
| 160 | C22 | — | 4 | 36 | no | — |
| 161 | C22 | — | 4 | 6 | no | — |
| 162 | C22 | — | 16 | 12 | no | — |
| 163 | C22 | — | 16 | 8 | no | — |
| 164 | C22 | C2×H(3,𝔽4) | 1 | 12 | yes | — |
| 165 | C22 | C22×H(5,𝔽2) | 1 | 3 | yes | — |
| 166 | C22 | 2−1+6 | 1 | 4 | yes | — |
| 167 | C4 | — | 4 | 36 | no | — |
| 168 | C4 | — | 4 | 6 | no | — |
| 169 | C4 | — | 4 | 12 | no | — |
| 170 | C2 | — | 2 | 12 | no | — |
| 171 | C2 | — | 8 | 6 | no | — |
| 172 | C2 | C22×H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 173 | C2 | #62871 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #52925 | C2 | semidirect |
| 2. #52914 | C2 | semidirect |
| 3. #35455 | C2 | semidirect |
| 5. C2×#2614 | C2 | semidirect |
| 7. C24×Q8 | C22 | semidirect |
| 8. C2×#543 | C22 | semidirect |
| 10. #2844 | C22 | semidirect |
| 11. #2840 | C22 | semidirect |
| 12. #2835 | C22 | semidirect |
| 13. #2832 | C22 | semidirect |
| 14. #2811 | C22 | semidirect |
| 15. #2785 | C22 | semidirect |
| 16. #2763 | C22 | semidirect |
| 17. #2736 | C22 | semidirect |
| 18. #2614 | C22 | semidirect |
| 19. #2561 | C22 | semidirect |
| 20. #2503 | C22 | semidirect |
| 21. #2488 | C22 | semidirect |
| 24. C2×#395 | C22 | semidirect |
| 25. C2×#393 | C22 | semidirect |
| 27. C2×#381 | C22 | semidirect |
| 28. #2273 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #52925 | 2 |
| #52914 | 2 |
| #35455 | 2 |
| #35446 | 2 |
| C2×#2614 | 2 |