Finite Group Databasebeta

← #9201

Subgroups — #9201

Order 256.

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

normal subgroups only

This group has 306 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#2626C211yesmaximal, characteristic
2#2312C211yesmaximal, characteristic
3#1520C212yesmaximal, complemented
4C2×#323C211yesmaximal, characteristic
5#1289C212yesmaximal, complemented
6C2×C22⋊Q8C2211yescharacteristic
7C22×C22⋊C4C2211yescharacteristic
8C2×C22⋊C8C2212yes—
9#399C412yescomplemented
10#396—22no—
11#393C412yescomplemented
12#388—24no—
13#385—21no—
14#380C2211yescharacteristic
15#380C2212yes—
16#379—21no—
17#379—21no—
18C4×C22⋊C4—21no—
19#328—24no—
20#323—22no—
21C25D411yescharacteristic
22C22×Q8C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26C22⋊Q8—22no—
27C22⋊Q8—22no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—21no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4C2×C412yes—
43C2×C22⋊C4D412yes—
44C2×C42—22no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47#95—22no—
48#95—22no—
49#95C2×C412yes—
50#95D414yes—
51C24—21no—
52C24—21no—
53C24—22no—
54C24—22no—
55C24—81no—
56C24—81no—
57C24C22⋊C412yes—
58C24C2×D411yescharacteristic
59C2×Q8—22no—
60C2×Q8—81no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—24no—
67C22×C4—81no—
68C22×C4—81no—
69C22×C4C22⋊C412yes—
70C22×C4C22×C411yescharacteristic
71C22×C4C2×D412yes—
72C22×C4C2×D411yescharacteristic
73C22×C4#4112yes—
74C2×C8—42no—
75C2×C8—82no—
76C4⋊C4—22no—
77C4⋊C4—42no—
78C4⋊C4—82no—
79C4⋊C4—81no—
80C4⋊C4—82no—
81C22⋊C4—24no—
82C22⋊C4—24no—
83C22⋊C4—41no—
84C22⋊C4—41no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C22⋊C4—42no—
88C22⋊C4—82no—
89C22⋊C4—82no—
90C22⋊C4—82no—
91C22⋊C4—82no—
92C42—82no—
93C23—22no—
94C23—22no—
95C23—22no—
96C23—41no—
97C23—41no—
98C23—42no—
99C23—42no—
100C23—42no—
101C23—42no—
102C23—42no—
103C23—42no—
104C23—81no—
105C23—81no—
106C23—84no—
107C23—84no—
108C23C2×C22⋊C411yescharacteristic
109C23#12011yescharacteristic
110C23#12311yescharacteristic
111C23#13512yes—
112C23#13712yes—
113Q8—82no—
114C2×C4—22no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—44no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—44no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—44no—
131C2×C4—81no—
132C2×C4C4×D412yes—
133C2×C4#12112yes—
134C8—82no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—21no—
141C22—42no—
142C22—42no—
143C22—44no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—82no—
148C22—84no—
149C22—84no—
150C22#38611yescharacteristic
151C22#41611yescharacteristic
152C22#41911yescharacteristic
153C22#45212yes—
154C22#48512yes—
155C4—22no—
156C4—22no—
157C4—42no—
158C4—82no—
159C4—84no—
160C4—81no—
161C4—81no—
162C2—21no—
163C2—22no—
164C2—21no—
165C2—22no—
166C2—82no—
167C2#187311yescharacteristic
168C2#191111yescharacteristic
169C2#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. #1520C2semidirect
5. #1289C2semidirect
9. #399C4semidirect
11. #393C4semidirect

Maximal subgroups

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

subgroupindex
#26262
#23122
#15202
C2×#3232
#12892