Finite Group Databasebeta

← #319277

Subgroups — #319277

Order 512.

1142 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 202 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#33207C211yesmaximal, characteristic, complemented
2#27960C212yesmaximal, complemented
3#25717C212yesmaximal
4C2×#1957C211yesmaximal, characteristic
5#10677C212yesmaximal, complemented
6#10315C212yesmaximal, complemented
7#9829C214yesmaximal, complemented
8#8144C211yesmaximal, characteristic, complemented
9C22×#127C2211yescharacteristic
10C2×#498C2211yescharacteristic, complemented
11C2×#492C2214yes—
12C2×#450—24no—
13C22×#102—22no—
14C22×C4×C8C2211yescharacteristic
15#2857C2212yes—
16C2×#384C2211yescharacteristic
17#2099—24no—
18#2014—24no—
19#1990C2214yescomplemented
20#1990C2212yescomplemented
21#1957—22no—
22#1957C2212yes—
23#1957C2212yes—
24#1957C2214yes—
25#1941—22no—
26#1844C2211yescharacteristic, complemented
27#1834C2214yescomplemented
28#1823—24no—
29#1807—22no—
30C4×C22⋊C8C2212yes—
31C2×#322C2212yescomplemented
32#1273C2212yescomplemented
33C23×D4—21no—
34C23×D4—22no—
35C22×SD16—28no—
36C23×C8D412yes—
37C2×C4⋊Q8C2312yes—
38C2×#127—24no—
39C2×#127—21no—
40C2×#127—22no—
41C2×#127—42no—
42C2×#127C2312yes—
43C2×C22⋊Q8—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C42C2311yescharacteristic
47#498—42no—
48#492—28no—
49#492—44no—
50#450—416no—
51C2×#106—22no—
52C2×#106—22no—
53C2×#102—24no—
54C2×#102—24no—
55C2×#102—28no—
56C2×#102—48no—
57C2×C22⋊C8D414yes—
58C2×C4×C8—42no—
59C2×C4×C8D412yes—
60C2×C4×C8C2314yes—
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#391—22no—
64#389—22no—
65#389—22no—
66#384—22no—
67#384—21no—
68#384C2312yes—
69#384C2312yes—
70#380—22no—
71#380—22no—
72#340—24no—
73#340—28no—
74#340—28no—
75#336D414yes—
76#322—42no—
77C25—41no—
78C22×Q8—22no—
79C22×D4—24no—
80C22×D4—24no—
81C22×D4—22no—
82C22×D4—48no—
83C22×D4—44no—
84C22×D4—42no—
85C22×D4—41no—
86C22×D4—42no—
87C22×D4—44no—
88C22×D4—48no—
89C22×D4—84no—
90C23×C4—41no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C2×SD16—416no—
94C2×SD16—416no—
95C2×SD16—88no—
96C22×C8—42no—
97C22×C8—42no—
98C22×C8SD1618yescomplemented
99C22×C8C2×D418yes—
100C22×C8C2×D412yes—
101C22×C8C2×D412yes—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104#127—24no—
105#127—22no—
106#127—24no—
107#127—48no—
108#127—42no—
109#127#4114yes—
110C22⋊Q8—48no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—48no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42C2×D412yes—
127C2×C42C2×D411yescharacteristic
128C2×C42C2411yescharacteristic
129#106—44no—
130#106—44no—
131#102—416no—
132#102—48no—
133#102—48no—
134C22⋊C8—28no—
135C4×C8—24no—
136C4×C8—22no—
137C4×C8—24no—
138C4×C8—22no—
139C4×C8—42no—
140#95—42no—
141#95—42no—
142C24—44no—
143C24—42no—
144C24—82no—
145C24—84no—
146C24—82no—
147C24—84no—
148C24#12711yescharacteristic
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×D4—24no—
153C2×D4—28no—
154C2×D4—416no—
155C2×D4—48no—
156C2×D4—44no—
157C2×D4—42no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—48no—
161C2×D4—44no—
162C2×D4—48no—
163C2×D4—832no—
164C2×D4—82no—
165C2×D4—88no—
166C2×D4—84no—
167C2×D4—82no—
168C2×D4—84no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—82no—
178C22×C4—84no—
179C22×C4#12711yescharacteristic
180C22×C4#12712yes—
181C22×C4C2×SD1614yes—
182C22×C4AGL(1,ℤ/8)18yes—
183C22×C4C22×D411yescharacteristic
184C22×C4C22×D412yes—
185SD16—832no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—28no—
191C2×C8—28no—
192C2×C8—28no—
193C2×C8—48no—
194C2×C8C2×SD1614yes—
195C2×C8C2×SD1614yes—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—48no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—82no—
205C4⋊C4—84no—
206C4⋊C4—84no—
207C22⋊C4—84no—
208C42—21no—
209C42—21no—
210C42—24no—
211C42C2×#4112yes—
212C42H(5,𝔽2)12yes—
213C23—21no—
214C23—22no—
215C23—21no—
216C23—44no—
217C23—88no—
218C23—816no—
219C23—82no—
220C23—84no—
221C23—82no—
222C23—84no—
223C23—164no—
224C23—168no—
225C23—164no—
226C23#49212yes—
227C23#49614yes—
228C23C2×#12711yescharacteristic
229Q8—88no—
230D4—48no—
231D4—416no—
232D4—88no—
233D4—816no—
234D4—84no—
235D4—88no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—216no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—84no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—164no—
252C2×C4#49212yes—
253C2×C4#49614yes—
254C2×C4#52511yescharacteristic
255C2×C4#53411yescharacteristic
256C2×C4#54614yes—
257C2×C4C22×SD1612yes—
258C2×C4C2×AGL(1,ℤ/8)14yes—
259C2×C4#57512yes—
260C2×C4#57714yes—
261C8—24no—
262C8—24no—
263C8—48no—
264C22—22no—
265C22—24no—
266C22—22no—
267C22—24no—
268C22—84no—
269C22—88no—
270C22—1616no—
271C22—162no—
272C22—162no—
273C22—164no—
274C22#171514yes—
275C22#253911yescharacteristic
276C22C2×#49211yescharacteristic
277C22C2×#49612yes—
278C22#315311yescharacteristic
279C22#315412yes—
280C4—22no—
281C4—24no—
282C4—162no—
283C4—162no—
284C4—162no—
285C4#310112yes—
286C4#318712yes—
287C4#331018yes—
288C2—24no—
289C2—164no—
290C2C2×#171511yescharacteristic
291C2#1965111yescharacteristic
292C2#1967812yes—
293C2#2830211yescharacteristic
294C2#2834412yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33207C2semidirect
2. #27960C2semidirect
5. #10677C2semidirect
6. #10315C2semidirect
7. #9829C2semidirect
8. #8144C2semidirect
10. C2×#498C22semidirect
19. #1990C22semidirect
20. #1990C22semidirect
26. #1844C22semidirect
27. #1834C22semidirect
31. C2×#322C22semidirect
32. #1273C22semidirect
98. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332072
#279602
#257172
C2×#19572
#106772
#103152
#98292
#81442