Finite Group Databasebeta

← #149573

Subgroups — #149573

Order 512.

500 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 82 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#21195C211yesmaximal, characteristic
2#11447C212yesmaximal, complemented
3#7355C212yesmaximal, complemented
4#7354C212yesmaximal, complemented
5C2×#532C411yescharacteristic
6C2×#416—22no—
7C22×C4×C8C411yescharacteristic
8#2285C2211yescharacteristic
9#1958—22no—
10#1957C411yescharacteristic
11#1957C2211yescharacteristic
12#1956C411yescharacteristic
13#1956C2211yescharacteristic
14#1458—22no—
15#1325C2214yes—
16#1324—22no—
17C22×#41—21no—
18C23×C8—21no—
19#532—22no—
20#532—42no—
21#532C814yescomplemented
22C2×C4⋊Q8C2×C411yescharacteristic
23C2×#127C2×C411yescharacteristic
24C2×#124—21no—
25C2×#121—41no—
26C2×C4×D4—41no—
27C2×#117D412yes—
28C22×C42C2×C411yescharacteristic
29#431—42no—
30#416—44no—
31#416—44no—
32C2×#103—22no—
33C2×#103—22no—
34C2×#102—22no—
35C2×#102—22no—
36C2×C4×C8—21no—
37C2×C4×C8—42no—
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8C2×C411yescharacteristic
40#384C2×C411yescharacteristic
41#384C2311yescharacteristic
42#382D412yes—
43#380—41no—
44C4×C4⋊C4—41no—
45#324—24no—
46#323—42no—
47C2×#41—24no—
48C2×#41—44no—
49C2×#41—42no—
50C2×#41—42no—
51C2×#41—42no—
52C22×Q8—21no—
53C22×D4—21no—
54C22×D4—41no—
55C22×D4—41no—
56C23×C4—21no—
57C23×C4—41no—
58C23×C4C22⋊C411yescharacteristic
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—42no—
64C22×C8—42no—
65C4⋊Q8—21no—
66C4⋊Q8—41no—
67C4⋊Q8C2×C812yes—
68#127—21no—
69#127—41no—
70#127C2×C812yes—
71#124—24no—
72#124—42no—
73#121—44no—
74#121—44no—
75C4×D4—44no—
76C4×D4—44no—
77#117—42no—
78#117—42no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4C22⋊C412yes—
83C2×C4⋊C4C2×D412yes—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—42no—
86C2×C42—21no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C2×C812yes—
91C2×C42M1612yes—
92C2×C42C22×C411yescharacteristic
93C4⋊C8—82no—
94#103—44no—
95#103—44no—
96#102—44no—
97#102—44no—
98C22⋊C8—44no—
99C4×C8—41no—
100C4×C8—41no—
101C4×C8—42no—
102C4×C8—82no—
103#95—42no—
104C24—41no—
105C24#9911yescharacteristic
106#41—48no—
107#41—48no—
108#41—84no—
109#41—84no—
110C2×Q8—22no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—41no—
114C2×D4—22no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—44no—
120C2×D4—41no—
121C2×D4—82no—
122C2×D4—82no—
123C2×D4—82no—
124C2×D4—82no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—44no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4—82no—
141C22×C4C22⋊C812yes—
142C22×C4#9911yescharacteristic
143C22×C4#10012yes—
144C22×C4C2×C22⋊C411yescharacteristic
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—84no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C22⋊C4—48no—
161C22⋊C4—82no—
162C42—22no—
163C42—22no—
164C42—21no—
165C42—21no—
166C42—42no—
167C42—42no—
168C42C22⋊C812yes—
169C42C22×C811yescharacteristic
170C42C2×M1611yescharacteristic
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23—42no—
176C23—82no—
177C23—82no—
178C23—81no—
179C23—81no—
180C23#32312yes—
181C23C2×#9911yescharacteristic
182Q8—44no—
183Q8—82no—
184D4—44no—
185D4—82no—
186D4—84no—
187D4—84no—
188D4—88no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4#32312yes—
214C2×C4#35311yescharacteristic
215C2×C4#35311yescharacteristic
216C2×C4#35512yes—
217C2×C4C2×C22⋊C811yescharacteristic
218C2×C4C2×#10011yescharacteristic
219C8—42no—
220C8—42no—
221C8—164no—
222C22—21no—
223C22—21no—
224C22—22no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22—84no—
230C22—82no—
231C22—162no—
232C22—161no—
233C22—161no—
234C22C2×#32311yescharacteristic
235C22C2×#35311yescharacteristic
236C22C2×#35511yescharacteristic
237C4—22no—
238C4—22no—
239C4—21no—
240C4—21no—
241C4—42no—
242C4—82no—
243C4—82no—
244C4—82no—
245C4#131812yes—
246C4#131812yes—
247C2—21no—
248C2—21no—
249C2—42no—
250C2—162no—
251C2C2×#131811yescharacteristic
252C2#1358311yescharacteristic
253C2#1358911yescharacteristic
subgroupG/H
centerC2×C4#353
commutatorC42C22×C8
Frattini, cosocle#384C23
socleC22C2×#353
prime radicalC2×#532C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11447C2semidirect
3. #7355C2semidirect
4. #7354C2semidirect
21. #532C8semidirect

Maximal subgroups

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

subgroupindex
#211952
#114472
#73552
#73542