Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
642 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 642 conjugacy classes of subgroups — too many to draw. Only its 111 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 | C2×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#454 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #2202 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 4 | C2×#355 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×2−1+4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×AGL(1,ℤ/8) | — | 2 | 3 | no | — |
| 7 | H(3,𝔽4) | — | 2 | 6 | no | — |
| 8 | H(3,𝔽4) | C22 | 1 | 4 | yes | complemented |
| 9 | C2×#124 | — | 2 | 3 | no | — |
| 10 | C2×#124 | C22 | 1 | 3 | yes | — |
| 11 | C2×#120 | — | 2 | 3 | no | — |
| 12 | #454 | — | 2 | 6 | no | — |
| 13 | #454 | C22 | 1 | 12 | yes | complemented |
| 14 | C2×#104 | — | 2 | 3 | no | — |
| 15 | C2×#101 | C22 | 1 | 3 | yes | complemented |
| 16 | #355 | C22 | 1 | 12 | yes | complemented |
| 17 | C25 | — | 2 | 1 | no | — |
| 18 | 2−1+4 | — | 2 | 3 | no | — |
| 19 | 2−1+4 | — | 4 | 2 | no | — |
| 20 | 2−1+4 | C23 | 1 | 2 | yes | complemented |
| 21 | C2×#41 | — | 2 | 3 | no | — |
| 22 | C2×#41 | — | 4 | 1 | no | — |
| 23 | C22×Q8 | — | 2 | 1 | no | — |
| 24 | C22×Q8 | — | 4 | 1 | no | — |
| 25 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 26 | C22×D4 | — | 2 | 3 | no | — |
| 27 | C22×D4 | D4 | 1 | 3 | yes | — |
| 28 | AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 29 | AGL(1,ℤ/8) | — | 4 | 6 | no | — |
| 30 | C2×SD16 | — | 4 | 3 | no | — |
| 31 | C2×D8 | — | 4 | 3 | no | — |
| 32 | C2×M16 | — | 2 | 3 | no | — |
| 33 | #124 | — | 2 | 12 | no | — |
| 34 | #124 | — | 2 | 6 | no | — |
| 35 | #124 | — | 2 | 12 | no | — |
| 36 | #124 | D4 | 1 | 6 | yes | — |
| 37 | #124 | C23 | 1 | 6 | yes | — |
| 38 | #120 | — | 2 | 12 | no | — |
| 39 | #120 | — | 2 | 12 | no | — |
| 40 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 41 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 42 | C2×C42 | D4 | 1 | 3 | yes | — |
| 43 | #104 | — | 2 | 12 | no | — |
| 44 | #101 | — | 2 | 3 | no | — |
| 45 | #101 | C23 | 1 | 6 | yes | complemented |
| 46 | C24 | — | 2 | 4 | no | — |
| 47 | C24 | — | 2 | 3 | no | — |
| 48 | C24 | — | 8 | 4 | no | — |
| 49 | C24 | — | 8 | 1 | no | — |
| 50 | C24 | — | 8 | 1 | no | — |
| 51 | #41 | — | 2 | 6 | no | — |
| 52 | #41 | — | 2 | 6 | no | — |
| 53 | #41 | — | 4 | 3 | no | — |
| 54 | #41 | — | 4 | 2 | no | — |
| 55 | #41 | — | 4 | 6 | no | — |
| 56 | #41 | — | 4 | 3 | no | — |
| 57 | C2×Q8 | — | 2 | 4 | no | — |
| 58 | C2×Q8 | — | 2 | 3 | no | — |
| 59 | C2×Q8 | — | 4 | 1 | no | — |
| 60 | C2×Q8 | — | 4 | 1 | no | — |
| 61 | C2×Q8 | — | 4 | 2 | no | — |
| 62 | C2×Q8 | — | 4 | 6 | no | — |
| 63 | C2×Q8 | — | 8 | 2 | no | — |
| 64 | C2×Q8 | C2×D4 | 1 | 3 | yes | — |
| 65 | C2×Q8 | C24 | 1 | 1 | yes | characteristic |
| 66 | C2×D4 | — | 2 | 3 | no | — |
| 67 | C2×D4 | — | 2 | 12 | no | — |
| 68 | C2×D4 | — | 2 | 3 | no | — |
| 69 | C2×D4 | — | 4 | 3 | no | — |
| 70 | C2×D4 | — | 4 | 3 | no | — |
| 71 | C2×D4 | — | 4 | 1 | no | — |
