Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2472 conjugacy classes of subgroups; only the 558 normal subgroups are shown.
This group has 2472 conjugacy classes of subgroups — too many to draw. Only its 558 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 | #61100 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | #31077 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #28250 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #24287 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #21709 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #3584 | C22 | 1 | 12 | yes | complemented |
| 7 | #3289 | — | 2 | 24 | no | — |
| 8 | #3288 | — | 2 | 24 | no | — |
| 9 | #3150 | C22 | 1 | 3 | yes | — |
| 10 | #3067 | C22 | 1 | 24 | yes | complemented |
| 11 | #3018 | C22 | 1 | 24 | yes | complemented |
| 12 | C2×#444 | C22 | 1 | 12 | yes | complemented |
| 13 | #2931 | C22 | 1 | 6 | yes | complemented |
| 14 | #2050 | C22 | 1 | 24 | yes | complemented |
| 15 | #2017 | C22 | 1 | 8 | yes | complemented |
| 16 | #1974 | C22 | 1 | 12 | yes | complemented |
| 17 | #1963 | C22 | 1 | 6 | yes | complemented |
| 18 | #1849 | C22 | 1 | 3 | yes | complemented |
| 19 | #1813 | C22 | 1 | 12 | yes | complemented |
| 20 | #1791 | C22 | 1 | 8 | yes | complemented |
| 21 | #1726 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | #585 | C23 | 1 | 8 | yes | — |
| 23 | #578 | — | 2 | 24 | no | — |
| 24 | #577 | — | 2 | 48 | no | — |
| 25 | #576 | — | 2 | 24 | no | — |
| 26 | #575 | — | 4 | 12 | no | — |
| 27 | #575 | C23 | 1 | 24 | yes | — |
| 28 | C2×#135 | — | 4 | 12 | no | — |
| 29 | C2×#135 | C23 | 1 | 12 | yes | — |
| 30 | C2×#131 | C23 | 1 | 12 | yes | — |
| 31 | #532 | C23 | 1 | 6 | yes | — |
| 32 | #497 | D4 | 1 | 12 | yes | complemented |
| 33 | #496 | D4 | 1 | 12 | yes | complemented |
| 34 | #473 | — | 2 | 24 | no | — |
| 35 | #472 | — | 2 | 24 | no | — |
| 36 | #471 | — | 2 | 48 | no | — |
| 37 | #456 | — | 2 | 24 | no | — |
| 38 | #455 | — | 2 | 24 | no | — |
| 39 | #454 | — | 2 | 24 | no | — |
| 40 | #453 | — | 2 | 24 | no | — |
| 41 | #444 | — | 2 | 48 | no | — |
| 42 | #436 | — | 4 | 6 | no | — |
| 43 | C8×D4 | — | 4 | 6 | no | — |
| 44 | C2×#108 | C23 | 1 | 12 | yes | — |
| 45 | #423 | C23 | 1 | 6 | yes | — |
| 46 | C2×#104 | C23 | 1 | 24 | yes | — |
| 47 | #413 | C23 | 1 | 24 | yes | — |
| 48 | #408 | C23 | 1 | 12 | yes | — |
| 49 | C2×#97 | C23 | 1 | 3 | yes | — |
| 50 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 51 | #376 | C23 | 1 | 1 | yes | characteristic |
| 52 | #338 | C23 | 1 | 8 | yes | — |
| 53 | 2−1+4 | — | 2 | 24 | no | — |
| 54 | H(5,𝔽2) | — | 2 | 24 | no | — |
| 55 | H(5,𝔽2) | — | 4 | 8 | no | — |
| 56 | C2×#41 | — | 4 | 24 | no | — |
| 57 | C2×#41 | C24 | 1 | 12 | yes | — |
| 58 | #137 | — | 2 | 48 | no | — |
| 59 | #137 | — | 4 | 48 | no | — |
| 60 | AGL(1,ℤ/8) | — | 2 | 48 | no | — |
| 61 | AGL(1,ℤ/8) | — | 4 | 48 | no | — |
| 62 | #135 | — | 2 | 24 | no | — |
| 63 | #135 | — | 2 | 24 | no | — |
| 64 | #135 | — | 4 | 24 | no | — |
| 65 | #135 | — | 4 | 24 | no | — |
| 66 | #135 | — | 4 | 24 | no | — |
| 67 | C2×Q16 | — | 4 | 12 | no | — |
| 68 | C2×Q16 | C2×D4 | 1 | 12 | yes | — |
| 69 | C2×SD16 | — | 4 | 24 | no | — |
| 70 | C2×SD16 | C2×D4 | 1 | 24 | yes | — |
| 71 | C2×D8 | — | 4 | 12 | no | — |
| 72 | C2×D8 | C2×D4 | 1 | 12 | yes | — |
| 73 | #131 | — | 2 | 24 | no | — |
| 74 | #131 | — | 2 | 24 | no | — |
| 75 | C2×M16 | — | 2 | 12 | no | — |
| 76 | C2×M16 | C24 | 1 | 12 | yes | — |
| 77 | C22×C8 | — | 2 | 12 | no | — |
| 78 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 79 | C22×C8 | C24 | 1 | 3 | yes | — |
| 80 | C4⋊Q8 | C2×D4 | 1 | 6 | yes | — |
| 81 | #127 | C2×D4 | 1 | 6 | yes | — |
| 82 | #124 | C2×D4 | 1 | 12 | yes | — |
| 83 | #121 | — | 4 | 6 | no | — |
| 84 | C4×D4 | — | 4 | 6 | no | — |
