Finite Group Databasebeta

← #492224

Subgroups — #492224

Order 512.

525 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 126 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#33901C211yesmaximal, characteristic, complemented
2D4×D16C211yesmaximal, characteristic, complemented
3#19883C212yesmaximal, complemented
4#12503C211yesmaximal, characteristic, complemented
5#12483C212yesmaximal, complemented
6#12451C211yesmaximal, characteristic, complemented
7#12430C211yesmaximal, characteristic, complemented
8#12368C212yesmaximal, complemented
9#12340C212yesmaximal, complemented
10#12081C211yesmaximal, characteristic, complemented
11C16×D8C211yesmaximal, characteristic, complemented
12C22×D16—21no—
13#3334C2212yescomplemented
14D4×D8C2212yes—
15#3272—22no—
16#2252C2211yescharacteristic, complemented
17#2246C2211yescharacteristic
18#2240C2212yescomplemented
19#2232—22no—
20#2224C2212yescomplemented
21#2215—21no—
22#2215—21no—
23#2206C2212yes—
24#2184—22no—
25C4×D16C2211yescharacteristic
26D4×C16C2211yescharacteristic
27D4×C16C2211yescharacteristic, complemented
28C2×#366—21no—
29#1706C2212yescomplemented
30#1685—22no—
31#1673C2212yes—
32#1667C2212yescomplemented
33#1639—22no—
34#1625—22no—
35#1589C2212yescomplemented
36C8×D8C2211yescharacteristic
37#1371C2211yescharacteristic, complemented
38#1370C2211yescharacteristic, complemented
39#1364C2212yescomplemented
40#1361C2212yescomplemented
41#1331C2212yescomplemented
42D4⋊C16C2211yescharacteristic
43D4⋊C16C2211yescharacteristic, complemented
44C8×C16C2211yescharacteristic, complemented
45C2×AGL(1,ℤ/8)—42no—
46C22×D8—22no—
47#547—22no—
48D42—22no—
49C2×D16—22no—
50C2×D16—21no—
51C2×D16—21no—
52C2×D16—44no—
53C2×D16D411yescharacteristic
54C22×C16—21no—
55C22×C16—21no—
56#500C2312yes—
57#493C2312yes—
58#492—22no—
59#486—22no—
60#480—42no—
61#474—22no—
62#468—42no—
63#466—22no—
64#466—22no—
65#461—22no—
66#459—22no—
67#459—22no—
68#450—42no—
69#449—42no—
70#447—42no—
71C4×D8D411yescharacteristic
72C4×D8C2312yes—
73#436—22no—
74C8×D4C2311yescharacteristic
75C8×D4C2311yescharacteristic
76C2×#107—22no—
77#366—21no—
78#366—21no—
79#366—22no—
80#366D411yescharacteristic
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#357—22no—
84#357—22no—
85#357—42no—
86#357D412yes—
87C22⋊C16—21no—
88C22⋊C16—21no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2312yes—
93#332—22no—
94#331—22no—
95D4⋊C8—22no—
96D4⋊C8D411yescharacteristic
97D4⋊C8D411yescharacteristic
98C82C2311yescharacteristic
99C2×#41—42no—
100C22×D4—42no—
101C22×D4—42no—
102AGL(1,ℤ/8)—88no—
103C2×SD16—42no—
104C2×SD16—42no—
105C2×SD16—42no—
106C2×D8—22no—
107C2×D8—22no—
108C2×D8—42no—
109C2×D8—44no—
110C2×D8—42no—
111C2×D8—82no—
112C2×D8D811yescharacteristic
113C2×D8C2×D412yes—
114C2×M16—42no—
115C22×C8—21no—
116C22×C8—21no—
117C4⋊Q8—22no—
118#127—22no—
119#123—82no—
120C22⋊Q8—42no—
121#121—42no—
122#121—42no—
123#121—42no—
124#120—82no—
125C4×D4—22no—
126C4×D4C2×D411yescharacteristic
127C4×D4C2×D411yescharacteristic
128C2×C4⋊C4—42no—
129D16—22no—
130D16—44no—
131D16—42no—
132D16—81no—
133C2×C16—21no—
134C2×C16—21no—
135C2×C16—21no—
136C2×C16—42no—
137C2×C16—42no—
138C2×C16C2×D411yescharacteristic
139C2×C16C2×D411yescharacteristic
140#107—22no—
141#107—22no—
142#107—42no—
143#107D811yescharacteristic
144#107C2×D412yes—
145C4⋊C8—22no—
