Finite Group Databasebeta

← #8532

Subgroups — #8532

Order 256.

400 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 75 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#3548C211yesmaximal, characteristic
2#1994C211yesmaximal, characteristic, complemented
3#1962C212yesmaximal, complemented
4#1561C211yesmaximal, characteristic, complemented
5#1560C212yesmaximal, complemented
6#562C414yescomplemented
7#554—22no—
8#550—22no—
9#547—22no—
10#540—22no—
11#537—24no—
12#533C2211yescharacteristic
13#532C2212yes—
14#423C2211yescharacteristic
15#423C2212yes—
16C2×#104—24no—
17C43C2211yescharacteristic
18#339—22no—
19#333—22no—
20#332—22no—
21#331—22no—
22C2×#41—24no—
23C2×#41D414yes—
24C2×M16—21no—
25C2×M16—22no—
26C4⋊Q8D412yes—
27C4⋊Q8D412yes—
28#127—22no—
29#127D412yes—
30#125—22no—
31#125C2×C412yes—
32#124C2×C414yes—
33#123—28no—
34C22⋊Q8—24no—
35C22⋊Q8—28no—
36C22⋊Q8—22no—
37#121—28no—
38#121—24no—
39#121—28no—
40#121—22no—
41#121—22no—
42C4×Q8—22no—
43C4×D4—24no—
44C4×D4—22no—
45C4×D4—22no—
46#117D412yes—
47C2×C4⋊C4—24no—
48C2×C42—21no—
49C2×C42—22no—
50C2×C42C2311yescharacteristic
51C4⋊C8—22no—
52C4⋊C8—22no—
53C4⋊C8—22no—
54#104—48no—
55#41—44no—
56#41—44no—
57#41—84no—
58#41—84no—
59C2×Q8—22no—
60C2×Q8C22⋊C414yes—
61C2×D4—28no—
62C2×D4—24no—
63C2×D4—22no—
64C2×D4—22no—
65C2×D4C22⋊C414yes—
66C22×C4—28no—
67C22×C4—24no—
68C22×C4—41no—
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D412yes—
71M16—44no—
72M16—42no—
73C2×C8—42no—
74C2×C8—41no—
75C4⋊C4—28no—
76C4⋊C4—24no—
77C4⋊C4—22no—
78C4⋊C4—22no—
79C4⋊C4—22no—
80C4⋊C4C22⋊C414yes—
81C22⋊C4—28no—
82C22⋊C4—22no—
83C22⋊C4—24no—
84C22⋊C4—24no—
85C42—22no—
86C42—24no—
87C42—22no—
88C42—41no—
89C42—42no—
90C42C22×C411yescharacteristic
91C42C2×D411yescharacteristic
92C42C2×D412yes—
93C23—24no—
94C23#12011yescharacteristic
95Q8—44no—
96Q8—42no—
97Q8—42no—
98D4—44no—
99D4—42no—
100D4—42no—
101D4—42no—
102D4—42no—
103D4—84no—
104D4—88no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—21no—
109C2×C4—24no—
110C2×C4—24no—
111C2×C4—44no—
112C2×C4—42no—
113C2×C4—44no—
114C2×C4—44no—
115C2×C4—88no—
116C2×C4C2×C22⋊C411yescharacteristic
117C2×C4C2×C22⋊C412yes—
118C2×C4#12012yes—
119C2×C4#12011yescharacteristic
120C8—82no—
121C8—81no—
122C22—22no—
123C22—21no—
124C22—44no—
125C22—44no—
126C22—44no—
127C22#37911yescharacteristic
128C4—22no—
129C4—21no—
130C4—44no—
131C4—44no—
132C4—44no—
133C4#45312yes—
134C4#45411yescharacteristic
135C4#45512yes—
136C4#45611yescharacteristic
137C2—41no—
138C2—84no—
139C2#179212yes—
140C2#179311yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1994C2semidirect
3. #1962C2semidirect
4. #1561C2semidirect
5. #1560C2semidirect
6. #562C4semidirect

Maximal subgroups

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

subgroupindex
#35482
#19942
#19622
#15612
#15602