Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1719 conjugacy classes of subgroups; only the 312 normal subgroups are shown.
This group has 1719 conjugacy classes of subgroups — too many to draw. Only its 312 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 | #20585 | C2 | 1 | 3 | yes | maximal |
| 2 | #10938 | C2 | 1 | 12 | yes | maximal, complemented |
| 3 | C22×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 4 | C22×#99 | C22 | 1 | 12 | yes | — |
| 5 | #2308 | C22 | 1 | 6 | yes | — |
| 6 | #2273 | C22 | 1 | 4 | yes | complemented |
| 7 | #1782 | C4 | 1 | 12 | yes | complemented |
| 8 | #1781 | C4 | 1 | 12 | yes | complemented |
| 9 | C2×#342 | C22 | 1 | 12 | yes | complemented |
| 10 | C2×H(5,𝔽2) | — | 4 | 12 | no | — |
| 11 | C22×#41 | C2×C4 | 1 | 6 | yes | — |
| 12 | C23×D4 | C2×C4 | 1 | 6 | yes | — |
| 13 | C23×D4 | C23 | 1 | 3 | yes | — |
| 14 | C22×C22⋊C4 | C23 | 1 | 12 | yes | — |
| 15 | C2×#99 | — | 4 | 24 | no | — |
| 16 | C2×#99 | C2×C4 | 1 | 48 | yes | — |
| 17 | #386 | — | 4 | 12 | no | — |
| 18 | #381 | C2×C4 | 1 | 12 | yes | — |
| 19 | #379 | C2×C4 | 1 | 12 | yes | — |
| 20 | C4×C22⋊C4 | — | 4 | 12 | no | — |
| 21 | #342 | — | 4 | 12 | no | — |
| 22 | C25 | C42 | 1 | 2 | yes | complemented |
| 23 | C25 | C22×C4 | 1 | 3 | yes | — |
| 24 | C25 | C24 | 1 | 1 | yes | characteristic |
| 25 | H(5,𝔽2) | — | 4 | 48 | no | — |
| 26 | H(5,𝔽2) | — | 4 | 16 | no | — |
| 27 | C2×#41 | — | 4 | 12 | no | — |
| 28 | C2×#41 | — | 4 | 24 | no | — |
| 29 | C22×Q8 | C42 | 1 | 2 | yes | complemented |
| 30 | C22×D4 | — | 2 | 12 | no | — |
| 31 | C22×D4 | — | 2 | 24 | no | — |
| 32 | C22×D4 | — | 4 | 24 | no | — |
| 33 | C22×D4 | — | 4 | 12 | no | — |
| 34 | C22×D4 | C42 | 1 | 6 | yes | complemented |
| 35 | C22×D4 | C22×C4 | 1 | 12 | yes | — |
| 36 | C23×C4 | — | 2 | 12 | no | — |
| 37 | C23×C4 | C42 | 1 | 6 | yes | complemented |
| 38 | C23×C4 | C22×C4 | 1 | 3 | yes | — |
| 39 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 40 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 41 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 42 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 43 | C2×C22⋊C4 | C22×C4 | 1 | 24 | yes | — |
| 44 | C2×C42 | — | 4 | 8 | no | — |
| 45 | #99 | — | 4 | 48 | no | — |
| 46 | #95 | — | 4 | 12 | no | — |
| 47 | #95 | — | 4 | 8 | no | — |
| 48 | C24 | — | 2 | 6 | no | — |
| 49 | C24 | — | 2 | 12 | no | — |
| 50 | C24 | — | 4 | 6 | no | — |
| 51 | C24 | — | 16 | 1 | no | — |
| 52 | C24 | — | 16 | 3 | no | — |
| 53 | C24 | — | 16 | 2 | no | — |
| 54 | C24 | C2×C42 | 1 | 6 | yes | — |
| 55 | C24 | C23×C4 | 1 | 3 | yes | — |
| 56 | C24 | H(5,𝔽2) | 1 | 12 | yes | — |
| 57 | #41 | — | 4 | 96 | no | — |
| 58 | C2×Q8 | — | 4 | 6 | no | — |
| 59 | C2×D4 | — | 2 | 24 | no | — |
