Finite Group Databasebeta

← #213800

Subgroups — #213800

Order 512.

895 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 895 conjugacy classes of subgroups — too many to draw. Only its 194 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#35624C211yesmaximal, characteristic
2#17369C212yesmaximal
3#9103C214yesmaximal, complemented
4#8961C214yesmaximal, complemented
5#8733C212yesmaximal, complemented
6#8327C212yesmaximal, complemented
7C2×#541C2211yescharacteristic
8C2×#518C2211yescharacteristic
9#2975C2214yes—
10#2566C412yes—
11#2547C412yes—
12#2499C412yes—
13#2480—24no—
14#2460—24no—
15#2435C412yes—
16#2326C2212yes—
17C4×#124C2212yes—
18#2298C2211yescharacteristic
19#1479C2218yescomplemented
20#1465—24no—
21#1462—24no—
22#1460C2214yes—
23#1458C2214yes—
24#1276C2214yescomplemented
25#1274C2214yescomplemented
26C23×D4C2×C411yescharacteristic
27#541—44no—
28C2×C4⋊Q8C2×C411yescharacteristic
29C2×#126—22no—
30C2×#125—22no—
31C2×#125C2×C411yescharacteristic
32C2×#124C2×C412yes—
33C2×#124C2×C412yes—
34C2×C22⋊Q8C2×C412yes—
35#518—44no—
36C2×C4×Q8C2312yes—
37C2×C4×D4C2×C412yes—
38C2×C4×D4D414yes—
39C2×C4×D4C2312yes—
40C2×#117C2×C411yescharacteristic
41C2×#117C2311yescharacteristic
42C22×C22⋊C4D414yes—
43#436—48no—
44#431C2318yes—
45#407—28no—
46#397—24no—
47#397—24no—
48#396—24no—
49#394—24no—
50#391C2×C412yes—
51#391C2×C412yes—
52#390C2×C412yes—
53#390C2×C412yes—
54#388—24no—
55#388—24no—
56#388—22no—
57#387—22no—
58#387—24no—
59#385—24no—
60#385—24no—
61#384C2×C412yes—
62#382C2×C412yes—
63#380—24no—
64#379—24no—
65C4×C4⋊C4C2×C412yes—
66C4×C22⋊C4—24no—
67C4×C22⋊C4C2×C412yes—
68#376C2311yescharacteristic
69C43C2311yescharacteristic
70Q8⋊C8—48no—
71#324—24no—
72#323—24no—
73#323—28no—
74C25C22⋊C412yes—
75C22×Q8C22×C412yes—
76C22×D4—42no—
77C22×D4—44no—
78C22×D4C22⋊C412yes—
79C22×D4C22×C412yes—
80C23×C4C22⋊C414yes—
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D414yes—
83C2×M16—48no—
84C4⋊Q8—42no—
85#126—48no—
86#125—42no—
87#125—48no—
88#124—44no—
89#124—44no—
90C22⋊Q8—44no—
91C4×Q8—28no—
92C4×D4—28no—
93C4×D4—44no—
94C4×D4—48no—
95#117—24no—
96#117—42no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4C22×C412yes—
104C2×C4⋊C4C22×C412yes—
105C2×C4⋊C4C2×D412yes—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—48no—
113C2×C22⋊C4C22⋊C412yes—
114C2×C22⋊C4C22⋊C414yes—
115C2×C22⋊C4C2×D412yes—
116C2×C22⋊C4C2×D414yes—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42C22⋊C412yes—
120C2×C42C22×C412yes—
121C2×C42C22×C412yes—
122C2×C42C22×C411yescharacteristic
123C2×C42C2411yescharacteristic
124C4⋊C8—28no—
125C22⋊C8—28no—
126C22⋊C8—48no—
127C4×C8—28no—
128#95—24no—
129#95—24no—
130#95—22no—
131#95—22no—
132#95—24no—
133#95—22no—
134C24—42no—
135C24—44no—
136C24—84no—
137C24C2×C22⋊C411yescharacteristic
138C24C2×C22⋊C414yes—
139C2×Q8—24no—
140C2×Q8—24no—
141C2×Q8—24no—
142C2×D4—24no—
143C2×D4—24no—
144C2×D4—24no—
145C2×D4—42no—
146C2×D4—44no—
147C2×D4—416no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—28no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—44no—
156C22×C4—84no—
157C22×C4—84no—
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4C2×C22⋊C414yes—
160C22×C4C2×C22⋊C412yes—
161C22×C4C23×C411yescharacteristic
162C22×C4C22×D412yes—
163M16—168no—
164C2×C8—48no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C22⋊C4—24no—
175C22⋊C4—28no—
176C22⋊C4—42no—
177C22⋊C4—44no—
178C22⋊C4—44no—
179C22⋊C4—48no—
180C22⋊C4—48no—
181C22⋊C4—84no—
182C22⋊C4—82no—
183C22⋊C4—84no—
184C42—22no—
185C42—24no—
186C42—24no—
187C42—42no—
188C42—44no—
189C42—82no—
190C42H(5,𝔽2)14yes—
191C23—21no—
192C23—24no—
193C23—24no—
194C23—22no—
195C23—24no—
196C23—44no—
197C23—44no—
198C23—42no—
199C23—48no—
200C23—164no—
201C23—168no—
202C23C22×C22⋊C411yescharacteristic
203Q8—48no—
204D4—48no—
205D4—48no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—88no—
226C2×C4—164no—
227C2×C4#51812yes—
228C2×C4#53414yes—
229C2×C4#56814yes—
230C8—88no—
231C22—22no—
232C22—48no—
233C22—44no—
234C22—44no—
235C22—42no—
236C22—41no—
237C22—44no—
238C22—84no—
239C22—82no—
240C22—168no—
241C22#230811yescharacteristic
242C22#287912yes—
243C4—24no—
244C4—44no—
245C4—42no—
246C4—41no—
247C4—82no—
248C4—82no—
249C4—82no—
250C4—82no—
251C4#297814yes—
252C2—22no—
253C2—84no—
254C2—81no—
255C2#2174311yescharacteristic
256C2#2178412yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9103C2semidirect
4. #8961C2semidirect
5. #8733C2semidirect
6. #8327C2semidirect
19. #1479C22semidirect
24. #1276C22semidirect
25. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#356242
#173692
#91032
#89612
#87332
#83272