Finite Group Databasebeta

← #9711

Subgroups — #9711

Order 256.

358 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 93 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#2857C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic
3C2×#328C214yesmaximal
4C4×#95C211yesmaximal, characteristic
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C22⋊Q8C2212yes—
7C22×C42—21no—
8C22×C42C2211yescharacteristic
9C2×C4⋊C8—21no—
10C2×C22⋊C8C2212yes—
11#391C412yescomplemented
12#389—21no—
13#389—21no—
14#384C411yescharacteristic, complemented
15#380—21no—
16#380—21no—
17C2×#95—21no—
18C2×#95C2212yes—
19#336—21no—
20#328—28no—
21C22×Q8C2×C412yes—
22C23×C4—22no—
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27C4⋊Q8—42no—
28C22⋊Q8—24no—
29C22⋊Q8—24no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4C2×C412yes—
36C2×C22⋊C4—22no—
37C2×C42—24no—
38C2×C42—42no—
39C2×C42—42no—
40C2×C42—42no—
41C2×C42C2×C411yescharacteristic
42C2×C42C2×C411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42D411yescharacteristic
45C4⋊C8—42no—
46C22⋊C8—24no—
47#95—24no—
48#95—22no—
49#95—22no—
50#95—22no—
51#95D414yes—
52C24C2×D411yescharacteristic
53C2×Q8—24no—
54C2×Q8—42no—
55C2×Q8—42no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—24no—
59C22×C4—24no—
60C22×C4—24no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4SD1612yes—
77C22×C4Q1612yes—
78C22×C4C22×C411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D412yes—
81C22×C4#4112yes—
82C2×C8—42no—
83C2×C8—42no—
84C2×C8—42no—
85C4⋊C4—24no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—42no—
92C22⋊C4—44no—
93C42—22no—
94C42—22no—
95C42—41no—
96C42—41no—
97C42—48no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23C2×C22⋊C411yescharacteristic
103C23#12112yes—
104C23C2×SD1611yescharacteristic
105C23C2×Q1611yescharacteristic
106C23#13712yes—
107Q8—84no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—24no—
117C2×C4—24no—
118C2×C4—28no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—44no—
125C2×C4—44no—
126C2×C4—44no—
127C2×C4—44no—
128C2×C4—44no—
129C2×C4—44no—
130C2×C4—44no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4#9912yes—
138C2×C4#10312yes—
139C2×C4#10312yes—
140C2×C4#10412yes—
141C2×C4#10412yes—
142C2×C4C4×D412yes—
143C2×C4#12411yescharacteristic
144C2×C4#12711yescharacteristic
145C8—82no—
146C22—22no—
147C22—22no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—22no—
153C22—22no—
154C22#39011yescharacteristic
155C22C2×#9911yescharacteristic
156C22#41011yescharacteristic
157C22C2×#10311yescharacteristic
158C22#41911yescharacteristic
159C22C2×#10411yescharacteristic
160C22C2×#10411yescharacteristic
161C22#46112yes—
162C22#46212yes—
163C4—21no—
164C4—22no—
165C4—21no—
166C4—22no—
167C4—22no—
168C4—48no—
169C4—41no—
170C4—82no—
171C4—82no—
172C4#32814yescomplemented
173C2—22no—
174C2—22no—
175C2C2×#32811yescharacteristic
176C2#150311yescharacteristic
177C2#150611yescharacteristic
178C2#150811yescharacteristic
179C2#196211yescharacteristic
180C2#196411yescharacteristic
181C2#198111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
11. #391C4semidirect
14. #384C4semidirect
172. C4#328semidirect

Maximal subgroups

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

subgroupindex
#28572
#17972
C2×#3282
C4×#952