Finite Group Databasebeta

← #9206

Subgroups — #9206

Order 256.

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

normal subgroups only

This group has 289 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#2618C211yesmaximal, characteristic
2#2313C211yesmaximal, characteristic, complemented
3#1519C212yesmaximal
4C2×#324C211yesmaximal, characteristic, complemented
5#1289C212yesmaximal, complemented
6C2×#123—21no—
7C2×#121C2211yescharacteristic
8C22×C4⋊C4C2211yescharacteristic
9C2×C22⋊C8C2212yes—
10#398C412yescomplemented
11#396—22no—
12#394C412yescomplemented
13#387—24no—
14#386—21no—
15#381—21no—
16#380C2211yescharacteristic
17#380C2212yes—
18C4×C22⋊C4—21no—
19#327—24no—
20#324—22no—
21C22×D4—21no—
22C22×D4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27#123—44no—
28#121—22no—
29#121—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C412yes—
37C2×C4⋊C4D412yes—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—22no—
43C2×C42—22no—
44C22⋊C8—22no—
45C22⋊C8—22no—
46#95—22no—
47#95—24no—
48#95—22no—
49#95C2×C412yes—
50#95D414yes—
51C24—21no—
52C24C2×D411yescharacteristic
53C2×D4—22no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4—42no—
57C2×D4—42no—
58C2×D4—81no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—24no—
68C22×C4—81no—
69C22×C4—81no—
70C22×C4C22⋊C412yes—
71C22×C4C22⋊C412yes—
72C22×C4C22×C411yescharacteristic
73C22×C4C2×D412yes—
74C22×C4C2×D411yescharacteristic
75C22×C4#4112yes—
76C2×C8—42no—
77C2×C8—82no—
78C4⋊C4—24no—
79C4⋊C4—24no—
80C4⋊C4—22no—
81C4⋊C4—44no—
82C4⋊C4—82no—
83C4⋊C4—81no—
84C22⋊C4—44no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C22⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—82no—
90C42—82no—
91C23—41no—
92C23—41no—
93C23—42no—
94C23—42no—
95C23—41no—
96C23—41no—
97C23—41no—
98C23C2×C22⋊C411yescharacteristic
99C23#12011yescharacteristic
100C23#12311yescharacteristic
101C23#13512yes—
102C23AGL(1,ℤ/8)12yes—
103D4—82no—
104D4—82no—
105D4—82no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—41no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—44no—
119C2×C4—42no—
120C2×C4—44no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—44no—
127C2×C4—81no—
128C2×C4C4×D412yes—
129C2×C4#12112yes—
130C8—82no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—42no—
138C22—42no—
139C22—82no—
140C22—82no—
141C22—81no—
142C22—81no—
143C22—81no—
144C22#38611yescharacteristic
145C22#41611yescharacteristic
146C22#41811yescharacteristic
147C22#44912yes—
148C22#48212yes—
149C4—22no—
150C4—22no—
151C4—42no—
152C4—42no—
153C4—82no—
154C4—84no—
155C4—81no—
156C2—21no—
157C2—22no—
158C2—21no—
159C2—22no—
160C2—161no—
161C2#187511yescharacteristic
162C2#191211yescharacteristic
163C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2313C2semidirect
4. C2×#324C2semidirect
5. #1289C2semidirect
10. #398C4semidirect
12. #394C4semidirect

Maximal subgroups

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

subgroupindex
#26182
#23132
#15192
C2×#3242
#12892