Finite Group Databasebeta

← #9458

Subgroups — #9458

Order 256.

268 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 63 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#2983C211yesmaximal, characteristic, complemented
2#2548C211yesmaximal, characteristic
3#1555C212yesmaximal
4#1467C211yesmaximal, characteristic, complemented
5#1285C212yesmaximal, complemented
6C2×#127C2211yescharacteristic
7C2×#124C412yes—
8C2×C4×Q8C2211yescharacteristic
9#446—24no—
10#432—21no—
11#432C2212yes—
12#432C2212yes—
13#431—21no—
14#394—21no—
15#394—21no—
16#390—21no—
17#390—21no—
18#390C412yes—
19C4×C4⋊C4C2211yescharacteristic
20#331—24no—
21#324—21no—
22#324—21no—
23C22×Q8C2×C411yescharacteristic
24C22×D4—21no—
25C22×D4—21no—
26C22×D4C2×C412yes—
27#127—22no—
28#127—22no—
29#124—44no—
30C4×Q8—22no—
31C4×Q8—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4D411yescharacteristic
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C42C2×C412yes—
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C2×C42C2311yescharacteristic
44C4⋊C8—22no—
45C4⋊C8—22no—
46C4⋊C8—22no—
47C4⋊C8D414yes—
48C22⋊C8—22no—
49C22⋊C8—22no—
50#97—22no—
51#97—22no—
52#97—22no—
53C4×C8—22no—
54#95—21no—
55C24—22no—
56C2×Q8—22no—
57C2×Q8—22no—
58C2×Q8—41no—
59C2×D4—24no—
60C2×D4—42no—
61C2×D4—42no—
62C2×D4—42no—
63C2×D4—42no—
64C2×D4—41no—
65C2×D4—42no—
66C2×D4—41no—
67C2×D4—82no—
68C22×C4—22no—
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22×C411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C2×C8—24no—
77C2×C8—42no—
78C4⋊C4—22no—
79C4⋊C4—22no—
80C4⋊C4—22no—
81C4⋊C4—42no—
82C22⋊C4—44no—
83C22⋊C4—44no—
84C22⋊C4—82no—
85C22⋊C4—82no—
86C42—21no—
87C42—21no—
88C42—21no—
89C42—21no—
90C42—42no—
91C42C2×D412yes—
92C42#4112yes—
93C23—42no—
94C23—42no—
95C23—42no—
96C23—42no—
97C23—42no—
98C23—42no—
99C23—42no—
100C23C2×C22⋊C411yescharacteristic
101Q8—44no—
102D4—84no—
103D4—82no—
104D4—84no—
105D4—82no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4#10012yes—
123C2×C4C4×D412yes—
124C2×C4#12011yescharacteristic
125C2×C4#12112yes—
126C2×C4#12311yescharacteristic
127C8—44no—
128C8—82no—
129C22—21no—
130C22—21no—
131C22—82no—
132C22—82no—
133C22—82no—
134C22—82no—
135C22—82no—
136C22—82no—
137C22—82no—
138C22#38611yescharacteristic
139C22C2×#10011yescharacteristic
140C22#41311yescharacteristic
141C4—21no—
142C4—22no—
143C4—22no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—41no—
148C4—82no—
149C4#45412yes—
150C2—21no—
151C2—21no—
152C2—162no—
153C2#188811yescharacteristic
154C2#190011yescharacteristic
155C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
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
1. #2983C2semidirect
4. #1467C2semidirect
5. #1285C2semidirect

Maximal subgroups

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

subgroupindex
#29832
#25482
#15552
#14672
#12852