Finite Group Databasebeta

← #11770

Subgroups — #11770

Order 256.

158 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3176C211yesmaximal, characteristic
2#2166C211yesmaximal, characteristic, complemented
3#1747C211yesmaximal, characteristic, complemented
4#1341C212yesmaximal, complemented
5#1339C212yesmaximal, complemented
6#540—21no—
7#499C411yescharacteristic, complemented
8#493C411yescharacteristic, complemented
9#490—21no—
10#469C2212yes—
11#466—21no—
12#442C412yescomplemented
13C4×D8—21no—
14#405—21no—
15#405C2211yescharacteristic
16#364—21no—
17#349C2212yes—
18#347—21no—
19#338C2211yescharacteristic
20H(3,ℤ/4)—21no—
21H(3,ℤ/4)C2211yescharacteristic
22C2×D8—21no—
23C2×D8C2×C412yes—
24C2×M16—21no—
25C2×M16C2311yescharacteristic
26C22×C8D411yescharacteristic
27#127—21no—
28#125—21no—
29#123—42no—
30#121—22no—
31#121—42no—
32C4×D4—22no—
33#117—21no—
34#117D411yescharacteristic
35M32—22no—
36#107—21no—
37#106C2×C412yes—
38#102—22no—
39#102—22no—
40#97—22no—
41#97C2×C411yescharacteristic
42#97D411yescharacteristic
43C4×C8—22no—
44C4×C8C2×C411yescharacteristic
45C4×C8D411yescharacteristic
46C2×D4—22no—
47C2×D4—42no—
48C2×D4—42no—
49C22×C4—42no—
50C22×C4C2×D411yescharacteristic
51D8—42no—
52D8—41no—
53D8—41no—
54M16—41no—
55M16—41no—
56M16—41no—
57C2×C8—21no—
58C2×C8—21no—
59C2×C8—22no—
60C2×C8C22⋊C411yescharacteristic
61C2×C8C22⋊C411yescharacteristic
62C2×C8C22×C411yescharacteristic
63C2×C8C2×D411yescharacteristic
64C4⋊C4—21no—
65C4⋊C4—22no—
66C4⋊C4—41no—
67C4⋊C4—42no—
68C22⋊C4—21no—
69C22⋊C4—42no—
70C22⋊C4—41no—
71C22⋊C4—42no—
72C42—22no—
73C42C22⋊C411yescharacteristic
74C42C22⋊C411yescharacteristic
75C16—42no—
76C23—42no—
77C23AGL(1,ℤ/8)11yescharacteristic
78D4—42no—
79D4—42no—
80D4—162no—
81D4—162no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—162no—
89C2×C4C2×C22⋊C411yescharacteristic
90C2×C4AGL(1,ℤ/8)11yescharacteristic
91C8—21no—
92C8—21no—
93C8—21no—
94C8—42no—
95C8#10411yescharacteristic
96C8#10411yescharacteristic
97C22—21no—
98C22—41no—
99C22—82no—
100C22—82no—
101C22—82no—
102C22#41811yescharacteristic
103C4—21no—
104C4—41no—
105C4—41no—
106C4—82no—
107C4—82no—
108C4#41011yescharacteristic
109C4C2×#10411yescharacteristic
110C2—21no—
111C2—41no—
112C2—162no—
113C2#150511yescharacteristic
subgroupG/H
center, socleC2#1505
commutatorC2×C8C22×C4
Frattini, cosocleC2×M16C23
prime radical#3176C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2166C2semidirect
3. #1747C2semidirect
4. #1341C2semidirect
5. #1339C2semidirect
7. #499C4semidirect
8. #493C4semidirect
12. #442C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#31762
#21662
#17472
#13412
#13392