Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5328 conjugacy classes of subgroups; only the 2870 normal subgroups are shown.
This group has 5328 conjugacy classes of subgroups — too many to draw. Only its 2870 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 | #58750 | C2 | 1 | 24 | yes | maximal, complemented |
| 2 | #46837 | C2 | 1 | 18 | yes | maximal, complemented |
| 3 | #36720 | C2 | 1 | 18 | yes | maximal, complemented |
| 4 | #35446 | C2 | 1 | 3 | yes | maximal |
| 5 | C22×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×#543 | C22 | 1 | 6 | yes | complemented |
| 7 | C22×#123 | C22 | 1 | 9 | yes | — |
| 8 | #2848 | C22 | 1 | 12 | yes | complemented |
| 9 | #2843 | C22 | 1 | 24 | yes | complemented |
| 10 | #2840 | C22 | 1 | 6 | yes | complemented |
| 11 | #2811 | C22 | 1 | 8 | yes | complemented |
| 12 | #2764 | C22 | 1 | 72 | yes | complemented |
| 13 | #2687 | C22 | 1 | 144 | yes | complemented |
| 14 | #2680 | C22 | 1 | 72 | yes | complemented |
| 15 | #2627 | C22 | 1 | 36 | yes | complemented |
| 16 | #2572 | C22 | 1 | 72 | yes | complemented |
| 17 | #2561 | C22 | 1 | 36 | yes | complemented |
| 18 | #2502 | C22 | 1 | 36 | yes | complemented |
| 19 | #2483 | C22 | 1 | 18 | yes | complemented |
| 20 | #2406 | C22 | 1 | 36 | yes | — |
| 21 | #2405 | C22 | 1 | 36 | yes | — |
| 22 | #2312 | C22 | 1 | 18 | yes | complemented |
| 23 | #2308 | C22 | 1 | 9 | yes | — |
| 24 | C2×H(5,𝔽2) | — | 4 | 12 | no | — |
| 25 | C22×#41 | C23 | 1 | 6 | yes | — |
| 26 | C23×D4 | C23 | 1 | 9 | yes | — |
| 27 | C24×C4 | C23 | 1 | 3 | yes | — |
| 28 | #543 | — | 4 | 24 | no | — |
| 29 | C2×#123 | — | 4 | 36 | no | — |
| 30 | C2×#123 | C23 | 1 | 72 | yes | — |
| 31 | C2×C22⋊Q8 | C23 | 1 | 36 | yes | — |
| 32 | C2×C22⋊Q8 | C23 | 1 | 36 | yes | — |
| 33 | C22×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 34 | C22×C22⋊C4 | C23 | 1 | 18 | yes | — |
| 35 | C22×C22⋊C4 | C23 | 1 | 9 | yes | — |
| 36 | #400 | C23 | 1 | 72 | yes | — |
| 37 | #398 | C23 | 1 | 144 | yes | — |
| 38 | #397 | C23 | 1 | 72 | yes | — |
| 39 | #395 | C23 | 1 | 24 | yes | — |
| 40 | #395 | C23 | 1 | 24 | yes | — |
| 41 | #394 | C23 | 1 | 72 | yes | — |
| 42 | #393 | C23 | 1 | 72 | yes | — |
| 43 | #391 | C23 | 1 | 72 | yes | — |
| 44 | #388 | C23 | 1 | 144 | yes | — |
| 45 | #386 | C23 | 1 | 72 | yes | — |
| 46 | #385 | C23 | 1 | 72 | yes | — |
| 47 | #385 | C23 | 1 | 72 | yes | — |
| 48 | #384 | C23 | 1 | 24 | yes | — |
| 49 | #383 | C23 | 1 | 48 | yes | — |
| 50 | #381 | C23 | 1 | 72 | yes | — |
| 51 | #381 | C23 | 1 | 18 | yes | — |
| 52 | #380 | C23 | 1 | 36 | yes | — |
| 53 | #379 | C23 | 1 | 24 | yes | — |
| 54 | #379 | C23 | 1 | 18 | yes | — |
| 55 | C4×C22⋊C4 | C23 | 1 | 36 | yes | — |
| 56 | C25 | C24 | 1 | 3 | yes | — |
| 57 | C25 | C24 | 1 | 3 | yes | — |
| 58 | H(5,𝔽2) | — | 4 | 64 | no | — |
| 59 | C2×#41 | — | 4 | 36 | no | — |
| 60 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 61 | C22×Q8 | C24 | 1 | 12 | yes | — |
| 62 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 63 | C22×D4 | — | 2 | 36 | no | — |
| 64 | C22×D4 | — | 4 | 36 | no | — |
| 65 | C22×D4 | C24 | 1 | 18 | yes | — |
| 66 | C23×C4 | — | 4 | 18 | no | — |
| 67 | C23×C4 | C24 | 1 | 18 | yes | — |
| 68 | C23×C4 | C24 | 1 | 18 | yes | — |
| 69 | C23×C4 | C24 | 1 | 9 | yes | — |
| 70 | #123 | — | 4 | 144 | no | — |
| 71 | C22⋊Q8 | — | 4 | 72 | no | — |
| 72 | C22⋊Q8 | — | 4 | 72 | no | — |
| 73 | C2×C4⋊C4 | — | 4 | 36 | no | — |
| 74 | C2×C4⋊C4 | C24 | 1 | 72 | yes | — |
