Finite Group Databasebeta

← #492513

Subgroups — #492513

Order 512.

535 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 124 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#33838C212yesmaximal, complemented
2#19968C212yesmaximal, complemented
3#12518C211yesmaximal, characteristic, complemented
4#12435C212yesmaximal, complemented
5#12428C212yesmaximal, complemented
6#12323C212yesmaximal, complemented
7#12316C212yesmaximal, complemented
8C8×D16C211yesmaximal, characteristic, complemented
9#12039C211yesmaximal, characteristic, complemented
10C2×#509—22no—
11#3294—22no—
12#3283C2212yes—
13D4×D8C2212yescomplemented
14#2248C2212yescomplemented
15#2247C2211yescharacteristic, complemented
16#2246C2211yescharacteristic
17#2220—22no—
18#2208C2212yescomplemented
19#2206C2212yes—
20#2187—22no—
21#2186—22no—
22C4×D16C2211yescharacteristic
23#2169C2212yescomplemented
24#1709C2212yescomplemented
25#1681—22no—
26#1669C2212yescomplemented
27#1667C2212yes—
28#1639—22no—
29#1625—22no—
30C8×D8C2211yescharacteristic
31C8×D8C2212yescomplemented
32#1371C2211yescharacteristic, complemented
33#1370C2211yescharacteristic, complemented
34#1362C2212yescomplemented
35#1361C2212yescomplemented
36#1331C2212yescomplemented
37D4⋊C16C2212yescomplemented
38C8×C16C2211yescharacteristic, complemented
39C2×#135—22no—
40C22×D8—22no—
41#547—22no—
42D42—22no—
43#509—48no—
44C2×Q32—41no—
45C2×SD32—22no—
46C2×SD32—22no—
47C2×SD32—22no—
48C2×D16—22no—
49C2×D16—41no—
50C2×D16D411yescharacteristic
51C2×M32—22no—
52#494C2312yes—
53#493C2312yes—
54#486—22no—
55#486—22no—
56#480—42no—
57#474—22no—
58#467—22no—
59#466—22no—
60#466—42no—
61#461—22no—
62#459—22no—
63#459—22no—
64#452—42no—
65#447—42no—
66#447—42no—
67C4×D8D411yescharacteristic
68C4×D8C2312yes—
69C8×D4—22no—
70C8×D4C2312yes—
71C2×#102—42no—
72#366D411yescharacteristic
73C4⋊C16C2312yes—
74#358—22no—
75#357—22no—
76#357D412yes—
77C22⋊C16—22no—
78C4×C16C2311yescharacteristic
79C4×C16C2311yescharacteristic
80#335D411yescharacteristic
81#335C2312yes—
82#332—22no—
83#331—22no—
84D4⋊C8—22no—
85D4⋊C8D412yes—
86C82C2311yescharacteristic
87C2×#41—42no—
88C22×D4—42no—
89C22×D4—42no—
90#135—44no—
91#135—82no—
92C2×Q16—22no—
93C2×Q16—22no—
94C2×SD16—42no—
95C2×D8—22no—
96C2×D8—22no—
97C2×D8—42no—
98C2×D8—44no—
99C2×D8—42no—
100C2×D8—82no—
101C2×D8D811yescharacteristic
102C2×D8C2×D412yes—
103C22×C8—22no—
104C22×C8—42no—
105C4⋊Q8—22no—
106#127—22no—
107#123—82no—
108C22⋊Q8—42no—
109#121—42no—
110#121—42no—
111#121—42no—
112#120—82no—
113C4×D4—22no—
114C4×D4C2×D412yes—
115C2×C4⋊C4—42no—
116Q32—82no—
117SD32—44no—
118SD32—44no—
119SD32—82no—
120D16—44no—
121D16—82no—
122D16D814yescomplemented
123M32—44no—
124C2×C16—21no—
125C2×C16—22no—
126C2×C16C2×D411yescharacteristic
127C2×C16C2×D411yescharacteristic
128#107—42no—
129#107D811yescharacteristic
130#107C2×D412yes—
131C4⋊C8—22no—
132C4⋊C8C2×D412yes—
133#103—42no—
134#102—22no—
