Finite Group Databasebeta

← #4462576

Subgroups — #4462576

Order 512.

3946 conjugacy classes of subgroups; only the 554 normal subgroups are shown.

normal subgroups only

This group has 3946 conjugacy classes of subgroups — too many to draw. Only its 554 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×#2656C211yesmaximal, characteristic, complemented
2C22×#396C211yesmaximal, characteristic, complemented
3#17204C218yesmaximal, complemented
4#17091C218yesmaximal, complemented
5#14719C216yesmaximal, complemented
6#14716C216yesmaximal, complemented
7#14349C211yesmaximal, characteristic, complemented
8C22×#121C2216yes—
9C22×#120C2214yes—
10C23×C4⋊C4C2213yes—
11C23×C22⋊C4C2213yes—
12#2656—28no—
13C2×#399C2214yescomplemented
14C2×#396—28no—
15C2×#396C2218yes—
16C2×#394C22112yescomplemented
17C2×#392C22112yes—
18C2×#392C2214yescomplemented
19C2×#386C22112yes—
20C22×#95C2211yescharacteristic
21#1443C2218yescomplemented
22#1441C22112yescomplemented
23#1439C22112yes—
24#1439C22124yescomplemented
25#1438C22124yes—
26C22⋊#95C2216yes—
27C26C2311yescharacteristic
28C23×D4C2314yes—
29C23×D4C2318yes—
30C23×D4C2316yes—
31C24×C4C2313yes—
32C24×C4C2313yes—
33C2×#121—224no—
34C2×#121—424no—
35C2×#120—424no—
36C22×C4⋊C4—212no—
37C22×C4⋊C4—412no—
38C22×C4⋊C4C23112yes—
39C22×C4⋊C4C2316yes—
40C22×C22⋊C4—212no—
41C22×C22⋊C4—412no—
42C22×C22⋊C4C2314yes—
43C22×C22⋊C4C23112yes—
44C22×C22⋊C4C23124yes—
45C22×C22⋊C4C23112yes—
46C22×C22⋊C4C2318yes—
47#399—216no—
48#396—232no—
49#394—424no—
50#392—216no—
51#392—424no—
52#386—424no—
53C2×#95D418yescomplemented
54C2×#95C23112yes—
55C2×#95C23112yes—
56C2×#95C23124yes—
57C2×#95C2314yes—
58C25—22no—
59C25—24no—
60C25—212no—
61C25—46no—
62C25C2414yes—
63C25C2412yes—
64C25C2411yescharacteristic
65C22×D4—28no—
66C22×D4—224no—
67C22×D4—224no—
68C22×D4—216no—
69C22×D4—212no—
70C22×D4—424no—
71C22×D4—412no—
72C22×D4—412no—
73C22×D4—424no—
74C22×D4—412no—
75C23×C4—26no—
76C23×C4—212no—
77C23×C4—212no—
78C23×C4—212no—
79C23×C4—46no—
80C23×C4—46no—
81C23×C4—46no—
82C23×C4C2×D4112yes—
83C23×C4C2414yes—
84C23×C4C2418yes—
85C23×C4C2416yes—
86C23×C4C2416yes—
87#121—496no—
88#120—464no—
89C2×C4⋊C4—224no—
90C2×C4⋊C4—212no—
91C2×C4⋊C4—224no—
92C2×C4⋊C4—412no—
93C2×C4⋊C4—412no—
94C2×C4⋊C4—448no—
95C2×C22⋊C4—224no—
96C2×C22⋊C4—224no—
97C2×C22⋊C4—224no—
98C2×C22⋊C4—224no—
99C2×C22⋊C4—248no—
100C2×C22⋊C4—412no—
101C2×C22⋊C4—424no—
102C2×C22⋊C4—412no—
103C2×C22⋊C4—412no—
104C2×C22⋊C4—448no—
105C2×C22⋊C4—424no—
106C2×C22⋊C4—424no—
107#95—424no—
108#95—412no—
109#95—412no—
110#95C2×D4116yes—
111C24—26no—
112C24—22no—
113C24—28no—
114C24—22no—
115C24—212no—
116C24—212no—
117C24—22no—
