Finite Group Databasebeta

← #4131001

Subgroups — #4131001

Order 512.

1881 conjugacy classes of subgroups; only the 530 normal subgroups are shown.

normal subgroups only

This group has 1881 conjugacy classes of subgroups — too many to draw. Only its 530 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#17301C211yesmaximal, characteristic, complemented
2#16007C215yesmaximal, complemented
3#15710C215yesmaximal
4#15616C215yesmaximal
5#14206C215yesmaximal
6#14092C215yesmaximal, complemented
7#14046C215yesmaximal, complemented
8C2×#396C2215yes—
9C2×#396C2215yes—
10C2×#389C2215yes—
11C2×#389C2215yes—
12C2×#389C2215yes—
13C2×#389C2215yes—
14C2×#388C2215yes—
15C2×#387C2215yes—
16C2×C4×C4⋊C4C2215yes—
17#1443C2215yes—
18#1443C2215yes—
19#1442C2215yes—
20#1442C2215yes—
21#1441C2215yes—
22#1441C2215yes—
23#1441C2215yes—
24#1441C2215yes—
25#1441C2215yes—
26#1441C2215yes—
27#1440C2215yes—
28#1440C2215yes—
29#1437C2215yes—
30#1437C2215yes—
31#1436C2215yes—
32#1436C2215yes—
33#1436C2215yes—
34#1436C2215yes—
35C4⋊#95C2215yes—
36C4⋊#95C2215yes—
37#1434C2215yes—
38C4×#95C2215yes—
39C22×C4⋊C4C2315yes—
40C22×C4⋊C4C2315yes—
41C22×C4⋊C4C2315yes—
42C22×C4⋊C4C2315yes—
43C22×C4⋊C4C2315yes—
44C22×C4⋊C4C2315yes—
45C22×C4⋊C4C2315yes—
46C22×C22⋊C4C2315yes—
47C22×C22⋊C4C2315yes—
48C22×C22⋊C4C2315yes—
49C22×C42C2315yes—
50C22×C42C2315yes—
51C22×C42C2315yes—
52#396—220no—
53#396—410no—
54#389—220no—
55#389—220no—
56#389—410no—
57#389—410no—
58#388—410no—
59#387—410no—
60C4×C4⋊C4—410no—
61C2×#95C2315yes—
62C2×#95C2315yes—
63C2×#95C2315yes—
64C2×#95C2315yes—
65C2×#95C2315yes—
66C2×#95C2315yes—
67C2×#95C2315yes—
68C2×#95C2315yes—
69C2×#95C2315yes—
70C2×#95C2315yes—
71C2×#95C2315yes—
72C2×#95C2315yes—
73C2×#95C2315yes—
74C2×#95C2315yes—
75C2×#95C2315yes—
76C2×#95C2315yes—
77C2×#95C2315yes—
78C2×#95C2315yes—
79C25C2411yescharacteristic
80C23×C4C2415yes—
81C23×C4C2415yes—
82C23×C4C2415yes—
83C23×C4C2415yes—
84C23×C4C2415yes—
85C23×C4C2415yes—
86C2×C4⋊C4—210no—
87C2×C4⋊C4—210no—
88C2×C4⋊C4—210no—
89C2×C4⋊C4—210no—
90C2×C4⋊C4—210no—
91C2×C4⋊C4—210no—
92C2×C4⋊C4—210no—
93C2×C4⋊C4—210no—
94C2×C4⋊C4—210no—
95C2×C4⋊C4—210no—
96C2×C4⋊C4—210no—
97C2×C4⋊C4—210no—
98C2×C4⋊C4—210no—
99C2×C4⋊C4—210no—
100C2×C4⋊C4—45no—
101C2×C4⋊C4—45no—
102C2×C4⋊C4—45no—
103C2×C4⋊C4—45no—
104C2×C4⋊C4—45no—
105C2×C4⋊C4—45no—
106C2×C4⋊C4#41120yes—
107C2×C22⋊C4—210no—
108C2×C22⋊C4—210no—
109C2×C22⋊C4—210no—
110C2×C22⋊C4—210no—
111C2×C22⋊C4—210no—
112C2×C22⋊C4—45no—
113C2×C22⋊C4—45no—
114C2×C22⋊C4—45no—
115C2×C22⋊C4—45no—
116C2×C42—210no—
117C2×C42—210no—
118C2×C42—210no—
119C2×C42—210no—
120C2×C42—45no—
121C2×C42—45no—
122C2×C42—45no—
123C2×C42—45no—
