Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
358 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 120 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 | #30238 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #30237 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #13246 | C2 | 1 | 3 | yes | maximal |
| 4 | #13173 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #13158 | C2 | 1 | 3 | yes | maximal |
| 6 | #13157 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #12542 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×SD32 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#368 | C22 | 1 | 3 | yes | — |
| 10 | C2×#362 | C22 | 1 | 3 | yes | — |
| 11 | C2×#361 | C22 | 1 | 3 | yes | — |
| 12 | #2110 | C22 | 1 | 3 | yes | — |
| 13 | #2108 | C22 | 1 | 3 | yes | — |
| 14 | #2082 | — | 2 | 3 | no | — |
| 15 | #2081 | — | 2 | 3 | no | — |
| 16 | #2070 | — | 2 | 1 | no | — |
| 17 | #2069 | — | 2 | 1 | no | — |
| 18 | #1979 | C22 | 1 | 3 | yes | — |
| 19 | #1978 | C22 | 1 | 3 | yes | — |
| 20 | #1953 | C22 | 1 | 3 | yes | — |
| 21 | #1953 | C22 | 1 | 3 | yes | — |
| 22 | #1747 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1388 | C22 | 1 | 6 | yes | — |
| 24 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 25 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 26 | C2×SD32 | — | 2 | 3 | no | — |
| 27 | C2×SD32 | — | 2 | 3 | no | — |
| 28 | C2×M32 | C23 | 1 | 3 | yes | — |
| 29 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 30 | #430 | — | 2 | 3 | no | — |
| 31 | #430 | — | 2 | 3 | no | — |
| 32 | C2×#108 | C23 | 1 | 3 | yes | — |
| 33 | C2×#108 | C23 | 1 | 3 | yes | — |
| 34 | #419 | — | 2 | 3 | no | — |
| 35 | #418 | — | 2 | 3 | no | — |
| 36 | C2×#101 | — | 2 | 3 | no | — |
| 37 | C2×#100 | — | 2 | 3 | no | — |
| 38 | #405 | C23 | 1 | 3 | yes | — |
| 39 | #368 | — | 2 | 3 | no | — |
| 40 | #368 | — | 2 | 3 | no | — |
| 41 | #362 | — | 2 | 6 | no | — |
| 42 | #361 | — | 2 | 6 | no | — |
| 43 | #347 | — | 2 | 6 | no | — |
| 44 | #346 | — | 2 | 3 | no | — |
| 45 | C4×C16 | — | 2 | 3 | no | — |
| 46 | #344 | — | 2 | 3 | no | — |
| 47 | #344 | — | 2 | 3 | no | — |
| 48 | #338 | D4 | 1 | 1 | yes | characteristic |
| 49 | H(3,ℤ/4) | D4 | 1 | 3 | yes | — |
| 50 | C22×Q8 | — | 2 | 1 | no | — |
| 51 | C22×D4 | — | 2 | 1 | no | — |
| 52 | C2×Q16 | — | 2 | 3 | no | — |
| 53 | C2×Q16 | — | 2 | 3 | no | — |
| 54 | C2×D8 | — | 2 | 3 | no | — |
| 55 | C2×D8 | — | 2 | 3 | no | — |
| 56 | C2×M16 | — | 2 | 3 | no | — |
| 57 | C2×M16 | — | 2 | 3 | no | — |
| 58 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 59 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 60 | #117 | C2×D4 | 1 | 3 | yes | — |
| 61 | SD32 | — | 4 | 1 | no | — |
| 62 | SD32 | — | 4 | 3 | no | — |
| 63 | M32 | — | 2 | 3 | no | — |
| 64 | M32 | #41 | 1 | 6 | yes | — |
| 65 | C2×C16 | — | 2 | 3 | no | — |
| 66 | C2×C16 | #41 | 1 | 3 | yes | — |
