Finite Group Databasebeta

← #341577

Subgroups — #341577

Order 512.

538 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 146 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#28792C212yesmaximal, complemented
2#26530C212yesmaximal
3#10671C211yesmaximal, characteristic
4#10657C212yesmaximal, complemented
5#10579C212yesmaximal, complemented
6#10351C211yesmaximal, characteristic
7#9563C212yesmaximal
8#9534C212yesmaximal, complemented
9#8920C211yesmaximal, characteristic
10#2571C2211yescharacteristic
11#2552C2212yes—
12#2366C2211yescharacteristic
13#2094—22no—
14#2085—22no—
15#2065—22no—
16#2059—22no—
17#1991—22no—
18#1982C2212yes—
19#1957C2212yes—
20#1957C2211yescharacteristic
21#1948C2212yes—
22#1947C2212yes—
23#1947C2212yes—
24#1942—22no—
25#1939C2212yes—
26#1864—22no—
27#1845C2211yescharacteristic
28#1834C2212yes—
29#1802—22no—
30C4×#106C2212yes—
31C8×C4⋊C4C2212yes—
32C2×#331C2211yescharacteristic
33#1296C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C2×#127C2311yescharacteristic
40C2×#125—22no—
41C2×C4×D4—22no—
42C2×#107—22no—
43C2×#107—22no—
44C2×#107—41no—
45C2×#106—22no—
46C2×#106D412yes—
47C2×#106D412yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#102—22no—
52C2×#102—22no—
53C2×#102—22no—
54C2×#102—22no—
55C2×#102—41no—
56C2×C4×C8—22no—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2312yes—
59#400—41no—
60#398—42no—
61#390—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#386—41no—
66#384—22no—
67#384C2311yescharacteristic
68#384C2312yes—
69#382—22no—
70C4×C4⋊C4C2312yes—
71C4×C4⋊C4C2312yes—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340D412yes—
78#340D412yes—
79#336—22no—
80#336—22no—
81#331Q814yes—
82C22×D4—22no—
83C22×D4—21no—
84C23×C4—41no—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8C2×D412yes—
89C22×C8C2×D412yes—
90#127—21no—
91#127—22no—
92#127C2×Q812yes—
93#125—44no—
94#125—82no—
95C4×D4—44no—
96C4×D4—82no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2×D412yes—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D412yes—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—41no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42C2411yescharacteristic
115#107—44no—
116#107—82no—
117#107—82no—
118#106—24no—
119#106—42no—
120#106—82no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—82no—
125C4⋊C8C2×Q814yes—
126#102—44no—
127#102—82no—
128#102—82no—
129#102—82no—
130#102—82no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—22no—
134C4×C8—82no—
135#95—22no—
136#95—22no—
137#95—41no—
138#95—42no—
139#95—42no—
140C24—41no—
141C2×D4—22no—
142C2×D4—24no—
143C2×D4—42no—
144C2×D4—41no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—81no—
150C2×D4—81no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—84no—
158C22×C4—161no—
159C22×C4C22×D412yes—
160C22×C4H(5,𝔽2)11yescharacteristic
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—24no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—81no—
190C4⋊C4—82no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C4⋊C4—162no—
194C4⋊C4—161no—
195C4⋊C4#13514yes—
196C22⋊C4—84no—
197C22⋊C4—161no—
198C42—42no—
199C42—42no—
200C42—42no—
201C42—42no—
202C42C22×Q811yescharacteristic
203C422−1+411yescharacteristic
204C422−1+412yes—
205C23—42no—
206C23—82no—
207C23—82no—
208C23—81no—
209C23—81no—
210C23D4211yescharacteristic
211D4—44no—
212D4—44no—
213D4—82no—
214D4—82no—
215D4—82no—
216D4—161no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4—82no—
236C2×C4—164no—
237C2×C4#53612yes—
238C2×C4#54411yescharacteristic
239C2×C4D4×Q812yes—
240C2×C4#56311yescharacteristic
241C2×C4C2×#13512yes—
242C2×C4#57512yes—
243C2×C4#57612yes—
244C2×C4#57712yes—
245C8—42no—
246C8—44no—
247C8—42no—
248C8—42no—
249C8—82no—
250C22—82no—
251C22—84no—
252C22—161no—
253C22—162no—
254C22—161no—
255C22#267711yescharacteristic
256C22#326812yes—
257C22#328212yes—
258C22#328911yescharacteristic
259C22#329211yescharacteristic
260C4—21no—
261C4—21no—
262C4—44no—
263C4—82no—
264C4—82no—
265C4—82no—
266C4—161no—
267C4—162no—
268C4#323312yes—
269C4#324112yes—
270C4#338012yes—
271C4#338812yes—
272C2—162no—
273C2#1983112yes—
274C2#1999611yescharacteristic
275C2#2013811yescharacteristic
276C2#3095812yes—
277C2#3097811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28792C2semidirect
4. #10657C2semidirect
5. #10579C2semidirect
8. #9534C2semidirect

Maximal subgroups

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

subgroupindex
#287922
#265302
#106712
#106572
#105792
#103512
#95632
#95342
#89202