Finite Group Databasebeta

← #135939

Subgroups — #135939

Order 512.

408 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 90 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#23269C211yesmaximal, characteristic
2C4×#354C211yesmaximal, characteristic
3#8957C211yesmaximal, characteristic
4#8851C212yesmaximal
5#7204C212yesmaximal
6#2972C2211yescharacteristic
7C4×#124C2211yescharacteristic
8C2×#354C411yescharacteristic
9C2×#354C411yescharacteristic
10#1984—21no—
11#1923—22no—
12#1869C2212yes—
13#1850—22no—
14C4×#99C2212yes—
15C8×C22⋊C4—22no—
16#1744—21no—
17#1479—21no—
18#1460C2211yescharacteristic
19#1393C412yes—
20#1282—22no—
21#1276—21no—
22C23×C8D412yes—
23C2×#124C2×C411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25C2×C4×D4C2311yescharacteristic
26C8×D4—42no—
27#431—21no—
28#431—21no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8—42no—
31C2×#99C2×C412yes—
32C2×#99C2×C412yes—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8D412yes—
36C2×C4×C8—22no—
37C2×C4×C8—21no—
38#391Q811yescharacteristic
39#390C2×C411yescharacteristic
40#388—22no—
41#382Q811yescharacteristic
42C4×C22⋊C4—22no—
43C4×C22⋊C4—22no—
44C43D411yescharacteristic
45#354—22no—
46#354C814yescomplemented
47#342—22no—
48#336—21no—
49#336—22no—
50#336—21no—
51#336—22no—
52#336—42no—
53Q8⋊C8—42no—
54#323—22no—
55C22×Q8C4⋊C411yescharacteristic
56C22×D4C22×C411yescharacteristic
57C23×C4C2×D412yes—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—44no—
63C22×C8—42no—
64#124—22no—
65#124C2×C812yes—
66#124M1612yes—
67C4×Q8—41no—
68C4×Q8—41no—
69C4×D4—41no—
70C4×D4—41no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C4⋊C411yescharacteristic
73C2×C4⋊C4C2×Q811yescharacteristic
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4#4112yes—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—42no—
83C2×C42C4⋊C411yescharacteristic
84C2×C42C4⋊C411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C4⋊C8—22no—
87C4⋊C8—41no—
88C4⋊C8—81no—
89C4⋊C8—84no—
90#99—22no—
91#99—22no—
92#99—22no—
93#99C2×C814yes—
94C22⋊C8—44no—
95C22⋊C8—42no—
96C22⋊C8—41no—
97C22⋊C8—42no—
98C22⋊C8—41no—
99C4×C8—41no—
100C4×C8—41no—
101C4×C8—44no—
102C4×C8—81no—
103#95—22no—
104#95—22no—
105#95—42no—
106C24C4×D412yes—
107C2×Q8—21no—
108C2×Q8—41no—
109C2×Q8—41no—
110C2×Q8C4⋊C812yescomplemented
111C2×D4—41no—
112C2×D4—41no—
113C2×D4C22×C811yescharacteristic
114C2×D4C2×M1611yescharacteristic
115C2×D4#13112yes—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C2×C4⋊C411yescharacteristic
127C22×C4#12011yescharacteristic
128C22×C4C22⋊Q812yes—
129C22×C4#12311yescharacteristic
130C2×C8—24no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8—44no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—82no—
139C2×C8—82no—
140C2×C8—82no—
141C2×C8—81no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—81no—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C22⋊C4—44no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—41no—
157C42—42no—
158C42—42no—
159C42—41no—
160C42—41no—
161C42—82no—
162C42—82no—
163C42C4⋊C812yescomplemented
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23#38511yescharacteristic
169C23C8×D412yes—
170C23#43512yes—
171Q8—42no—
172Q8—81no—
173D4—41no—
174D4—41no—
175D4—42no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4C2×C4⋊C811yescharacteristic
204C2×C4#42411yescharacteristic
205C2×C4B(4,𝔽2)11yescharacteristic
206C2×C4#45811yescharacteristic
207C8—44no—
208C8—82no—
209C8—82no—
210C8—161no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—22no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—82no—
220C22#185011yescharacteristic
221C22#186711yescharacteristic
222C22#186811yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—42no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—41no—
231C4—84no—
232C4—81no—
233C4—162no—
234C4#219712yes—
235C2—21no—
236C2—21no—
237C2—42no—
238C2—42no—
239C2#885111yescharacteristic
240C2#1296511yescharacteristic
241C2#1296811yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×D4C22×C8
Frattini, cosocleC2×C4×D4C23
socleC22#1867
prime radicalC22×D4C22×C4
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
46. #354C8semidirect
110. C2×Q8C4⋊C8semidirect
163. C42C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
#232692
C4×#3542
#89572
#88512
#72042