Finite Group Databasebeta

← #12213

Subgroups — #12213

Order 256.

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

normal subgroups only

This group has 190 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#2734C211yesmaximal, characteristic
2#2028C211yesmaximal, characteristic
3#1611C212yesmaximal
4#1346C212yesmaximal
5#1319C211yesmaximal, characteristic
6#544C2211yescharacteristic
7C2×C4⋊Q8—21no—
8#483—22no—
9#419—21no—
10#398C2212yes—
11#395—22no—
12#391C2211yescharacteristic
13#389—21no—
14#387—22no—
15#343C2211yescharacteristic
16#339—21no—
17#328—22no—
18#324C2212yes—
19C22×Q8—21no—
20C22×Q8D411yescharacteristic
21C2×M16—21no—
22C4⋊Q8—41no—
23C4⋊Q8—42no—
24C4⋊Q8—42no—
25#125—41no—
26C22⋊Q8—22no—
27#117—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4D412yes—
33C2×C4⋊C4C2311yescharacteristic
34C2×C42—21no—
35C2×C42D411yescharacteristic
36#106—42no—
37#103—41no—
38#103—41no—
39#103—42no—
40C22⋊C8—22no—
41#95—22no—
42#95—22no—
43#95—22no—
44#95D412yes—
45C2×Q8—21no—
46C2×Q8—22no—
47C2×Q8—42no—
48C2×Q8—41no—
49C2×Q8—41no—
50C2×Q8—81no—
51C2×Q8—81no—
52C2×Q8SD1612yes—
53C22×C4—21no—
54C22×C4—22no—
55C22×C4—21no—
56C22×C4—22no—
57C22×C4C2×D411yescharacteristic
58C22×C4C2×D412yes—
59M16—41no—
60M16—41no—
61C2×C8—42no—
62C2×C8—41no—
63C4⋊C4—24no—
64C4⋊C4—21no—
65C4⋊C4—21no—
66C4⋊C4—22no—
67C4⋊C4—41no—
68C4⋊C4—42no—
69C4⋊C4—44no—
70C4⋊C4—42no—
71C4⋊C4—42no—
72C4⋊C4—42no—
73C4⋊C4—81no—
74C22⋊C4—41no—
75C42—41no—
76C42—42no—
77C42—41no—
78C23#12011yescharacteristic
79Q8—42no—
80Q8—81no—
81Q8—82no—
82Q8—84no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—22no—
86C2×C4—24no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—41no—
90C2×C4—42no—
91C2×C4—42no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—42no—
101C2×C4C2×SD1611yescharacteristic
102C2×C4AGL(1,ℤ/8)11yescharacteristic
103C8—81no—
104C8—82no—
105C22—21no—
106C22—21no—
107C22#45011yescharacteristic
108C22#45511yescharacteristic
109C22B(4,𝔽2)11yescharacteristic
110C4—21no—
111C4—21no—
112C4—44no—
113C4—42no—
114C4—42no—
115C4—42no—
116C4—82no—
117C4—81no—
118C4—81no—
119C2—21no—
120C2—21no—
121C2#160111yescharacteristic
122C2#220111yescharacteristic
123C2#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
#27342
#20282
#16112
#13462
#13192