Finite Group Databasebeta

← #492576

Subgroups — #492576

Order 512.

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

normal subgroups only

This group has 489 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#33947C211yesmaximal, characteristic, complemented
2D4×D16C211yesmaximal, characteristic
3#19976C212yesmaximal, complemented
4#12503C211yesmaximal, characteristic, complemented
5#12452C211yesmaximal, characteristic, complemented
6#12439C211yesmaximal, characteristic, complemented
7#12418C212yesmaximal, complemented
8#12372C212yesmaximal, complemented
9#12355C212yesmaximal, complemented
10C16×SD16C211yesmaximal, characteristic, complemented
11C8×D16C211yesmaximal, characteristic, complemented
12C22×D16—21no—
13#3380C2212yescomplemented
14#3295—22no—
15D4×D8C2212yes—
16#2253—21no—
17#2252C2211yescharacteristic, complemented
18#2246C2211yescharacteristic
19#2240C2212yescomplemented
20#2228—22no—
21#2226C2212yescomplemented
22#2215—21no—
23#2206C2212yes—
24#2184—22no—
25Q8×C16C2211yescharacteristic, complemented
26C4×D16—21no—
27C4×D16C2211yescharacteristic
28D4×C16C2211yescharacteristic
29#1706C2212yescomplemented
30#1677—22no—
31#1675C2212yes—
32#1668C2212yescomplemented
33#1629—22no—
34#1622—22no—
35C8×SD16C2211yescharacteristic
36C8×D8C2212yescomplemented
37#1371C2211yescharacteristic, complemented
38#1367C2211yescharacteristic, complemented
39#1365C2212yescomplemented
40#1361C2212yescomplemented
41Q8⋊C16C2211yescharacteristic, complemented
42#1331C2212yescomplemented
43D4⋊C16C2211yescharacteristic
44C8×C16C2211yescharacteristic, complemented
45C22×D8—22no—
46#554—22no—
47D42—22no—
48C2×D16—22no—
49C2×D16—21no—
50C2×D16—21no—
51C2×D16—44no—
52C2×D16D411yescharacteristic
53C22×C16—21no—
54#500C2312yes—
55#499—22no—
56#493C2312yes—
57#492—22no—
58#486—22no—
59#481—42no—
60#478—42no—
61#476—22no—
62#476—22no—
63#466—22no—
64#465—42no—
65#460—22no—
66#459—22no—
67#450—42no—
68#447—42no—
69C8×Q8C2311yescharacteristic
70C4×SD16D411yescharacteristic
71C4×D8—22no—
72C4×D8C2312yes—
73C8×D4—22no—
74C8×D4C2311yescharacteristic
75C2×#102—42no—
76#367—22no—
77#366—21no—
78#366—21no—
79#366D411yescharacteristic
80C4⋊C16—21no—
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#357—22no—
84#357—22no—
85#357—42no—
86#357D412yes—
87C22⋊C16—21no—
88C4×C16—21no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335C2312yes—
92#334D411yescharacteristic
93#333—22no—
94#331—22no—
95Q8⋊C8D411yescharacteristic
96D4⋊C8—22no—
97D4⋊C8D411yescharacteristic
98C82C2311yescharacteristic
99C22×D4—42no—
100C22×D4—42no—
101C2×SD16—42no—
102C2×SD16—82no—
103C2×SD16D811yescharacteristic
104C2×D8—22no—
105C2×D8—22no—
106C2×D8—42no—
107C2×D8—44no—
108C2×D8—82no—
109C2×D8C2×D412yes—
110C22×C8—21no—
111C22×C8—42no—
112C4⋊Q8—22no—
113#127—22no—
114#125—42no—
115C22⋊Q8—42no—
116C22⋊Q8—82no—
117#121—42no—
118#121—42no—
119#120—82no—
120C4×Q8C2×D411yescharacteristic
121C4×D4—22no—
122C4×D4—42no—
123C4×D4C2×D411yescharacteristic
124C2×C4⋊C4—42no—
125D16—44no—
126D16—42no—
127D16—81no—
128D16SD1614yescomplemented
129C2×C16—21no—
130C2×C16—21no—
131C2×C16—21no—
132C2×C16—42no—
133C2×C16C2×D411yescharacteristic
134C2×C16C2×D411yescharacteristic
135#107—22no—
136#107—22no—
137#107C2×D412yes—
138#106—42no—
139#106—42no—
