Finite Group Databasebeta

← #482956

Subgroups — #482956

Order 512.

946 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 276 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#62245C211yesmaximal, characteristic
2#33486C211yesmaximal, characteristic, complemented
3#22705C211yesmaximal, characteristic, complemented
4#11886C214yesmaximal, complemented
5#11856C212yesmaximal, complemented
6#11742C212yesmaximal, complemented
7C2×#1347C214yesmaximal, complemented
8#3496—21no—
9#3334C414yescomplemented
10#3294—24no—
11D4×D8C414yescomplemented
12#3245—21no—
13#3186—22no—
14#3169C2212yes—
15C2×#466C2214yes—
16C2×C8×D4C2211yescharacteristic
17D4×C4⋊C4—21no—
18D4×C16—24no—
19#2162C2212yescomplemented
20C2×C22⋊C16C2214yescomplemented
21#1947—21no—
22#1852—22no—
23#1834C2211yescharacteristic, complemented
24#1808C2212yescomplemented
25C4×#107C2211yescharacteristic, complemented
26C2×#340C2212yescomplemented
27#1347C2218yescomplemented
28#1347C2214yescomplemented
29#1347C2214yescomplemented
30C2×H(5,𝔽2)—41no—
31C22×D8C2×C414yescomplemented
32C23×C8C2312yes—
33#547—24no—
34#546—44no—
35D42—24no—
36#537—41no—
37#534—42no—
38#532—22no—
39C2×#121—24no—
40C2×C4×D4—41no—
41C2×C4×D4D411yescharacteristic
42C22×C4⋊C4—22no—
43C22×C16—24no—
44#500C2×C412yescomplemented
45#493C2×C412yescomplemented
46#486—22no—
47#480—28no—
48#474—24no—
49#466—22no—
50#466—24no—
51#466—24no—
52#466C2×C418yescomplemented
53#466C2314yes—
54#459—24no—
55#447—28no—
56C4×D8—22no—
57C4×D8C2×C414yescomplemented
58C8×D4C2×C414yes—
59C8×D4D414yes—
60C2×#107—21no—
61C2×#107—22no—
62C2×#107C2×C414yescomplemented
63C2×C4⋊C8D411yescharacteristic
64C2×#102—24no—
65C2×C22⋊C8D412yes—
66C2×C4×C8C2311yescharacteristic
67#389—41no—
68#385—42no—
69#384—21no—
70#380—22no—
71C4×C4⋊C4—21no—
72C4⋊C16—22no—
73C22⋊C16—24no—
74C22⋊C16C2314yes—
75C4×C16—22no—
76#340—21no—
77#340—22no—
78#340C2311yescharacteristic
79#340C2312yes—
80#340C2311yescharacteristic
81H(5,𝔽2)—84no—
82H(5,𝔽2)—84no—
83C2×#41—44no—
84C2×#41—42no—
85C22×D4—24no—
86C22×D4—44no—
87C22×D4—44no—
88C22×D4D811yescharacteristic
89C23×C4—42no—
90C23×C4C2×D412yes—
91C2×D8—24no—
92C2×D8—22no—
93C2×D8—84no—
94C2×D8C22×C414yes—
95C22×C8—82no—
96C22×C8C22×C414yes—
97C22×C8C2×D412yes—
98C22×C8C2×D411yescharacteristic
99C22×C8C2×D414yes—
100C22×C8C2411yescharacteristic
101C4⋊Q8—22no—
102#127—22no—
103#124—42no—
104#123—48no—
105C22⋊Q8—44no—
106#121—28no—
107#121—24no—
108#121—44no—
109#121—42no—
110#121—44no—
111#121—44no—
112#120—48no—
113C4×D4—24no—
114C4×D4—42no—
115C4×D4—84no—
116C4×D4C22⋊C414yes—
117C4×D4SD1614yes—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4D811yescharacteristic
128C2×C22⋊C4—44no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4D812yes—
131C2×C42—41no—
132C2×C42C2×D411yescharacteristic
133C2×C16—24no—
134C2×C16—84no—
135#107—24no—
136#107—22no—
137#107—42no—
138#107—82no—
139#107C22×C414yes—
140C4⋊C8C22⋊C414yes—
141#102—28no—
142#102—24no—
143#102—24no—
144C22⋊C8C22⋊C418yes—
145C4×C8C22×C412yes—
146C4×C8C2×D412yes—
147#95—41no—
148#95—42no—
149C24—44no—
150C24C2×D812yes—
151#41—816no—
152#41—84no—
