Finite Group Databasebeta

← #138221

Subgroups — #138221

Order 512.

397 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×#2165C211yesmaximal, characteristic
2#33176C211yesmaximal, characteristic
3#12791C211yesmaximal, characteristic
4C2×#1358C212yesmaximal, complemented
5#7268C212yesmaximal, complemented
6C2×#494C2211yescharacteristic
7C2×#470—21no—
8C2×#470—21no—
9C2×C4×M16C2211yescharacteristic
10#2857C412yes—
11#2165—24no—
12C2×#364C2212yes—
13C2×#364C2212yes—
14C2×#349—21no—
15C2×#349—21no—
16C2×#347—21no—
17#1971—21no—
18#1971—21no—
19#1956C412yes—
20#1844C2211yescharacteristic
21#1375—21no—
22#1358—24no—
23C22×Q16—21no—
24C22×Q16—21no—
25C22×M16D411yescharacteristic
26C22×M16D411yescharacteristic
27C2×C4⋊Q8C2×C412yes—
28C2×C22⋊Q8—22no—
29C22×C42C2×C411yescharacteristic
30C2×M32—22no—
31C2×M32—22no—
32#494—22no—
33#494—22no—
34#470—44no—
35#470—44no—
36C2×#106—21no—
37C2×#106—21no—
38C2×#103—22no—
39C2×#103—22no—
40C4×M16—22no—
41C4×M16—22no—
42C2×#97D411yescharacteristic
43C2×#97D411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#391—22no—
47#389—42no—
48#384C2×C412yes—
49#380—42no—
50#364—22no—
51#364—22no—
52#364—22no—
53#364D414yes—
54#349—24no—
55#349—42no—
56#347—24no—
57C22×Q8—22no—
58C23×C4—21no—
59C23×C4C22⋊C411yescharacteristic
60C2×Q16—41no—
61C2×Q16—42no—
62C2×Q16—41no—
63C2×Q16—42no—
64C2×Q16—41no—
65C2×Q16—42no—
66C2×Q16—42no—
67C2×Q16—41no—
68C2×M16—21no—
69C2×M16—22no—
70C2×M16—21no—
71C2×M16—22no—
72C2×M16—21no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—21no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C4⋊Q8—24no—
81C4⋊Q8—42no—
82C22⋊Q8—44no—
83C22⋊Q8—44no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C42—22no—
90C2×C42—21no—
91C2×C42—41no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22×C411yescharacteristic
94M32—24no—
95M32—42no—
96M32—42no—
97M32—42no—
98C2×C16—42no—
99C2×C16—42no—
100#106—42no—
101#106—42no—
102#103—44no—
103#103—44no—
104#97—21no—
105#97—22no—
106#97—21no—
107C4×C8—22no—
108C4×C8C2×D412yes—
109C4×C8#4112yes—
110#95—44no—
111C24#10011yescharacteristic
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—42no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4#10011yescharacteristic
131C22×C4C2×C22⋊C411yescharacteristic
132Q16—82no—
133Q16—82no—
134Q16—82no—
135Q16—82no—
136M16—42no—
137M16—42no—
138M16—42no—
139M16—42no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8#12011yescharacteristic
147C2×C8#12112yes—
148C2×C8#12311yescharacteristic
149C2×C8#13512yes—
150C2×C8AGL(1,ℤ/8)12yes—
151C2×C8AGL(1,ℤ/8)12yes—
152C2×C8AGL(1,ℤ/8)11yescharacteristic
153C2×C8#13711yescharacteristic
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—84no—
157C4⋊C4—84no—
158C4⋊C4—84no—
159C22⋊C4—84no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—42no—
164C42C4×D412yes—
165C16—44no—
166C16—82no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—42no—
171C23C2×#10011yescharacteristic
172Q8—82no—
173Q8—82no—
174Q8—82no—
175Q8—82no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—84no—
188C2×C4—84no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4#38611yescharacteristic
194C2×C4#41311yescharacteristic
195C2×C4#41611yescharacteristic
196C2×C4#41711yescharacteristic
197C2×C4#41811yescharacteristic
198C2×C4#41811yescharacteristic
199C8—41no—
200C8—41no—
201C8#44912yes—
202C8#46812yes—
203C8#46912yes—
204C8#48212yes—
205C22—22no—
206C22—41no—
207C22—42no—
208C22—81no—
209C22—81no—
210C22#155111yescharacteristic
211C22#155511yescharacteristic
212C22#190011yescharacteristic
213C4—24no—
214C4—41no—
215C4—81no—
216C4—162no—
217C4—162no—
218C4#155311yescharacteristic
219C4#155711yescharacteristic
220C4#187511yescharacteristic
221C4#189511yescharacteristic
222C2—21no—
223C2—21no—
224C2—81no—
225C2#914211yescharacteristic
226C2#1199312yes—
subgroupG/H
center, socleC22#1555
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1358C2semidirect
5. #7268C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21652
#331762
#127912
C2×#13582
#72682