Finite Group Databasebeta

← #12256

Subgroups — #12256

Order 256.

221 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 37 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#2551C211yesmaximal, characteristic
2#2060C211yesmaximal, characteristic
3#1605C212yesmaximal
4#1354C212yesmaximal
5#1325C211yesmaximal, characteristic
6#544C2211yescharacteristic
7C2×C4⋊Q8—22no—
8#451—22no—
9#428—21no—
10#395—21no—
11#395C2212yes—
12#389—22no—
13#384C2211yescharacteristic
14C4×C4⋊C4—21no—
15#343C2211yescharacteristic
16#339—21no—
17#328—22no—
18#324C2212yes—
19C22×Q8—21no—
20C22×Q8D412yes—
21C2×Q16—42no—
22C2×M16—21no—
23C4⋊Q8—41no—
24C4⋊Q8—48no—
25#125—41no—
26C22⋊Q8—22no—
27#117—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4D411yescharacteristic
34C2×C4⋊C4C2311yescharacteristic
35C2×C42—22no—
36C2×C42D411yescharacteristic
37#107—41no—
38#106—41no—
39#103—42no—
40C22⋊C8—22no—
41#95—21no—
42#95—22no—
43#95D412yes—
44C2×Q8—22no—
45C2×Q8—44no—
46C2×Q8—44no—
47C2×Q8—42no—
48C2×Q8—81no—
49C22×C4—21no—
50C22×C4—22no—
51C22×C4—22no—
52C22×C4—21no—
53C22×C4C2×D411yescharacteristic
54C22×C4C2×D412yes—
55Q16—84no—
56M16—22no—
57M16—41no—
58C2×C8—41no—
59C2×C8—42no—
60C4⋊C4—24no—
61C4⋊C4—24no—
62C4⋊C4—22no—
63C4⋊C4—21no—
64C4⋊C4—21no—
65C4⋊C4—41no—
66C4⋊C4—41no—
67C4⋊C4—44no—
68C4⋊C4—44no—
69C4⋊C4—44no—
70C4⋊C4—42no—
71C22⋊C4—41no—
72C42—21no—
73C42—44no—
74C42—41no—
75C42Q1612yescomplemented
76C23#12011yescharacteristic
77Q8—44no—
78Q8—82no—
79Q8—84no—
80Q8—82no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—22no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—24no—
87C2×C4—24no—
88C2×C4—22no—
89C2×C4—42no—
90C2×C4—42no—
91C2×C4—42no—
92C2×C4—42no—
93C2×C4—42no—
94C2×C4—42no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4C2×Q1611yescharacteristic
99C2×C4#13711yescharacteristic
100C8—42no—
101C8—82no—
102C22—21no—
103C22—21no—
104C22#45111yescharacteristic
105C22#45611yescharacteristic
106C22B(4,𝔽2)11yescharacteristic
107C4—24no—
108C4—21no—
109C4—21no—
110C4—44no—
111C4—42no—
112C4—44no—
113C4—81no—
114C4—81no—
115C4—81no—
116C2—21no—
117C2—21no—
118C2#160511yescharacteristic
119C2#219611yescharacteristic
120C2#220511yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC23#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
75. C42Q16semidirect

Maximal subgroups

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

subgroupindex
#25512
#20602
#16052
#13542
#13252