| 75 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 76 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 77 | C2×C22⋊C4 | — | 2 | 72 | no | — |
| 78 | C2×C22⋊C4 | — | 4 | 18 | no | — |
| 79 | C2×C22⋊C4 | C24 | 1 | 72 | yes | — |
| 80 | C2×C22⋊C4 | C24 | 1 | 36 | yes | — |
| 81 | C2×C22⋊C4 | C24 | 1 | 72 | yes | — |
| 82 | C2×C42 | C24 | 1 | 24 | yes | — |
| 83 | #95 | C24 | 1 | 72 | yes | — |
| 84 | #95 | C24 | 1 | 72 | yes | — |
| 85 | #95 | C24 | 1 | 72 | yes | — |
| 86 | C24 | — | 2 | 18 | no | — |
| 87 | C24 | — | 4 | 6 | no | — |
| 88 | C24 | — | 4 | 12 | no | — |
| 89 | C24 | — | 4 | 12 | no | — |
| 90 | C24 | H(5,𝔽2) | 1 | 12 | yes | — |
| 91 | C24 | C25 | 1 | 9 | yes | — |
| 92 | #41 | — | 4 | 96 | no | — |
| 93 | C2×Q8 | — | 4 | 6 | no | — |
| 94 | C2×Q8 | — | 4 | 12 | no | — |
| 95 | C2×Q8 | — | 4 | 12 | no | — |
| 96 | C2×Q8 | — | 4 | 24 | no | — |
| 97 | C2×D4 | — | 2 | 72 | no | — |
| 98 | C2×D4 | — | 4 | 18 | no | — |
| 99 | C2×D4 | — | 4 | 144 | no | — |
| 100 | C22×C4 | — | 2 | 18 | no | — |
| 101 | C22×C4 | — | 2 | 36 | no | — |
| 102 | C22×C4 | — | 2 | 36 | no | — |
| 103 | C22×C4 | — | 4 | 18 | no | — |
| 104 | C22×C4 | — | 4 | 36 | no | — |
| 105 | C22×C4 | — | 4 | 36 | no | — |
| 106 | C22×C4 | — | 8 | 24 | no | — |
| 107 | C22×C4 | C25 | 1 | 36 | yes | — |
| 108 | C22×C4 | C25 | 1 | 6 | yes | — |
| 109 | C22×C4 | C25 | 1 | 12 | yes | — |
| 110 | C4⋊C4 | — | 4 | 72 | no | — |
| 111 | C4⋊C4 | — | 4 | 36 | no | — |
| 112 | C4⋊C4 | — | 4 | 36 | no | — |
| 113 | C22⋊C4 | — | 2 | 144 | no | — |
| 114 | C22⋊C4 | — | 4 | 36 | no | — |
| 115 | C22⋊C4 | — | 4 | 72 | no | — |
| 116 | C42 | — | 4 | 24 | no | — |
| 117 | C23 | — | 2 | 24 | no | — |
| 118 | C23 | — | 2 | 18 | no | — |
| 119 | C23 | — | 2 | 36 | no | — |
| 120 | C23 | — | 4 | 12 | no | — |
| 121 | C23 | — | 4 | 24 | no | — |
| 122 | C23 | — | 8 | 36 | no | — |
| 123 | C23 | — | 16 | 12 | no | — |
| 124 | C23 | C2×H(5,𝔽2) | 1 | 18 | yes | — |
| 125 | C23 | C26 | 1 | 1 | yes | characteristic |
| 126 | Q8 | — | 4 | 8 | no | — |
| 127 | Q8 | — | 16 | 4 | no | — |
| 128 | Q8 | — | 16 | 12 | no | — |
| 129 | D4 | — | 4 | 72 | no | — |
| 130 | C2×C4 | — | 2 | 36 | no | — |
| 131 | C2×C4 | — | 2 | 36 | no | — |
| 132 | C2×C4 | — | 2 | 18 | no | — |
| 133 | C2×C4 | — | 2 | 72 | no | — |
| 134 | C2×C4 | — | 4 | 36 | no | — |
| 135 | C2×C4 | — | 8 | 36 | no | — |
| 136 | C2×C4 | — | 8 | 36 | no | — |
| 137 | C22 | — | 2 | 18 | no | — |
| 138 | C22 | — | 2 | 36 | no | — |
| 139 | C22 | — | 4 | 36 | no | — |
| 140 | C22 | — | 8 | 24 | no | — |
| 141 | C22 | — | 16 | 12 | no | — |
| 142 | C22 | C22×H(5,𝔽2) | 1 | 3 | yes | — |
| 143 | C22 | 2−1+6 | 1 | 4 | yes | — |
| 144 | C4 | — | 4 | 6 | no | — |
| 145 | C4 | — | 4 | 12 | no | — |
| 146 | C4 | — | 4 | 36 | no | — |
| 147 | C2 | — | 4 | 18 | no | — |
| 148 | C2 | #62817 | 1 | 1 | yes | characteristic |
| 149 | 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. #58750 | C2 | semidirect |
| 2. #46837 | C2 | semidirect |
| 3. #36720 | C2 | semidirect |
| 6. C2×#543 | C22 | semidirect |
| 8. #2848 | C22 | semidirect |
| 9. #2843 | C22 | semidirect |
| 10. #2840 | C22 | semidirect |
| 11. #2811 | C22 | semidirect |
| 12. #2764 | C22 | semidirect |
| 13. #2687 | C22 | semidirect |
| 14. #2680 | C22 | semidirect |
| 15. #2627 | C22 | semidirect |
| 16. #2572 | C22 | semidirect |
| 17. #2561 | C22 | semidirect |
| 18. #2502 | C22 | semidirect |
| 19. #2483 | C22 | semidirect |
| 22. #2312 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #58750 | 2 |
| #46837 | 2 |
| #36720 | 2 |
| #35446 | 2 |