135#102—42no—
136#102—42no—
137#102—42no—
138#102—42no—
139#102—42no—
140#102—82no—
141#102D812yes—
142C22⋊C8—22no—
143C22⋊C8—42no—
144C4×C8C2×D411yescharacteristic
145C4×C8C2×D411yescharacteristic
146C4×C8C2411yescharacteristic
147C24—82no—
148#41—84no—
149#41—162no—
150C2×Q8—42no—
151C2×D4—22no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—82no—
156C2×D4—84no—
157C2×D4—82no—
158C2×D4—84no—
159C2×D4—82no—
160C2×D4—162no—
161C2×D4—162no—
162C2×D4C2×D812yes—
163C22×C4—22no—
164C22×C4—42no—
165C22×C4—82no—
166Q16—42no—
167Q16—42no—
168Q16—82no—
169SD16—84no—
170D8—22no—
171D8—41no—
172D8—42no—
173D8—42no—
174D8—82no—
175D8—84no—
176D8—162no—
177D8C2×D814yes—
178C2×C8—22no—
179C2×C8—21no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—162no—
184C2×C8C2×D811yescharacteristic
185C2×C8C2×D811yescharacteristic
186C2×C8C22×D411yescharacteristic
187C2×C8H(5,𝔽2)11yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—82no—
192C4⋊C4C2×D812yes—
193C22⋊C4—22no—
194C22⋊C4—42no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C42C22×D411yescharacteristic
198C16—22no—
199C16—21no—
200C16—42no—
201C16C2×D812yes—
202C23—22no—
203C23—42no—
204C23—82no—
205C23—164no—
206C23—164no—
207C23—322no—
208Q8—82no—
209Q8—82no—
210D4—22no—
211D4—22no—
212D4—24no—
213D4—42no—
214D4—82no—
215D4—82no—
216D4—82no—
217D4—82no—
218D4—162no—
219D4—164no—
220D4—164no—
221D4—164no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—164no—
230C2×C4D4211yescharacteristic
231C2×C4C22×D811yescharacteristic
232C2×C4#57611yescharacteristic
233C8—21no—
234C8—21no—
235C8—21no—
236C8—41no—
237C8—42no—
238C8—82no—
239C8C22×D811yescharacteristic
240C8#57611yescharacteristic
241C22—42no—
242C22—42no—
243C22—42no—
244C22—44no—
245C22—82no—
246C22—82no—
247C22—162no—
248C22—162no—
249C22—162no—
250C22—328no—
251C22D4×D811yescharacteristic
252C4—21no—
253C4—42no—
254C4—82no—
255C4—162no—
256C4D4×D811yescharacteristic
257C4#329211yescharacteristic
258C4#341411yescharacteristic
259C4#341511yescharacteristic
260C2—82no—
261C2—84no—
262C2—322no—
263C2D8211yescharacteristic
264C2#3383911yescharacteristic
265C2#3385711yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33838C2semidirect
2. #19968C2semidirect
3. #12518C2semidirect
4. #12435C2semidirect
5. #12428C2semidirect
6. #12323C2semidirect
7. #12316C2semidirect
8. C8×D16C2semidirect
9. #12039C2semidirect
13. D4×D8C22semidirect
14. #2248C22semidirect
15. #2247C22semidirect
18. #2208C22semidirect
23. #2169C22semidirect
24. #1709C22semidirect
26. #1669C22semidirect
31. C8×D8C22semidirect
32. #1371C22semidirect
33. #1370C22semidirect
34. #1362C22semidirect
35. #1361C22semidirect
36. #1331C22semidirect
37. D4⋊C16C22semidirect
38. C8×C16C22semidirect
122. D16D8semidirect

Maximal subgroups

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

subgroupindex
#338382
#199682
#125182
#124352
#124282
#123232
#123162
C8×D162
#120392