Finite Group Databasebeta

← #9411

Subgroups — #9411

Order 256.

303 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 61 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#2561C211yesmaximal, characteristic
2#2312C211yesmaximal, characteristic
3#1508C212yesmaximal, complemented
4#1462C211yesmaximal, characteristic
5#1297C212yesmaximal, complemented
6C2×C22⋊Q8C2211yescharacteristic
7C22×C22⋊C4C2211yescharacteristic
8#407C2212yes—
9#397C412yescomplemented
10#393C412yescomplemented
11#388—22no—
12#388—24no—
13#381—21no—
14#380—21no—
15#379—21no—
16#379—21no—
17C4×C22⋊C4C2211yescharacteristic
18C4×C22⋊C4C2212yes—
19#328—24no—
20#323—22no—
21C25D411yescharacteristic
22C22×Q8C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C2×M16—22no—
26C22⋊Q8—22no—
27C22⋊Q8—22no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C22⋊C4—22no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4C2×C412yes—
41C2×C22⋊C4D412yes—
42C2×C42—21no—
43C2×C42—22no—
44C22⋊C8—22no—
45C22⋊C8—22no—
46#95—22no—
47#95—22no—
48#95—22no—
49#95C2×C412yes—
50#95D414yes—
51C24—21no—
52C24—21no—
53C24—42no—
54C24—42no—
55C24—81no—
56C24C22⋊C412yes—
57C24C2×D411yescharacteristic
58C2×Q8—22no—
59C2×Q8—81no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—24no—
66C22×C4—42no—
67C22×C4—81no—
68C22×C4C22⋊C412yes—
69C22×C4C22×C411yescharacteristic
70C22×C4C2×D412yes—
71C22×C4C2×D411yescharacteristic
72C22×C4#4112yes—
73M16—82no—
74C2×C8—42no—
75C4⋊C4—22no—
76C4⋊C4—42no—
77C4⋊C4—82no—
78C22⋊C4—24no—
79C22⋊C4—24no—
80C22⋊C4—44no—
81C22⋊C4—42no—
82C22⋊C4—44no—
83C22⋊C4—41no—
84C22⋊C4—41no—
85C22⋊C4—82no—
86C22⋊C4—82no—
87C22⋊C4—82no—
88C42—81no—
89C42—82no—
90C23—22no—
91C23—22no—
92C23—22no—
93C23—22no—
94C23—22no—
95C23—41no—
96C23—42no—
97C23—42no—
98C23—42no—
99C23—42no—
100C23—42no—
101C23—42no—
102C23—81no—
103C23—81no—
104C23—88no—
105C23C2×C22⋊C411yescharacteristic
106C23#12011yescharacteristic
107C23#12311yescharacteristic
108Q8—82no—
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—44no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—44no—
126C2×C4—81no—
127C2×C4C4×D412yes—
128C2×C4#12112yes—
129C8—82no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—44no—
138C22—44no—
139C22—42no—
140C22—42no—
141C22—42no—
142C22—82no—
143C22—84no—
144C22—84no—
145C22#38611yescharacteristic
146C22#42112yes—
147C22#45512yes—
148C4—22no—
149C4—22no—
150C4—42no—
151C4—84no—
152C4—82no—
153C4—81no—
154C4—81no—
155C2—22no—
156C2—22no—
157C2—41no—
158C2—82no—
159C2#188511yescharacteristic
160C2#189711yescharacteristic
161C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
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
3. #1508C2semidirect
5. #1297C2semidirect
9. #397C4semidirect
10. #393C4semidirect

Maximal subgroups

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

subgroupindex
#25612
#23122
#15082
#14622
#12972