124C2×C42—45no—
125#95—210no—
126#95—210no—
127#95—210no—
128#95—210no—
129#95—210no—
130#95—210no—
131#95—210no—
132#95—210no—
133#95—210no—
134#95—45no—
135#95—45no—
136#95—45no—
137#95—45no—
138#95—45no—
139#95—45no—
140#95—45no—
141#95—45no—
142#95—45no—
143C24—25no—
144C24—25no—
145C24—25no—
146C24C2511yescharacteristic
147C22×C4—25no—
148C22×C4—25no—
149C22×C4—25no—
150C22×C4—25no—
151C22×C4—25no—
152C22×C4—25no—
153C22×C4—25no—
154C22×C4—25no—
155C22×C4—25no—
156C22×C4—25no—
157C22×C4—25no—
158C22×C4—25no—
159C22×C4—25no—
160C22×C4—25no—
161C22×C4—25no—
162C22×C4—25no—
163C22×C4—25no—
164C22×C4—25no—
165C22×C4—25no—
166C22×C4—25no—
167C22×C4—25no—
168C22×C4—25no—
169C22×C4—25no—
170C22×C4—25no—
171C22×C4—25no—
172C22×C4—25no—
173C22×C4—25no—
174C22×C4—25no—
175C22×C4—25no—
176C22×C4—25no—
177C22×C4—25no—
178C22×C4—25no—
179C22×C4—25no—
180C22×C4—25no—
181C22×C4—25no—
182C22×C4—25no—
183C22×C4—25no—
184C22×C4C2×#41110yes—
185C22×C4C2×#41110yes—
186C22×C4C2×#41110yes—
187C22×C4H(5,𝔽2)110yes—
188C22×C42−1+4110yes—
189C4⋊C4—240no—
190C4⋊C4—420no—
191C4⋊C4—420no—
192C4⋊C4—420no—
193C4⋊C4—420no—
194C4⋊C4—810no—
195C4⋊C4—810no—
196C22⋊C4—420no—
197C22⋊C4—810no—
198C22⋊C4—810no—
199C42—420no—
200C42—810no—
201C42—810no—
202C23—45no—
203C23—45no—
204C23—45no—
205C23—45no—
206C23—45no—
207C23—45no—
208C23—45no—
209C23C22×#4115yes—
210C23C2×H(5,𝔽2)15yes—
211C23C2×2−1+415yes—
212C2×C4—210no—
213C2×C4—210no—
214C2×C4—210no—
215C2×C4—210no—
216C2×C4—210no—
217C2×C4—210no—
218C2×C4—210no—
219C2×C4—210no—
220C2×C4—210no—
221C2×C4—210no—
222C2×C4—210no—
223C2×C4—210no—
224C2×C4—45no—
225C2×C4—45no—
226C2×C4—45no—
227C2×C4—45no—
228C2×C4—45no—
229C2×C4—45no—
230C2×C4—45no—
231C2×C4—45no—
232C2×C4—45no—
233C2×C4—45no—
234C2×C4—45no—
235C2×C4—45no—
236C2×C4—45no—
237C2×C4—45no—
238C2×C4—45no—
239C2×C4—45no—
240C2×C4—45no—
241C2×C4—45no—
242C2×C4—45no—
243C2×C4—45no—
244C2×C4—45no—
245C2×C4—45no—
246C2×C4—45no—
247C2×C4—45no—
248C2×C4—45no—
249C2×C4—45no—
250C2×C4—45no—
251C2×C4—45no—
252C2×C4—45no—
253C2×C4#553120yes—
254C22—85no—
255C22—85no—
256C22—85no—
257C22C2×#55315yes—
258C22#348215yes—
259C22#351115yes—
260C22#351915yes—
261C22#352315yes—
262C22#353615yes—
263C22#355215yes—
264C4—220no—
265C4—410no—
266C4—410no—
267C4—85no—
268C4—85no—
269C4—85no—
270C2—161no—
271C2#4680015yes—
272C2#5967715yes—
273C2#5969515yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17301C2semidirect
2. #16007C2semidirect
6. #14092C2semidirect
7. #14046C2semidirect

Maximal subgroups

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

subgroupindex
#173012
#160072
#157102
#156162
#142062
#140922
#140462