118C24—24no—
119C24—224no—
120C24—212no—
121C24—212no—
122C24—28no—
123C24—448no—
124C24—412no—
125C24—412no—
126C24—416no—
127C24—824no—
128C24C22×D412yes—
129C24H(5,𝔽2)112yes—
130C242−1+414yes—
131C24C2511yescharacteristic
132C2×D4—424no—
133C2×D4—432no—
134C2×D4—448no—
135C2×D4—416no—
136C2×D4—448no—
137C2×D4—812no—
138C2×D4—812no—
139C2×D4—848no—
140C2×D4—848no—
141C2×D4—848no—
142C22×C4—224no—
143C22×C4—26no—
144C22×C4—212no—
145C22×C4—212no—
146C22×C4—212no—
147C22×C4—212no—
148C22×C4—26no—
149C22×C4—212no—
150C22×C4—224no—
151C22×C4—24no—
152C22×C4—224no—
153C22×C4—212no—
154C22×C4—224no—
155C22×C4—212no—
156C22×C4—28no—
157C22×C4—412no—
158C22×C4—412no—
159C22×C4—448no—
160C22×C4—412no—
161C22×C4—412no—
162C22×C4—824no—
163C22×C4C22×D4112yes—
164C22×C4H(5,𝔽2)112yes—
165C4⋊C4—448no—
166C4⋊C4—424no—
167C22⋊C4—448no—
168C22⋊C4—448no—
169C22⋊C4—432no—
170C22⋊C4—416no—
171C22⋊C4—848no—
172C23—26no—
173C23—22no—
174C23—224no—
175C23—224no—
176C23—24no—
177C23—224no—
178C23—448no—
179C23—46no—
180C23—46no—
181C23—48no—
182C23—412no—
183C23—424no—
184C23—412no—
185C23—412no—
186C23—44no—
187C23—44no—
188C23—412no—
189C23—424no—
190C23—424no—
191C23—424no—
192C23—424no—
193C23—864no—
194C23—824no—
195C23—88no—
196C23#535112yes—
197C23#560116yes—
198C23C23×D411yescharacteristic
199C23C2×H(5,𝔽2)16yes—
200C23C2×H(5,𝔽2)16yes—
201C23C2×2−1+412yes—
202D4—848no—
203D4—864no—
204D4—832no—
205C2×C4—212no—
206C2×C4—212no—
207C2×C4—212no—
208C2×C4—212no—
209C2×C4—224no—
210C2×C4—448no—
211C2×C4—46no—
212C2×C4—46no—
213C2×C4—412no—
214C2×C4—424no—
215C2×C4—412no—
216C2×C4—44no—
217C2×C4—48no—
218C2×C4—412no—
219C2×C4—424no—
220C2×C4—424no—
221C2×C4—824no—
222C22—212no—
223C22—212no—
224C22—232no—
225C22—412no—
226C22—416no—
227C22—832no—
228C22—82no—
229C22—86no—
230C22—88no—
231C22—84no—
232C22—812no—
233C22—812no—
234C22—88no—
235C22—824no—
236C22#281818yes—
237C22C2×#53513yes—
238C22#348413yes—
239C22#348611yescharacteristic
240C22C2×#56014yes—
241C22#3536112yes—
242C22#3537112yes—
243C4—412no—
244C4—412no—
245C4—84no—
246C4—88no—
247C2—28no—
248C2—84no—
249C2—164no—
250C2C2×#281811yescharacteristic
251C2#3670713yes—
252C2#3671713yes—
253C2#5247218yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2656C2semidirect
2. C22×#396C2semidirect
3. #17204C2semidirect
4. #17091C2semidirect
5. #14719C2semidirect
6. #14716C2semidirect
7. #14349C2semidirect
13. C2×#399C22semidirect
16. C2×#394C22semidirect
18. C2×#392C22semidirect
21. #1443C22semidirect
22. #1441C22semidirect
24. #1439C22semidirect
53. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26562
C22×#3962
#172042
#170912
#147192
#147162
#143492