| 85 | C2×C42 | C24 | 1 | 3 | yes | — |
| 86 | #108 | — | 2 | 24 | no | — |
| 87 | C4⋊C8 | — | 4 | 6 | no | — |
| 88 | #104 | — | 2 | 48 | no | — |
| 89 | #101 | — | 2 | 24 | no | — |
| 90 | #100 | — | 2 | 24 | no | — |
| 91 | C22⋊C8 | — | 4 | 12 | no | — |
| 92 | #97 | C2×D4 | 1 | 12 | yes | — |
| 93 | C4×C8 | — | 2 | 6 | no | — |
| 94 | #95 | — | 4 | 1 | no | — |
| 95 | #41 | — | 2 | 24 | no | — |
| 96 | #41 | — | 2 | 24 | no | — |
| 97 | #41 | — | 4 | 48 | no | — |
| 98 | #41 | — | 8 | 24 | no | — |
| 99 | #41 | — | 8 | 8 | no | — |
| 100 | C2×Q8 | — | 4 | 24 | no | — |
| 101 | C2×Q8 | C22×D4 | 1 | 12 | yes | — |
| 102 | C2×D4 | — | 2 | 12 | no | — |
| 103 | C2×D4 | — | 4 | 6 | no | — |
| 104 | C2×D4 | — | 4 | 24 | no | — |
| 105 | C2×D4 | — | 8 | 24 | no | — |
| 106 | C2×D4 | C22×D4 | 1 | 12 | yes | — |
| 107 | C22×C4 | — | 2 | 3 | no | — |
| 108 | C22×C4 | — | 2 | 12 | no | — |
| 109 | C22×C4 | — | 8 | 8 | no | — |
| 110 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 111 | Q16 | — | 2 | 24 | no | — |
| 112 | Q16 | — | 4 | 24 | no | — |
| 113 | SD16 | — | 2 | 48 | no | — |
| 114 | SD16 | — | 4 | 48 | no | — |
| 115 | D8 | — | 2 | 24 | no | — |
| 116 | D8 | — | 4 | 24 | no | — |
| 117 | M16 | — | 2 | 24 | no | — |
| 118 | M16 | — | 4 | 24 | no | — |
| 119 | C2×C8 | — | 2 | 3 | no | — |
| 120 | C2×C8 | — | 2 | 12 | no | — |
| 121 | C2×C8 | — | 4 | 12 | no | — |
| 122 | C2×C8 | — | 4 | 12 | no | — |
| 123 | C2×C8 | C22×D4 | 1 | 12 | yes | — |
| 124 | C2×C8 | H(5,𝔽2) | 1 | 6 | yes | — |
| 125 | C4⋊C4 | — | 4 | 6 | no | — |
| 126 | C22⋊C4 | — | 4 | 12 | no | — |
| 127 | C42 | — | 2 | 3 | no | — |
| 128 | C42 | C22×D4 | 1 | 6 | yes | — |
| 129 | C23 | — | 2 | 12 | no | — |
| 130 | C23 | — | 8 | 8 | no | — |
| 131 | C23 | #585 | 1 | 1 | yes | characteristic |
| 132 | Q8 | — | 2 | 24 | no | — |
| 133 | Q8 | — | 8 | 8 | no | — |
| 134 | D4 | — | 2 | 24 | no | — |
| 135 | D4 | — | 4 | 24 | no | — |
| 136 | D4 | — | 8 | 24 | no | — |
| 137 | D4 | — | 8 | 12 | no | — |
| 138 | C2×C4 | — | 2 | 6 | no | — |
| 139 | C2×C4 | — | 2 | 12 | no | — |
| 140 | C2×C4 | — | 4 | 12 | no | — |
| 141 | C2×C4 | — | 4 | 12 | no | — |
| 142 | C2×C4 | — | 8 | 3 | no | — |
| 143 | C2×C4 | — | 8 | 24 | no | — |
| 144 | C2×C4 | C23×D4 | 1 | 3 | yes | — |
| 145 | C2×C4 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 146 | C8 | — | 4 | 12 | no | — |
| 147 | C8 | — | 4 | 1 | no | — |
| 148 | C8 | D42 | 1 | 12 | yes | — |
| 149 | C22 | — | 4 | 1 | no | — |
| 150 | C22 | — | 4 | 12 | no | — |
| 151 | C22 | — | 8 | 12 | no | — |
| 152 | C22 | — | 32 | 8 | no | — |
| 153 | C22 | #3496 | 1 | 3 | yes | — |
| 154 | C4 | — | 4 | 6 | no | — |
| 155 | C4 | — | 4 | 12 | no | — |
| 156 | C4 | C2×D42 | 1 | 3 | yes | — |
| 157 | C4 | #3549 | 1 | 1 | yes | characteristic |
| 158 | C2 | — | 2 | 3 | no | — |
| 159 | C2 | — | 8 | 12 | no | — |
| 160 | C2 | #59816 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3549 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #59816 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #61100 | C2 | semidirect |
| 2. #31077 | C2 | semidirect |
| 3. #28250 | C2 | semidirect |
| 4. #24287 | C2 | semidirect |
| 5. #21709 | C2 | semidirect |
| 6. #3584 | C22 | semidirect |
| 10. #3067 | C22 | semidirect |
| 11. #3018 | C22 | semidirect |
| 12. C2×#444 | C22 | semidirect |
| 13. #2931 | C22 | semidirect |
| 14. #2050 | C22 | semidirect |
| 15. #2017 | C22 | semidirect |
| 16. #1974 | C22 | semidirect |
| 17. #1963 | C22 | semidirect |
| 18. #1849 | C22 | semidirect |
| 19. #1813 | C22 | semidirect |
| 20. #1791 | C22 | semidirect |
| 21. #1726 | C22 | semidirect |
| 32. #497 | D4 | semidirect |
| 33. #496 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #61100 | 2 |
| #31077 | 2 |
| #28250 | 2 |
| #24287 | 2 |
| #21709 | 2 |