| 72 | C2×D4 | — | 8 | 6 | no | — |
| 73 | C2×D4 | C2×D4 | 1 | 6 | yes | — |
| 74 | C22×C4 | — | 2 | 3 | no | — |
| 75 | C22×C4 | — | 4 | 3 | no | — |
| 76 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 77 | SD16 | — | 4 | 12 | no | — |
| 78 | D8 | — | 4 | 12 | no | — |
| 79 | M16 | — | 2 | 6 | no | — |
| 80 | M16 | — | 4 | 3 | no | — |
| 81 | C2×C8 | — | 4 | 3 | no | — |
| 82 | C22⋊C4 | — | 2 | 12 | no | — |
| 83 | C22⋊C4 | — | 2 | 12 | no | — |
| 84 | C22⋊C4 | — | 2 | 12 | no | — |
| 85 | C42 | — | 2 | 3 | no | — |
| 86 | C42 | C2×D4 | 1 | 6 | yes | — |
| 87 | C23 | — | 2 | 6 | no | — |
| 88 | C23 | — | 4 | 12 | no | — |
| 89 | C23 | — | 4 | 3 | no | — |
| 90 | C23 | — | 4 | 1 | no | — |
| 91 | C23 | — | 4 | 3 | no | — |
| 92 | C23 | — | 8 | 8 | no | — |
| 93 | C23 | — | 8 | 4 | no | — |
| 94 | C23 | — | 8 | 8 | no | — |
| 95 | C23 | — | 8 | 8 | no | — |
| 96 | C23 | #120 | 1 | 1 | yes | characteristic |
| 97 | Q8 | — | 2 | 6 | no | — |
| 98 | Q8 | — | 4 | 3 | no | — |
| 99 | Q8 | — | 4 | 1 | no | — |
| 100 | Q8 | — | 4 | 3 | no | — |
| 101 | Q8 | — | 8 | 4 | no | — |
| 102 | D4 | — | 2 | 6 | no | — |
| 103 | D4 | — | 4 | 6 | no | — |
| 104 | D4 | — | 4 | 3 | no | — |
| 105 | D4 | — | 4 | 2 | no | — |
| 106 | D4 | — | 4 | 2 | no | — |
| 107 | D4 | — | 8 | 3 | no | — |
| 108 | D4 | — | 8 | 12 | no | — |
| 109 | C2×C4 | — | 2 | 3 | no | — |
| 110 | C2×C4 | — | 2 | 3 | no | — |
| 111 | C2×C4 | — | 2 | 6 | no | — |
| 112 | C2×C4 | — | 4 | 6 | no | — |
| 113 | C2×C4 | — | 4 | 6 | no | — |
| 114 | C2×C4 | — | 4 | 6 | no | — |
| 115 | C2×C4 | — | 4 | 1 | no | — |
| 116 | C2×C4 | #120 | 1 | 3 | yes | — |
| 117 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 118 | C8 | — | 4 | 6 | no | — |
| 119 | C22 | — | 2 | 2 | no | — |
| 120 | C22 | — | 4 | 12 | no | — |
| 121 | C22 | — | 4 | 2 | no | — |
| 122 | C22 | — | 4 | 6 | no | — |
| 123 | C22 | — | 8 | 6 | no | — |
| 124 | C22 | — | 8 | 16 | no | — |
| 125 | C22 | — | 8 | 2 | no | — |
| 126 | C22 | — | 8 | 6 | no | — |
| 127 | C22 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 128 | C22 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 129 | C22 | C2×#120 | 1 | 1 | yes | characteristic |
| 130 | C4 | — | 2 | 3 | no | — |
| 131 | C4 | — | 2 | 3 | no | — |
| 132 | C4 | — | 4 | 6 | no | — |
| 133 | C4 | — | 4 | 2 | no | — |
| 134 | C2 | — | 2 | 1 | no | — |
| 135 | C2 | — | 2 | 1 | no | — |
| 136 | C2 | — | 8 | 6 | no | — |
| 137 | C2 | — | 8 | 2 | no | — |
| 138 | C2 | #2202 | 1 | 2 | yes | direct product factor |
| 139 | C2 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×H(3,𝔽4) | C2 | semidirect |
| 2. C2×#454 | C2 | semidirect |
| 3. #2202 | C2 | direct (N × H) |
| 4. C2×#355 | C2 | semidirect |
| 5. C2×2−1+4 | C22 | semidirect |
| 8. H(3,𝔽4) | C22 | semidirect |
| 13. #454 | C22 | semidirect |
| 15. C2×#101 | C22 | semidirect |
| 16. #355 | C22 | semidirect |
| 20. 2−1+4 | C23 | semidirect |
| 45. #101 | C23 | semidirect |
| 138. C2 | #2202 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(3,𝔽4) | 2 |
| C2×#454 | 2 |
| #2202 | 2 |
| C2×#355 | 2 |