140#106D811yescharacteristic
141C4⋊C8—21no—
142C4⋊C8—22no—
143C4⋊C8C2×D411yescharacteristic
144C4⋊C8C2×D411yescharacteristic
145#103D811yescharacteristic
146#102—22no—
147#102—42no—
148#102—42no—
149#102—42no—
150#102—42no—
151#102—42no—
152#102—42no—
153#102D811yescharacteristic
154C22⋊C8—21no—
155C22⋊C8—42no—
156C4×C8—21no—
157C4×C8C2×D411yescharacteristic
158C4×C8C2×D411yescharacteristic
159C4×C8C2411yescharacteristic
160C24—82no—
161C2×Q8—42no—
162C2×Q8C2×D811yescharacteristic
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—82no—
168C2×D4—84no—
169C2×D4—82no—
170C2×D4—84no—
171C2×D4—162no—
172C2×D4—162no—
173C2×D4C2×D811yescharacteristic
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—82no—
177SD16—22no—
178SD16—164no—
179SD16—162no—
180D8—22no—
181D8—42no—
182D8—42no—
183D8—82no—
184D8—84no—
185D8C2×SD1614yes—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—162no—
193C2×C8C2×D811yescharacteristic
194C2×C8C2×D811yescharacteristic
195C2×C8C22×D411yescharacteristic
196C2×C8H(5,𝔽2)11yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4C2×D811yescharacteristic
204C4⋊C4C2×D811yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—42no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C42—21no—
210C42C22×D411yescharacteristic
211C16—21no—
212C16—41no—
213C16—41no—
214C16—41no—
215C16C2×SD1612yes—
216C23—21no—
217C23—42no—
218C23—82no—
219C23—164no—
220C23—164no—
221C23—322no—
222Q8—21no—
223Q8—21no—
224Q8—162no—
225D4—21no—
226D4—24no—
227D4—42no—
228D4—82no—
229D4—82no—
230D4—82no—
231D4—82no—
232D4—162no—
233D4—164no—
234D4—164no—
235D4#50812yes—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—322no—
246C2×C4D4211yescharacteristic
247C2×C4C22×D811yescharacteristic
248C2×C4#57611yescharacteristic
249C8—22no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—41no—
254C8—41no—
255C8—82no—
256C8C22×SD1611yescharacteristic
257C8#57711yescharacteristic
258C22—22no—
259C22—41no—
260C22—41no—
261C22—44no—
262C22—82no—
263C22—82no—
264C22—162no—
265C22—162no—
266C22—162no—
267C22—328no—
268C22D4×D811yescharacteristic
269C4—21no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—82no—
274C4—162no—
275C4D4×SD1611yescharacteristic
276C4#329011yescharacteristic
277C4C2×#50811yescharacteristic
278C4#341511yescharacteristic
279C2—42no—
280C2—84no—
281C2—322no—
282C2D8×SD1611yescharacteristic
283C2#3384611yescharacteristic
284C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalD4×D16C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33947C2semidirect
3. #19976C2semidirect
4. #12503C2semidirect
5. #12452C2semidirect
6. #12439C2semidirect
7. #12418C2semidirect
8. #12372C2semidirect
9. #12355C2semidirect
10. C16×SD16C2semidirect
11. C8×D16C2semidirect
13. #3380C22semidirect
17. #2252C22semidirect
19. #2240C22semidirect
21. #2226C22semidirect
25. Q8×C16C22semidirect
29. #1706C22semidirect
32. #1668C22semidirect
36. C8×D8C22semidirect
37. #1371C22semidirect
38. #1367C22semidirect
39. #1365C22semidirect
40. #1361C22semidirect
41. Q8⋊C16C22semidirect
42. #1331C22semidirect
44. C8×C16C22semidirect
128. D16SD16semidirect

Maximal subgroups

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

subgroupindex
#339472
D4×D162
#199762
#125032
#124522
#124392
#124182
#123722
#123552
C16×SD162
C8×D162