| 60 | C2×D4 | — | 2 | 24 | no | — |
| 61 | C2×D4 | — | 2 | 24 | no | — |
| 62 | C2×D4 | — | 4 | 12 | no | — |
| 63 | C2×D4 | — | 4 | 6 | no | — |
| 64 | C2×D4 | — | 4 | 96 | no | — |
| 65 | C2×D4 | — | 4 | 36 | no | — |
| 66 | C2×D4 | — | 4 | 12 | no | — |
| 67 | C22×C4 | — | 2 | 6 | no | — |
| 68 | C22×C4 | — | 2 | 12 | no | — |
| 69 | C22×C4 | — | 2 | 24 | no | — |
| 70 | C22×C4 | — | 4 | 6 | no | — |
| 71 | C22×C4 | — | 4 | 12 | no | — |
| 72 | C22×C4 | — | 4 | 12 | no | — |
| 73 | C22×C4 | — | 4 | 12 | no | — |
| 74 | C22×C4 | — | 16 | 12 | no | — |
| 75 | C22×C4 | C2×C42 | 1 | 6 | yes | — |
| 76 | C22⋊C4 | — | 2 | 48 | no | — |
| 77 | C22⋊C4 | — | 4 | 24 | no | — |
| 78 | C22⋊C4 | — | 16 | 12 | no | — |
| 79 | C42 | — | 16 | 8 | no | — |
| 80 | C23 | — | 2 | 6 | no | — |
| 81 | C23 | — | 2 | 12 | no | — |
| 82 | C23 | — | 4 | 8 | no | — |
| 83 | C23 | — | 4 | 12 | no | — |
| 84 | C23 | — | 8 | 12 | no | — |
| 85 | C23 | — | 8 | 3 | no | — |
| 86 | C23 | — | 8 | 6 | no | — |
| 87 | C23 | — | 8 | 12 | no | — |
| 88 | C23 | — | 16 | 12 | no | — |
| 89 | C23 | — | 16 | 12 | no | — |
| 90 | C23 | C22×C42 | 1 | 1 | yes | characteristic |
| 91 | C23 | #518 | 1 | 12 | yes | — |
| 92 | C23 | #518 | 1 | 6 | yes | — |
| 93 | C23 | #560 | 1 | 12 | yes | — |
| 94 | C23 | H(3,𝔽4) | 1 | 4 | yes | — |
| 95 | Q8 | — | 4 | 8 | no | — |
| 96 | D4 | — | 4 | 24 | no | — |
| 97 | D4 | — | 4 | 48 | no | — |
| 98 | C2×C4 | — | 2 | 6 | no | — |
| 99 | C2×C4 | — | 2 | 12 | no | — |
| 100 | C2×C4 | — | 4 | 24 | no | — |
| 101 | C2×C4 | — | 4 | 12 | no | — |
| 102 | C2×C4 | — | 4 | 24 | no | — |
| 103 | C2×C4 | — | 8 | 24 | no | — |
| 104 | C2×C4 | — | 8 | 24 | no | — |
| 105 | C2×C4 | — | 16 | 12 | no | — |
| 106 | C22 | — | 4 | 4 | no | — |
| 107 | C22 | — | 4 | 6 | no | — |
| 108 | C22 | — | 4 | 12 | no | — |
| 109 | C22 | — | 4 | 1 | no | — |
| 110 | C22 | — | 4 | 12 | no | — |
| 111 | C22 | — | 4 | 12 | no | — |
| 112 | C22 | — | 4 | 6 | no | — |
| 113 | C22 | — | 8 | 12 | no | — |
| 114 | C22 | — | 16 | 12 | no | — |
| 115 | C22 | — | 16 | 8 | no | — |
| 116 | C22 | #2273 | 1 | 3 | yes | — |
| 117 | C22 | #2375 | 1 | 12 | yes | — |
| 118 | C4 | — | 4 | 6 | no | — |
| 119 | C4 | — | 8 | 24 | no | — |
| 120 | C2 | — | 2 | 12 | no | — |
| 121 | C2 | — | 2 | 3 | no | — |
| 122 | C2 | — | 8 | 6 | no | — |
| 123 | C2 | #13827 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13827 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10938 | C2 | semidirect |
| 6. #2273 | C22 | semidirect |
| 7. #1782 | C4 | semidirect |
| 8. #1781 | C4 | semidirect |
| 9. C2×#342 | C22 | semidirect |
| 22. C25 | C42 | semidirect |
| 29. C22×Q8 | C42 | semidirect |
| 34. C22×D4 | C42 | semidirect |
| 37. C23×C4 | C42 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20585 | 2 |
| #10938 | 2 |