146C4⋊C8C2×D411yescharacteristic
147C4⋊C8C2×D411yescharacteristic
148#103—42no—
149#102—22no—
150#102—42no—
151#102—42no—
152#102—42no—
153#102—82no—
154#102D811yescharacteristic
155#102D811yescharacteristic
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—42no—
159C4×C8C2×D411yescharacteristic
160C4×C8C2×D411yescharacteristic
161C4×C8C2411yescharacteristic
162C24—82no—
163#41—84no—
164#41—162no—
165C2×Q8—42no—
166C2×D4—22no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—82no—
171C2×D4—84no—
172C2×D4—82no—
173C2×D4—84no—
174C2×D4—82no—
175C2×D4—162no—
176C2×D4—162no—
177C2×D4C2×D811yescharacteristic
178C2×D4C2×D811yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—82no—
183SD16—84no—
184SD16—84no—
185SD16—162no—
186D8—22no—
187D8—22no—
188D8—42no—
189D8—42no—
190D8—82no—
191D8—84no—
192D8—84no—
193D8D1614yescomplemented
194M16—84no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8C2×D811yescharacteristic
204C2×C8C2×D811yescharacteristic
205C2×C8C22×D411yescharacteristic
206C2×C8H(5,𝔽2)11yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—82no—
211C4⋊C4C2×D811yescharacteristic
212C4⋊C4C2×D811yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C42C22×D411yescharacteristic
219C16—21no—
220C16—41no—
221C16—41no—
222C16—41no—
223C16AGL(1,ℤ/8)12yescomplemented
224C23—21no—
225C23—21no—
226C23—42no—
227C23—82no—
228C23—164no—
229C23—164no—
230C23—322no—
231Q8—82no—
232Q8—82no—
233D4—21no—
234D4—21no—
235D4—42no—
236D4—42no—
237D4—82no—
238D4—82no—
239D4—82no—
240D4—82no—
241D4—162no—
242D4—164no—
243D4—164no—
244D4—164no—
245D4C2×D1612yes—
246D4C2×D1612yescomplemented
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—164no—
258C2×C4D4211yescharacteristic
259C2×C4C22×D811yescharacteristic
260C2×C4#57611yescharacteristic
261C8—21no—
262C8—21no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8—82no—
267C8C2×D1612yescomplemented
268C8C2×AGL(1,ℤ/8)11yescharacteristic
269C8#57711yescharacteristic
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—162no—
280C22—162no—
281C22—162no—
282C22—328no—
283C22D4×D811yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—82no—
288C4—162no—
289C4#327211yescharacteristic
290C4#329011yescharacteristic
291C4C22×D1611yescharacteristic
292C4#341511yescharacteristic
293C2—42no—
294C2—42no—
295C2—162no—
296C2—322no—
297C2#1988311yescharacteristic
298C2D4×D1611yescharacteristic
299C2#3385911yescharacteristic
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. #33901C2semidirect
2. D4×D16C2semidirect
3. #19883C2semidirect
4. #12503C2semidirect
5. #12483C2semidirect
6. #12451C2semidirect
7. #12430C2semidirect
8. #12368C2semidirect
9. #12340C2semidirect
10. #12081C2semidirect
11. C16×D8C2semidirect
13. #3334C22semidirect
16. #2252C22semidirect
18. #2240C22semidirect
20. #2224C22semidirect
27. D4×C16C22semidirect
29. #1706C22semidirect
32. #1667C22semidirect
35. #1589C22semidirect
37. #1371C22semidirect
38. #1370C22semidirect
39. #1364C22semidirect
40. #1361C22semidirect
41. #1331C22semidirect
43. D4⋊C16C22semidirect
44. C8×C16C22semidirect
193. D8D16semidirect
223. C16AGL(1,ℤ/8)semidirect
246. D4C2×D16semidirect
267. C8C2×D16semidirect

Maximal subgroups

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

subgroupindex
#339012
D4×D162
#198832
#125032
#124832
#124512
#124302
#123682
#123402
#120812
C16×D82