Finite Group Databasebeta

← #213798

Subgroups — #213798

Order 512.

1035 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1035 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#35537C211yesmaximal, characteristic
2#17369C212yesmaximal, complemented
3#9105C214yesmaximal, complemented
4#8963C214yesmaximal, complemented
5#8733C212yesmaximal, complemented
6#8326C212yesmaximal, complemented
7C2×#540C2211yescharacteristic
8C2×#518C2211yescharacteristic
9#2975C2214yes—
10#2566C412yes—
11#2541C412yes—
12#2500C412yes—
13#2456—24no—
14#2433C412yes—
15#2424—24no—
16#2328C2212yes—
17C4×#127C2212yes—
18#2298C2211yescharacteristic
19#1478C2218yescomplemented
20#1465—24no—
21#1462—24no—
22#1460C2214yes—
23#1458C2214yes—
24#1276C2214yescomplemented
25#1274C2214yescomplemented
26C23×D4C2×C411yescharacteristic
27#540—44no—
28C2×#127C2×C412yes—
29C2×#127C2×C412yes—
30C2×#127C2×C411yescharacteristic
31C2×#125C2×C411yescharacteristic
32C2×#123—22no—
33C2×#121—24no—
34C2×#121—22no—
35C2×#121C2×C412yes—
36C2×#120—24no—
37#518—44no—
38C2×C4×D4—24no—
39C2×C4×D4C2×C412yes—
40C2×C4×D4D414yes—
41C2×C4×D4C2312yes—
42C2×C4×D4C2312yes—
43C2×#117C2×C411yescharacteristic
44C2×#117C2311yescharacteristic
45C22×C22⋊C4D414yes—
46#436—48no—
47#431C2318yes—
48#407—28no—
49#396—24no—
50#394—24no—
51#392—24no—
52#392—24no—
53#390—24no—
54#390C2×C412yes—
55#390C2×C412yes—
56#390C2×C412yes—
57#390C2×C412yes—
58#388—22no—
59#388—24no—
60#387—22no—
61#386—24no—
62#386—24no—
63#385—24no—
64#384C2×C412yes—
65#382C2×C412yes—
66C4×C4⋊C4C2×C412yes—
67C4×C22⋊C4C2×C412yes—
68#376C2311yescharacteristic
69C43C2311yescharacteristic
70D4⋊C8—48no—
71#324—24no—
72#323—24no—
73#323—28no—
74C25C22⋊C412yes—
75C22×D4—24no—
76C22×D4—22no—
77C22×D4—24no—
78C22×D4—22no—
79C22×D4—42no—
80C22×D4—44no—
81C22×D4C22⋊C412yes—
82C22×D4C22×C412yes—
83C22×D4C22×C412yes—
84C23×C4—22no—
85C23×C4C22⋊C414yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D414yes—
88C2×M16—48no—
89#127—44no—
90#127—44no—
91#127—42no—
92#125—42no—
93#123—48no—
94#121—48no—
95#121—44no—
96#121—88no—
97#120—88no—
98C4×D4—28no—
99C4×D4—28no—
100C4×D4—44no—
101C4×D4—48no—
102C4×D4—88no—
103#117—24no—
104#117—42no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4C22×C412yes—
109C2×C4⋊C4C22×C412yes—
110C2×C4⋊C4C2×D412yes—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—48no—
119C2×C22⋊C4C22⋊C412yes—
120C2×C22⋊C4C22⋊C414yes—
121C2×C22⋊C4C2×D412yes—
122C2×C22⋊C4C2×D414yes—
123C2×C42—22no—
124C2×C42C22⋊C412yes—
125C2×C42C22×C412yes—
126C2×C42C22×C412yes—
127C2×C42C22×C411yescharacteristic
128C2×C42C2411yescharacteristic
129C4⋊C8—28no—
130C22⋊C8—28no—
131C22⋊C8—48no—
132C4×C8—28no—
133#95—22no—
134#95—22no—
135#95—24no—
136#95—22no—
137C24—22no—
138C24—42no—
139C24—44no—
140C24—84no—
141C24C2×C22⋊C411yescharacteristic
142C24C2×C22⋊C414yes—
143C2×D4—24no—
144C2×D4—24no—
145C2×D4—24no—
146C2×D4—24no—
147C2×D4—24no—
148C2×D4—24no—
149C2×D4—42no—
150C2×D4—44no—
151C2×D4—48no—
152C2×D4—44no—
153C2×D4—416no—
154C2×D4—48no—
155C2×D4—44no—
156C2×D4—48no—
157C2×D4—84no—
158C2×D4—84no—
159C2×D4—84no—
160C2×D4—84no—
161C2×D4—84no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—28no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—84no—
171C22×C4—84no—
172C22×C4—84no—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C2×C22⋊C414yes—
175C22×C4C2×C22⋊C412yes—
176C22×C4C23×C411yescharacteristic
177C22×C4C22×D412yes—
178M16—168no—
179C2×C8—48no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—84no—
186C22⋊C4—24no—
187C22⋊C4—28no—
188C22⋊C4—42no—
189C22⋊C4—44no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—44no—
195C22⋊C4—84no—
196C22⋊C4—84no—
197C42—22no—
198C42—24no—
199C42—24no—
200C42—42no—
201C42—44no—
202C42H(5,𝔽2)14yes—
203C23—21no—
204C23—24no—
205C23—24no—
206C23—22no—
207C23—24no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23—44no—
212C23—44no—
213C23—48no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—164no—
218C23—168no—
219C23C22×C22⋊C411yescharacteristic
220D4—48no—
221D4—48no—
222D4—48no—
223D4—816no—
224D4—164no—
225D4—168no—
226D4—164no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—88no—
244C2×C4—164no—
245C2×C4—164no—
246C2×C4#51812yes—
247C2×C4#53414yes—
248C2×C4#56814yes—
249C8—88no—
250C22—22no—
251C22—48no—
252C22—44no—
253C22—44no—
254C22—42no—
255C22—41no—
256C22—44no—
257C22—84no—
258C22—82no—
259C22—84no—
260C22—84no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—168no—
265C22#230811yescharacteristic
266C22#287912yes—
267C4—24no—
268C4—44no—
269C4—44no—
270C4—42no—
271C4—41no—
272C4—82no—
273C4—82no—
274C4#297814yes—
275C2—22no—
276C2—84no—
277C2—81no—
278C2—162no—
279C2#2174311yescharacteristic
280C2#2178212yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35537C2
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
2. #17369C2semidirect
3. #9105C2semidirect
4. #8963C2semidirect
5. #8733C2semidirect
6. #8326C2semidirect
19. #1478C22semidirect
24. #1276C22semidirect
25. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#355372
#173692
#91052
#89632
#87332
#83262