| 67 | C2×C16 | #41 | 1 | 3 | yes | — |
| 68 | #108 | — | 2 | 3 | no | — |
| 69 | #108 | — | 2 | 3 | no | — |
| 70 | #108 | — | 2 | 3 | no | — |
| 71 | #108 | — | 2 | 3 | no | — |
| 72 | #108 | — | 8 | 3 | no | — |
| 73 | #108 | — | 8 | 3 | no | — |
| 74 | #103 | — | 8 | 3 | no | — |
| 75 | #102 | — | 8 | 3 | no | — |
| 76 | #101 | — | 8 | 3 | no | — |
| 77 | #100 | — | 8 | 3 | no | — |
| 78 | #97 | — | 2 | 3 | no | — |
| 79 | C4×C8 | — | 2 | 3 | no | — |
| 80 | C24 | — | 4 | 1 | no | — |
| 81 | C2×Q8 | — | 4 | 3 | no | — |
| 82 | C2×Q8 | — | 4 | 3 | no | — |
| 83 | C2×D4 | — | 4 | 3 | no | — |
| 84 | C2×D4 | — | 4 | 3 | no | — |
| 85 | C22×C4 | — | 4 | 1 | no | — |
| 86 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 87 | Q16 | — | 4 | 1 | no | — |
| 88 | Q16 | — | 4 | 3 | no | — |
| 89 | D8 | — | 4 | 1 | no | — |
| 90 | D8 | — | 4 | 3 | no | — |
| 91 | M16 | — | 2 | 3 | no | — |
| 92 | M16 | — | 2 | 3 | no | — |
| 93 | M16 | — | 4 | 3 | no | — |
| 94 | M16 | — | 4 | 3 | no | — |
| 95 | M16 | — | 4 | 3 | no | — |
| 96 | M16 | — | 4 | 3 | no | — |
| 97 | C2×C8 | — | 2 | 3 | no | — |
| 98 | C2×C8 | — | 4 | 3 | no | — |
| 99 | C2×C8 | — | 4 | 3 | no | — |
| 100 | C2×C8 | C2×#41 | 1 | 3 | yes | — |
| 101 | C2×C8 | C2×#41 | 1 | 3 | yes | — |
| 102 | C4⋊C4 | — | 2 | 3 | no | — |
| 103 | C22⋊C4 | — | 2 | 3 | no | — |
| 104 | C42 | — | 2 | 3 | no | — |
| 105 | C16 | — | 2 | 6 | no | — |
| 106 | C16 | #126 | 1 | 1 | yes | characteristic |
| 107 | C16 | #126 | 1 | 3 | yes | — |
| 108 | C23 | — | 8 | 3 | no | — |
| 109 | C23 | — | 32 | 1 | no | — |
| 110 | C23 | #578 | 1 | 1 | yes | characteristic |
| 111 | Q8 | — | 8 | 1 | no | — |
| 112 | Q8 | — | 8 | 3 | no | — |
| 113 | D4 | — | 8 | 1 | no | — |
| 114 | D4 | — | 8 | 3 | no | — |
| 115 | C2×C4 | — | 4 | 3 | no | — |
| 116 | C2×C4 | — | 8 | 3 | no | — |
| 117 | C2×C4 | #532 | 1 | 3 | yes | — |
| 118 | C2×C4 | #576 | 1 | 3 | yes | — |
| 119 | C8 | — | 4 | 3 | no | — |
| 120 | C8 | — | 8 | 3 | no | — |
| 121 | C8 | — | 8 | 3 | no | — |
| 122 | C8 | C2×#126 | 1 | 1 | yes | characteristic |
| 123 | C8 | #529 | 1 | 3 | yes | — |
| 124 | C22 | — | 4 | 1 | no | — |
| 125 | C22 | — | 16 | 1 | no | — |
| 126 | C22 | — | 32 | 3 | no | — |
| 127 | C22 | #3175 | 1 | 3 | yes | — |
| 128 | C4 | — | 8 | 3 | no | — |
| 129 | C4 | — | 16 | 1 | no | — |
| 130 | C4 | #2860 | 1 | 1 | yes | characteristic |
| 131 | C4 | #3176 | 1 | 3 | yes | — |
| 132 | C2 | — | 2 | 3 | no | — |
| 133 | C2 | — | 32 | 1 | no | — |
| 134 | C2 | #33272 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #33272 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30238 | C2 | semidirect |
| 4. #13173 | C2 | semidirect |
| 6. #13157 | C2 | semidirect |
| 7. #12542 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30238 | 2 |
| #30237 | 2 |
| #13246 | 2 |
| #13173 | 2 |
| #13158 | 2 |
| #13157 | 2 |
| #12542 | 2 |