Finite Group Databasebeta

← #9472

Subgroups — #9472

Order 256.

456 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 69 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#2973C211yesmaximal, characteristic, complemented
2#2408C211yesmaximal, characteristic
3#1799C211yesmaximal, characteristic, complemented
4#1786C212yesmaximal, complemented
5#1289C212yesmaximal, complemented
6C23×D4C2211yescharacteristic
7C2×#121—21no—
8C2×#120—21no—
9C2×C4×D4C2211yescharacteristic
10#435—24no—
11#432—21no—
12C2×#102—24no—
13C2×#99—21no—
14#407—21no—
15C2×C22⋊C8C2212yes—
16#394C412yescomplemented
17#392C412yescomplemented
18#390—21no—
19#380C2212yes—
20#380C2211yescharacteristic
21#379—21no—
22#342—21no—
23C25—21no—
24C22×D4—22no—
25C22×D4—22no—
26C22×D4—21no—
27C22×D4—22no—
28C22×D4—44no—
29C22×D4C2×C412yes—
30C22×D4D411yescharacteristic
31C23×C4D411yescharacteristic
32C23×C4C2311yescharacteristic
33C2×M16—22no—
34C22×C8D414yescomplemented
35#121—44no—
36#120—44no—
37C4×D4—22no—
38C4×D4—22no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4C2×C411yescharacteristic
48C2×C22⋊C4D411yescharacteristic
49C2×C42D411yescharacteristic
50C4⋊C8—22no—
51#102—48no—
52#99—42no—
53C22⋊C8—22no—
54C22⋊C8—22no—
55C22⋊C8—22no—
56#97—22no—
57#95—22no—
58#95C2×C412yes—
59C24—22no—
60C24—44no—
61C24—42no—
62C24—42no—
63C24—41no—
64C24—41no—
65C24—82no—
66C24C22⋊C411yescharacteristic
67C24C2×D411yescharacteristic
68C2×D4—28no—
69C2×D4—24no—
70C2×D4—22no—
71C2×D4—21no—
72C2×D4—22no—
73C2×D4—21no—
74C2×D4—24no—
75C2×D4—44no—
76C2×D4—42no—
77C2×D4—44no—
78C2×D4—48no—
79C2×D4—44no—
80C2×D4—44no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—82no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4C22⋊C411yescharacteristic
93C22×C4C22×C411yescharacteristic
94C22×C4C2×D412yes—
95C22×C4C2×D411yescharacteristic
96C22×C4#4112yes—
97M16—82no—
98C2×C8—24no—
99C2×C8—44no—
100C4⋊C4—22no—
101C4⋊C4—42no—
102C4⋊C4—44no—
103C22⋊C4—22no—
104C22⋊C4—22no—
105C22⋊C4—44no—
106C22⋊C4—44no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C22⋊C4—82no—
111C42—21no—
112C42—21no—
113C23—24no—
114C23—21no—
115C23—21no—
116C23—21no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—42no—
122C23—48no—
123C23—42no—
124C23—42no—
125C23—42no—
126C23—42no—
127C23—42no—
128C23—84no—
129C23—88no—
130C23—84no—
131C23—84no—
132C23—82no—
133C23—82no—
134C23C2×C22⋊C411yescharacteristic
135C23#12112yes—
136C23AGL(1,ℤ/8)11yescharacteristic
137C23AGL(1,ℤ/8)12yes—
138C23#13711yescharacteristic
139D4—28no—
140D4—44no—
141D4—44no—
142D4—42no—
143D4—42no—
144D4—44no—
145D4—88no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—81no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4C4×D412yes—
166C2×C4#12011yescharacteristic
167C2×C4#12311yescharacteristic
168C8—44no—
169C22—21no—
170C22—22no—
171C22—21no—
172C22—22no—
173C22—42no—
174C22—41no—
175C22—41no—
176C22—42no—
177C22—44no—
178C22—44no—
179C22—41no—
180C22—44no—
181C22—41no—
182C22—41no—
183C22—81no—
184C22—82no—
185C22—82no—
186C22—88no—
187C22—88no—
188C22—82no—
189C22—82no—
190C22#38611yescharacteristic
191C22#41711yescharacteristic
192C22#41811yescharacteristic
193C22#46812yes—
194C22#46912yes—
195C4—21no—
196C4—22no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—82no—
201C4—162no—
202C4B(4,𝔽2)12yes—
203C2—21no—
204C2—22no—
205C2—21no—
206C2—22no—
207C2—81no—
208C2—84no—
209C2#188911yescharacteristic
210C2#189511yescharacteristic
211C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#2408C2
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
1. #2973C2semidirect
3. #1799C2semidirect
4. #1786C2semidirect
5. #1289C2semidirect
16. #394C4semidirect
17. #392C4semidirect
34. C22×C8D4semidirect

Maximal subgroups

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

subgroupindex
#29732
#24082
#17992
#17862
#12892