Finite Group Databasebeta

← #4131148

Subgroups — #4131148

Order 512.

1551 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 1551 conjugacy classes of subgroups — too many to draw. Only its 490 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#16233C215yesmaximal
3#15920C215yesmaximal, complemented
4#15734C215yesmaximal
5#14245C215yesmaximal, complemented
6#14109C215yesmaximal, complemented
7#14013C215yesmaximal
8C2×#400C2215yes—
9C2×#388C2215yes—
10C2×#383C2215yes—
11#1443C2215yes—
12#1443C2215yes—
13#1443C2215yes—
14#1443C2215yes—
15#1443C2215yes—
16#1442C2215yes—
17#1442C2215yes—
18#1441C2215yes—
19#1441C2215yes—
20#1440C2215yes—
21#1440C2215yes—
22#1439C2215yes—
23#1439C2215yes—
24#1437C2215yes—
25#1437C2215yes—
26#1436C2215yes—
27#1436C2215yes—
28#1436C2215yes—
29#1436C2215yes—
30#1436C2215yes—
31#1436C2215yes—
32#1436C2215yes—
33#1436C2215yes—
34C4⋊#95C2215yes—
35#1434C2215yes—
36#1434C2215yes—
37C4×#95C2215yes—
38C4×#95C2215yes—
39C22×C4⋊C4C2315yes—
40C22×C4⋊C4C2315yes—
41C22×C4⋊C4C2315yes—
42C22×C22⋊C4C2315yes—
43C22×C22⋊C4C2315yes—
44C22×C22⋊C4C2315yes—
45C22×C42C2315yes—
46C22×C42C2315yes—
47C22×C42C2315yes—
48#400—220no—
49#388—220no—
50#383—220no—
51C2×#95C2315yes—
52C2×#95C2315yes—
53C2×#95C2315yes—
54C2×#95C2315yes—
55C2×#95C2315yes—
56C2×#95C2315yes—
57C2×#95C2315yes—
58C2×#95C2315yes—
59C2×#95C2315yes—
60C2×#95C2315yes—
61C2×#95C2315yes—
62C2×#95C2315yes—
63C2×#95C2315yes—
64C2×#95C2315yes—
65C2×#95C2315yes—
66C2×#95C2315yes—
67C2×#95C2315yes—
68C2×#95C2315yes—
69C2×#95C2315yes—
70C2×#95C2315yes—
71C2×#95C2315yes—
72C2×#95C2315yes—
73C25C2411yescharacteristic
74C23×C4C2415yes—
75C23×C4C2415yes—
76C23×C4C2415yes—
77C23×C4C2415yes—
78C23×C4C2415yes—
79C23×C4C2415yes—
80C2×C4⋊C4—210no—
81C2×C4⋊C4—210no—
82C2×C4⋊C4—210no—
83C2×C4⋊C4—45no—
84C2×C4⋊C4—45no—
85C2×C4⋊C4—45no—
86C2×C4⋊C4—45no—
87C2×C4⋊C4—45no—
88C2×C4⋊C4—45no—
89C2×C22⋊C4—210no—
90C2×C22⋊C4—210no—
91C2×C22⋊C4—210no—
92C2×C22⋊C4—45no—
93C2×C22⋊C4—45no—
94C2×C22⋊C4—45no—
95C2×C22⋊C4—45no—
96C2×C22⋊C4—45no—
97C2×C22⋊C4—45no—
98C2×C42—210no—
99C2×C42—210no—
100C2×C42—45no—
101C2×C42—45no—
102C2×C42—45no—
103C2×C42—45no—
104C2×C42—45no—
105C2×C42—45no—
106C2×C42—45no—
107#95—210no—
108#95—210no—
109#95—210no—
110#95—210no—
111#95—210no—
112#95—210no—
113#95—210no—
114#95—210no—
115#95—210no—
116#95—210no—
117#95—45no—
118#95—45no—
119#95—45no—
120#95—45no—
121#95—45no—
122#95—45no—
123#95—45no—
124#95—45no—
125#95—45no—
126#95—45no—
127#95—45no—
128#95#41120yes—
129C24—25no—
130C24—25no—
131C24—25no—
132C24C2511yescharacteristic
133C22×C4—25no—
134C22×C4—25no—
135C22×C4—25no—
136C22×C4—25no—
137C22×C4—25no—
138C22×C4—25no—
139C22×C4—25no—
140C22×C4—25no—
141C22×C4—25no—
142C22×C4—25no—
143C22×C4—25no—
144C22×C4—25no—
145C22×C4—25no—
146C22×C4—25no—
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×C4C2×#41110yes—
173C22×C4C2×#41110yes—
174C22×C4C2×#41110yes—
175C4⋊C4—810no—
176C4⋊C4—810no—
177C4⋊C4—810no—
178C22⋊C4—420no—
179C22⋊C4—810no—
180C22⋊C4—810no—
181C42—420no—
182C42—810no—
183C42—810no—
184C23—45no—
185C23—45no—
186C23—45no—
187C23—45no—
188C23—45no—
189C23—45no—
190C23—45no—
191C23C22×#4115yes—
192C23#58515yes—
193C23#58515yes—
194C2×C4—210no—
195C2×C4—210no—
196C2×C4—210no—
197C2×C4—210no—
198C2×C4—210no—
199C2×C4—210no—
200C2×C4—210no—
201C2×C4—210no—
202C2×C4—210no—
203C2×C4—45no—
204C2×C4—45no—
205C2×C4—45no—
206C2×C4—45no—
207C2×C4—45no—
208C2×C4—45no—
209C2×C4—45no—
210C2×C4—45no—
211C2×C4—45no—
212C2×C4—45no—
213C2×C4—45no—
214C2×C4—45no—
215C2×C4—45no—
216C2×C4—45no—
217C2×C4—45no—
218C2×C4—45no—
219C2×C4—45no—
220C2×C4—45no—
221C2×C4—45no—
222C2×C4—45no—
223C2×C4—45no—
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—
236C22—85no—
237C22—85no—
238C22—85no—
239C22#347515yes—
240C22#348015yes—
241C22#351215yes—
242C22#352115yes—
243C22#352115yes—
244C22#356615yes—
245C22#356815yes—
246C4—410no—
247C4—410no—
248C4—410no—
249C4—85no—
250C4—85no—
251C4—85no—
252C2—161no—
253C2#4730915yes—
254C2#5977415yes—
255C2#5990715yes—
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
3. #15920C2semidirect
5. #14245C2semidirect
6. #14109C2semidirect

Maximal subgroups

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

subgroupindex
#173012
#162332
#159202
#157342
#142452
#141092
#140132