Finite Group Databasebeta

← #8918

Subgroups — #8918

Order 256.

196 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 59 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#2980C211yesmaximal, characteristic
2#1963C211yesmaximal, characteristic
3#1569C212yesmaximal, complemented
4#1525C211yesmaximal, characteristic
5#1300C212yesmaximal, complemented
6#532C2211yescharacteristic
7#435—22no—
8#435—22no—
9#432C2212yes—
10#423C2211yescharacteristic
11C2×#104—21no—
12C2×#104—21no—
13#408—21no—
14#408—21no—
15C4×M16C2212yes—
16#376C2211yescharacteristic
17#335—22no—
18#334—22no—
19#329C411yescharacteristic, complemented
20#329C412yescomplemented
21#329C411yescharacteristic, complemented
22C2×#41D411yescharacteristic
23C2×#41D411yescharacteristic
24C2×M16—22no—
25C2×M16—21no—
26C2×M16—21no—
27C2×M16—21no—
28C22×C8—21no—
29C22×C8—21no—
30C4⋊Q8Q811yescharacteristic
31#127Q811yescharacteristic
32#124C2×C411yescharacteristic
33#124C2×C411yescharacteristic
34#121—22no—
35C4×D4—22no—
36C2×C42—21no—
37C2×C42C2311yescharacteristic
38C4⋊C8—22no—
39C4⋊C8D412yes—
40C4⋊C8D412yes—
41#104—42no—
42#104—42no—
43C22⋊C8—22no—
44C22⋊C8—22no—
45C22⋊C8—22no—
46#97—22no—
47#97C2×C412yes—
48#97C2×C412yes—
49C4×C8—22no—
50#95—22no—
51#41—41no—
52#41—41no—
53#41—41no—
54#41—41no—
55C2×Q8C4⋊C411yescharacteristic
56C2×Q8C4⋊C411yescharacteristic
57C2×D4—21no—
58C2×D4—21no—
59C2×D4—22no—
60C2×D4C4⋊C411yescharacteristic
61C2×D4C4⋊C411yescharacteristic
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—41no—
65C22×C4C2×D411yescharacteristic
66C22×C4#4112yes—
67M16—44no—
68M16—41no—
69M16—41no—
70M16—81no—
71M16—81no—
72C2×C8—22no—
73C2×C8—22no—
74C2×C8—22no—
75C2×C8—22no—
76C2×C8—41no—
77C2×C8—81no—
78C2×C8—81no—
79C4⋊C4—22no—
80C22⋊C4—22no—
81C22⋊C4—22no—
82C42—22no—
83C42—41no—
84C42C22×C411yescharacteristic
85C42C2×D412yes—
86C42C2×Q811yescharacteristic
87C23—21no—
88C23—21no—
89C23#12311yescharacteristic
90Q8—41no—
91Q8—41no—
92D4—41no—
93D4—41no—
94D4—82no—
95D4—81no—
96D4—81no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—42no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—42no—
107C2×C4—82no—
108C2×C4C2×C4⋊C411yescharacteristic
109C2×C4C4×D412yes—
110C2×C4#12011yescharacteristic
111C2×C4C22⋊Q812yes—
112C8—24no—
113C8—42no—
114C8—42no—
115C8—42no—
116C8—81no—
117C22—21no—
118C22—22no—
119C22—41no—
120C22—41no—
121C22—42no—
122C22—42no—
123C22#38511yescharacteristic
124C4—21no—
125C4—22no—
126C4—22no—
127C4—41no—
128C4—41no—
129C4—82no—
130C4#45311yescharacteristic
131C4#45511yescharacteristic
132C2—41no—
133C2—81no—
134C2—81no—
135C2#185611yescharacteristic
136C2#186112yes—
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#385
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1569C2semidirect
5. #1300C2semidirect
19. #329C4semidirect
20. #329C4semidirect
21. #329C4semidirect

Maximal subgroups

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

subgroupindex
#29802
#19632
#15692
#15252
#13002