Finite Group Databasebeta

← #492260

Subgroups — #492260

Order 512.

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

normal subgroups only

This group has 515 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#33861C212yesmaximal, complemented
2#19883C212yesmaximal, complemented
3#12503C211yesmaximal, characteristic, complemented
4#12483C212yesmaximal, complemented
5#12467C212yesmaximal, complemented
6#12392C212yesmaximal, complemented
7#12316C212yesmaximal, complemented
8#12081C211yesmaximal, characteristic, complemented
9C16×D8C211yesmaximal, characteristic, complemented
10#3334C2212yes—
11D4×D8C2212yescomplemented
12#3272—22no—
13#2252C2211yescharacteristic, complemented
14#2246C2211yescharacteristic
15#2240C2212yescomplemented
16#2232—22no—
17#2224C2212yescomplemented
18#2215—22no—
19#2206C2212yes—
20#2184—22no—
21C4×D16C2211yescharacteristic
22D4×C16C2212yescomplemented
23C2×#357—22no—
24#1706C2212yescomplemented
25#1685—22no—
26#1673C2212yescomplemented
27#1667C2212yes—
28#1639—22no—
29#1625—22no—
30#1589C2212yescomplemented
31C8×D8C2211yescharacteristic
32#1371C2211yescharacteristic, complemented
33#1370C2211yescharacteristic, complemented
34#1364C2212yescomplemented
35#1361C2212yescomplemented
36#1331C2212yescomplemented
37D4⋊C16C2212yescomplemented
38C8×C16C2211yescharacteristic, complemented
39C2×AGL(1,ℤ/8)—42no—
40C22×D8—22no—
41#547—22no—
42D42—22no—
43C2×D16—22no—
44C2×D16—41no—
45C2×D16D411yescharacteristic
46C22×C16—22no—
47#500C2312yes—
48#493C2312yes—
49#492—22no—
50#486—22no—
51#480—42no—
52#474—22no—
53#468—42no—
54#466—22no—
55#466—22no—
56#461—22no—
57#459—22no—
58#459—22no—
59#450—42no—
60#449—42no—
61#447—42no—
62C4×D8D411yescharacteristic
63C4×D8C2312yes—
64#436—22no—
65C8×D4C2312yes—
66C2×#107—22no—
67#366—22no—
68#366—41no—
69#366D411yescharacteristic
70C4⋊C16C2312yes—
71#357—22no—
72#357—22no—
73#357—22no—
74#357—22no—
75#357D412yes—
76C22⋊C16—22no—
77C4×C16C2311yescharacteristic
78C4×C16C2311yescharacteristic
79#335D411yescharacteristic
80#335C2312yes—
81#332—22no—
82#331—22no—
83D4⋊C8—22no—
84D4⋊C8D412yes—
85C82C2311yescharacteristic
86C2×#41—42no—
87C22×D4—42no—
88C22×D4—42no—
89AGL(1,ℤ/8)—88no—
90C2×SD16—42no—
91C2×SD16—42no—
92C2×SD16—42no—
93C2×D8—22no—
94C2×D8—22no—
95C2×D8—42no—
96C2×D8—44no—
97C2×D8—42no—
98C2×D8—82no—
99C2×D8D811yescharacteristic
100C2×D8C2×D412yes—
101C2×M16—42no—
102C22×C8—22no—
103C4⋊Q8—22no—
104#127—22no—
105#123—82no—
106C22⋊Q8—42no—
107#121—42no—
108#121—42no—
109#121—42no—
110#120—82no—
111C4×D4—22no—
112C4×D4C2×D412yes—
113C2×C4⋊C4—42no—
114D16—41no—
115D16—82no—
116D16—82no—
117C2×C16—21no—
118C2×C16—22no—
119C2×C16—82no—
120C2×C16C2×D411yescharacteristic
121C2×C16C2×D411yescharacteristic
122#107—22no—
123#107—22no—
124#107—42no—
125#107D811yescharacteristic
126#107C2×D412yes—
127C4⋊C8—22no—
128C4⋊C8C2×D412yes—
129#103—42no—
130#102—22no—