153#41—84no—
154C2×Q8—42no—
155C2×D4—24no—
156C2×D4—44no—
157C2×D4—48no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—84no—
162C2×D4—84no—
163C2×D4—88no—
164C2×D4—88no—
165C2×D4—816no—
166C2×D4#10214yes—
167C2×D4D1614yes—
168C2×D4SD3214yes—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—84no—
177C22×C4—84no—
178C22×C4—162no—
179C22×C4C2×C22⋊C414yes—
180C22×C4C2×D811yescharacteristic
181C22×C4C2×D812yes—
182C22×C4C2×D811yescharacteristic
183C22×C4C2×SD1614yes—
184C22×C4C22×D411yescharacteristic
185D8—44no—
186D8—82no—
187D8—84no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—44no—
191C2×C8—41no—
192C2×C8—44no—
193C2×C8C2×C22⋊C412yes—
194C2×C8C2×C22⋊C414yes—
195C2×C8C23×C411yescharacteristic
196C2×C8C22×D411yescharacteristic
197C4⋊C4—22no—
198C4⋊C4—24no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—84no—
207C4⋊C4—82no—
208C4⋊C4—161no—
209C4⋊C4#10214yes—
210C22⋊C4—44no—
211C22⋊C4—48no—
212C22⋊C4—44no—
213C22⋊C4—84no—
214C22⋊C4#10218yes—
215C42—82no—
216C42C2×C22⋊C412yes—
217C42C2×SD1612yes—
218C16—44no—
219C23—42no—
220C23—42no—
221C23—44no—
222C23—88no—
223C23—164no—
224C23—164no—
225C23C2×#10214yes—
226C23C2×D1614yes—
227C23C2×SD3214yes—
228C23C22×D811yescharacteristic
229Q8—84no—
230D4—44no—
231D4—44no—
232D4—84no—
233D4—84no—
234D4—816no—
235D4—84no—
236D4—162no—
237D4—164no—
238D4#357116yes—
239C2×C4—21no—
240C2×C4—24no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—88no—
251C2×C4—88no—
252C2×C4—82no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—168no—
259C2×C4C2×#10212yes—
260C2×C4C2×#10214yes—
261C2×C4C2×#10212yes—
262C2×C4C2×D1612yes—
263C2×C4C2×SD3212yes—
264C2×C4C22×C22⋊C411yescharacteristic
265C2×C4C22×SD1611yescharacteristic
266C8—21no—
267C8—21no—
268C8—24no—
269C8—22no—
270C22—21no—
271C22—24no—
272C22—21no—
273C22—24no—
274C22—44no—
275C22—44no—
276C22—84no—
277C22—84no—
278C22—84no—
279C22—328no—
280C22C2×#35718yes—
281C22C22×#10211yescharacteristic
282C22C22×D1611yescharacteristic
283C22C22×SD3211yescharacteristic
284C4—22no—
285C4—21no—
286C4—24no—
287C4—82no—
288C4—84no—
289C4—161no—
290C4—162no—
291C4C2×#35714yes—
292C4#288311yescharacteristic
293C4#290111yescharacteristic
294C2—21no—
295C2—21no—
296C2—28no—
297C2—164no—
298C2#1761311yescharacteristic
299C2C22×#35711yescharacteristic
300C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62245C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33486C2semidirect
3. #22705C2semidirect
4. #11886C2semidirect
5. #11856C2semidirect
6. #11742C2semidirect
7. C2×#1347C2semidirect
9. #3334C4semidirect
11. D4×D8C4semidirect
19. #2162C22semidirect
20. C2×C22⋊C16C22semidirect
23. #1834C22semidirect
24. #1808C22semidirect
25. C4×#107C22semidirect
26. C2×#340C22semidirect
27. #1347C22semidirect
28. #1347C22semidirect
29. #1347C22semidirect
31. C22×D8C2×C4semidirect
44. #500C2×C4semidirect
45. #493C2×C4semidirect
52. #466C2×C4semidirect
57. C4×D8C2×C4semidirect
62. C2×#107C2×C4semidirect

Maximal subgroups

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

subgroupindex
#622452
#334862
#227052
#118862
#118562
#117422
C2×#13472