Finite Group Databasebeta

← #9202

Subgroups — #9202

Order 256.

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

normal subgroups only

This group has 258 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#2627C211yesmaximal, characteristic
2#2313C211yesmaximal, characteristic
3#1520C212yesmaximal
4C2×#324C211yesmaximal, characteristic
5#1289C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8C2×C22⋊C8C2212yes—
9#400C412yescomplemented
10#398—22no—
11#395C412yescomplemented
12#387—24no—
13#385—21no—
14#381—21no—
15#381—21no—
16#380C2211yescharacteristic
17#380C2212yes—
18C4×C22⋊C4—21no—
19#328—24no—
20#324—22no—
21C22×Q8C2×C411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26C22⋊Q8—22no—
27C22⋊Q8—22no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C412yes—
36C2×C4⋊C4D412yes—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—21no—
40C2×C42—22no—
41C22⋊C8—22no—
42C22⋊C8—22no—
43#95—22no—
44#95—24no—
45#95—22no—
46#95—22no—
47#95—22no—
48#95C2×C412yes—
49#95D414yes—
50C24C2×D411yescharacteristic
51C2×Q8—22no—
52C2×Q8—81no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—21no—
62C22×C4—24no—
63C22×C4—81no—
64C22×C4—81no—
65C22×C4C22⋊C412yes—
66C22×C4C22⋊C412yes—
67C22×C4C22×C411yescharacteristic
68C22×C4C2×D412yes—
69C22×C4C2×D411yescharacteristic
70C22×C4#4112yes—
71C2×C8—42no—
72C2×C8—82no—
73C4⋊C4—24no—
74C4⋊C4—24no—
75C4⋊C4—22no—
76C4⋊C4—42no—
77C4⋊C4—82no—
78C4⋊C4—81no—
79C4⋊C4—82no—
80C22⋊C4—41no—
81C22⋊C4—41no—
82C22⋊C4—42no—
83C22⋊C4—42no—
84C22⋊C4—42no—
85C42—82no—
86C23—41no—
87C23—41no—
88C23C2×C22⋊C411yescharacteristic
89C23#12011yescharacteristic
90C23#12311yescharacteristic
91C23#13512yes—
92C23#13712yes—
93Q8—82no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—41no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—44no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—44no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—44no—
116C2×C4—81no—
117C2×C4C4×D412yes—
118C2×C4#12112yes—
119C8—82no—
120C22—21no—
121C22—21no—
122C22—21no—
123C22—21no—
124C22—21no—
125C22—21no—
126C22—42no—
127C22—42no—
128C22#38611yescharacteristic
129C22#41611yescharacteristic
130C22#41911yescharacteristic
131C22#45212yes—
132C22#48512yes—
133C4—22no—
134C4—22no—
135C4—42no—
136C4—42no—
137C4—82no—
138C4—84no—
139C4—81no—
140C4—81no—
141C2—21no—
142C2—22no—
143C2—21no—
144C2—22no—
145C2#187311yescharacteristic
146C2#191211yescharacteristic
147C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
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
9. #400C4semidirect
11. #395C4semidirect

Maximal subgroups

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

subgroupindex
#26272
#23132
#15202
C2×#3242
#12892