131#102—42no—
132#102—42no—
133#102—42no—
134#102—82no—
135#102D812yes—
136C22⋊C8—22no—
137C22⋊C8—42no—
138C4×C8C2×D411yescharacteristic
139C4×C8C2×D411yescharacteristic
140C4×C8C2411yescharacteristic
141C24—82no—
142#41—84no—
143#41—162no—
144C2×Q8—42no—
145C2×D4—22no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—82no—
150C2×D4—84no—
151C2×D4—82no—
152C2×D4—84no—
153C2×D4—82no—
154C2×D4—162no—
155C2×D4—162no—
156C2×D4C2×D812yes—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—82no—
160SD16—84no—
161SD16—84no—
162SD16—162no—
163D8—22no—
164D8—22no—
165D8—42no—
166D8—42no—
167D8—82no—
168D8—84no—
169D8—84no—
170D8D1614yescomplemented
171M16—84no—
172C2×C8—22no—
173C2×C8—21no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8C2×D811yescharacteristic
178C2×C8C2×D811yescharacteristic
179C2×C8C22×D411yescharacteristic
180C2×C8H(5,𝔽2)11yescharacteristic
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—82no—
185C4⋊C4C2×D812yes—
186C22⋊C4—22no—
187C22⋊C4—42no—
188C22⋊C4—82no—
189C22⋊C4—82no—
190C42C22×D411yescharacteristic
191C16—22no—
192C16—21no—
193C16—21no—
194C16—42no—
195C23—22no—
196C23—42no—
197C23—82no—
198C23—164no—
199C23—164no—
200C23—322no—
201Q8—82no—
202Q8—82no—
203D4—22no—
204D4—42no—
205D4—42no—
206D4—82no—
207D4—82no—
208D4—82no—
209D4—82no—
210D4—162no—
211D4—164no—
212D4—164no—
213D4—164no—
214D4C2×D1614yes—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—164no—
223C2×C4D4211yescharacteristic
224C2×C4C22×D811yescharacteristic
225C2×C4#57611yescharacteristic
226C8—21no—
227C8—21no—
228C8—41no—
229C8—42no—
230C8—82no—
231C8C2×D1612yes—
232C8#57511yescharacteristic
233C8#57611yescharacteristic
234C22—24no—
235C22—42no—
236C22—42no—
237C22—82no—
238C22—82no—
239C22—82no—
240C22—162no—
241C22—162no—
242C22—162no—
243C22—328no—
244C22D4×D811yescharacteristic
245C4—21no—
246C4—42no—
247C4—82no—
248C4—162no—
249C4#327311yescharacteristic
250C4#329111yescharacteristic
251C4C22×D1611yescharacteristic
252C4#341511yescharacteristic
253C2—44no—
254C2—162no—
255C2—322no—
256C2#1988411yescharacteristic
257C2D4×D1611yescharacteristic
258C2#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. #33861C2semidirect
2. #19883C2semidirect
3. #12503C2semidirect
4. #12483C2semidirect
5. #12467C2semidirect
6. #12392C2semidirect
7. #12316C2semidirect
8. #12081C2semidirect
9. C16×D8C2semidirect
11. D4×D8C22semidirect
13. #2252C22semidirect
15. #2240C22semidirect
17. #2224C22semidirect
22. D4×C16C22semidirect
24. #1706C22semidirect
26. #1673C22semidirect
30. #1589C22semidirect
32. #1371C22semidirect
33. #1370C22semidirect
34. #1364C22semidirect
35. #1361C22semidirect
36. #1331C22semidirect
37. D4⋊C16C22semidirect
38. C8×C16C22semidirect
170. D8D16semidirect

Maximal subgroups

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

subgroupindex
#338612
#198832
#125032
#124832
#124672
#123922
#123162
#120812
C16×D82