Finite Group Databasebeta

← #12247

Subgroups — #12247

Order 256.

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

normal subgroups only

This group has 197 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#2582C211yesmaximal, characteristic
2#2060C211yesmaximal, characteristic
3#1611C212yesmaximal
4#1346C212yesmaximal
5#1325C211yesmaximal, characteristic
6#544C2211yescharacteristic
7C2×C4×Q8—21no—
8#483—22no—
9#428—21no—
10#398—21no—
11#398C2212yes—
12#391—22no—
13#389—22no—
14#384C2211yescharacteristic
15#343C2211yescharacteristic
16#339—21no—
17#328—22no—
18#324C2212yes—
19C22×Q8—21no—
20C2×M16—21no—
21C4⋊Q8—41no—
22#125—41no—
23C22⋊Q8—22no—
24C4×Q8—42no—
25C4×Q8—42no—
26#117—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4D411yescharacteristic
32C2×C4⋊C4D412yes—
33C2×C4⋊C4C2311yescharacteristic
34C2×C42—22no—
35C2×C42D411yescharacteristic
36#107—41no—
37#106—41no—
38#106—42no—
39#103—42no—
40C22⋊C8—22no—
41#95—22no—
42#95—21no—
43#95—22no—
44#95D412yes—
45C2×Q8—24no—
46C2×Q8—22no—
47C2×Q8—42no—
48C2×Q8—42no—
49C22×C4—21no—
50C22×C4—22no—
51C22×C4—21no—
52C22×C4—22no—
53C22×C4C2×D411yescharacteristic
54C22×C4C2×D412yes—
55M16—22no—
56M16—41no—
57C2×C8—41no—
58C2×C8—42no—
59C4⋊C4—21no—
60C4⋊C4—22no—
61C4⋊C4—21no—
62C4⋊C4—41no—
63C4⋊C4—42no—
64C4⋊C4—41no—
65C4⋊C4—44no—
66C4⋊C4—42no—
67C4⋊C4—42no—
68C4⋊C4—42no—
69C4⋊C4—42no—
70C4⋊C4—81no—
71C22⋊C4—41no—
72C42—21no—
73C42—44no—
74C42—41no—
75C42SD1612yes—
76C23#12011yescharacteristic
77Q8—48no—
78Q8—82no—
79C2×C4—22no—
80C2×C4—22no—
81C2×C4—22no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—24no—
85C2×C4—22no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—42no—
91C2×C4—42no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—42no—
95C2×C4—41no—
96C2×C4C2×SD1611yescharacteristic
97C2×C4AGL(1,ℤ/8)11yescharacteristic
98C8—42no—
99C8—82no—
100C22—21no—
101C22—21no—
102C22#45011yescharacteristic
103C22#45611yescharacteristic
104C22B(4,𝔽2)11yescharacteristic
105C4—24no—
106C4—21no—
107C4—21no—
108C4—44no—
109C4—42no—
110C4—82no—
111C4—81no—
112C4—81no—
113C4—81no—
114C2—21no—
115C2—21no—
116C2#160411yescharacteristic
117C2#219811yescharacteristic
118C2#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

Maximal subgroups

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

subgroupindex
#25822
#20602
#16112
#13462
#13252