Finite Group Databasebeta

← #9192

Subgroups — #9192

Order 256.

229 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 67 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#2921C211yesmaximal, characteristic
2#2628C211yesmaximal, characteristic
3#1548C212yesmaximal, complemented
4#1458C211yesmaximal, characteristic
5#1286C212yesmaximal, complemented
6C2×C4⋊Q8C2211yescharacteristic
7C2×#117C2211yescharacteristic
8#433—24no—
9#431C2212yes—
10#423C2212yes—
11#407—21no—
12C4×M16—21no—
13#400—21no—
14#399—21no—
15#395C412yes—
16#393C412yes—
17#389—21no—
18#382C2211yescharacteristic
19#380—21no—
20#333—24no—
21#324—21no—
22#323—21no—
23C22×Q8C2×C412yes—
24C23×C4D411yescharacteristic
25C2×M16—22no—
26C4⋊Q8—22no—
27C4⋊Q8—22no—
28#117—22no—
29#117—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—22no—
37C2×C22⋊C4C2×C411yescharacteristic
38C2×C22⋊C4D411yescharacteristic
39C2×C42D411yescharacteristic
40C2×C42C2311yescharacteristic
41C4⋊C8—22no—
42C4⋊C8—22no—
43C4⋊C8D414yes—
44C22⋊C8—24no—
45C22⋊C8—22no—
46#97—22no—
47C4×C8—22no—
48C4×C8—22no—
49#95—22no—
50#95—22no—
51#95—21no—
52#95C2×C412yes—
53C24C22⋊C411yescharacteristic
54C2×Q8—24no—
55C2×Q8—82no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22×C411yescharacteristic
66C22×C4C2×D411yescharacteristic
67C22×C4C2×D411yescharacteristic
68M16—42no—
69M16—42no—
70C2×C8—24no—
71C2×C8—42no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—41no—
75C4⋊C4—42no—
76C4⋊C4—41no—
77C4⋊C4—42no—
78C4⋊C4—82no—
79C22⋊C4—22no—
80C22⋊C4—22no—
81C22⋊C4—82no—
82C42—21no—
83C42—21no—
84C42C2×D412yes—
85C42#4112yes—
86C23—21no—
87C23—21no—
88C23—21no—
89C23—41no—
90C23—41no—
91C23C2×C22⋊C411yescharacteristic
92Q8—84no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4C4×D412yes—
111C2×C4#12011yescharacteristic
112C2×C4#12112yes—
113C2×C4#12311yescharacteristic
114C2×C4#13512yes—
115C2×C4#13712yes—
116C8—44no—
117C8—82no—
118C22—21no—
119C22—21no—
120C22—42no—
121C22—41no—
122C22—41no—
123C22—41no—
124C22—81no—
125C22#38611yescharacteristic
126C22#41611yescharacteristic
127C22#41911yescharacteristic
128C4—22no—
129C4—22no—
130C4—41no—
131C4—41no—
132C4—41no—
133C4—82no—
134C4—82no—
135C4#45212yes—
136C4#48512yes—
137C2—21no—
138C2—21no—
139C2—81no—
140C2#187311yescharacteristic
141C2#190911yescharacteristic
142C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ 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
3. #1548C2semidirect
5. #1286C2semidirect

Maximal subgroups

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

subgroupindex
#29212
#26282
#15482
#14582
#12862