Finite Group Databasebeta

← #46848048

Subgroups — #46848048

Order 34688.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#22576183C211yesmaximal, characteristic, complemented
2C2×D8×D271C211yesmaximal, characteristic, complemented
3#31774540C212yesmaximal, complemented
4#31774135C211yesmaximal, characteristic, complemented
5#31774130C211yesmaximal, characteristic, complemented
6#31774109C212yesmaximal, complemented
7#31774046C212yesmaximal, complemented
8#31774041C212yesmaximal, complemented
9C4×D2168C211yesmaximal, characteristic, complemented
10C8×D1084C211yesmaximal, characteristic, complemented
11#495×C271C211yesmaximal, characteristic, complemented
12C2×#16864655C2212yescomplemented
13C2×D4×D271C2212yes—
14#22576183—24no—
15C2×#16864641C2212yescomplemented
16C2×D2168C2211yescharacteristic
17C2×#16864639C2212yes—
18D8×D271—24no—
19C2×C8×D271C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic, complemented
21#22576142—22no—
22#22576136C2212yes—
23#22576133—22no—
24#22576131C2212yescomplemented
25#22576130—22no—
26C4×D1084C2211yescharacteristic
27C4×D1084C2212yescomplemented
28C22⋊C4×D271—22no—
29#22576094C2211yescharacteristic, complemented
30C4⋊(C271⋊C8)C2211yescharacteristic, complemented
31#22576084C2212yescomplemented
32#22576082C2212yescomplemented
33#22576081C2212yescomplemented
34(C2×D271)⋊C8C2211yescharacteristic
35(C2×D271)⋊C8C2211yescharacteristic, complemented
36C2×Q16×C271C2211yescharacteristic, complemented
37C2×SD16×C271C2212yescomplemented
38C2×D8×C271C2211yescharacteristic
39#124×C271C2212yescomplemented
40C4×C8×C271C2211yescharacteristic, complemented
41C23×D271—22no—
42#16864655—24no—
43#16864655—42no—
44C2×D1084—22no—
45C2×D1084D411yescharacteristic
46C2×D1084C2312yes—
47C2×C22⋊D271—22no—
48C2×C22⋊D271—22no—
49D4×D271—24no—
50D4×D271—42no—
51C2×C4×D271—22no—
52C2×C4×D271—22no—
53C2×C4×D271C2311yescharacteristic
54C2×C4×D271C2311yescharacteristic
55#16864641—24no—
56D2168D414yescomplemented
57#16864639—24no—
58C2×C271⋊C8C2311yescharacteristic
59C2×C271⋊C8C2311yescharacteristic
60C8×D271—22no—
61C8×D271—22no—
62C4⋊Dic271D411yescharacteristic
63C4⋊Dic271C2312yes—
64Dic271⋊C4—22no—
65C22⋊Dic271—22no—
66(C2×D271)⋊C4—22no—
67(C2×D271)⋊C4—22no—
68(C2×D271)⋊C4D411yescharacteristic
69(C2×D271)⋊C4D411yescharacteristic
70C2×Q8×C271C2312yes—
71C2×D4×C271C2312yes—
72Q16×C271—22no—
73SD16×C271—42no—
74D8×C271—22no—
75C2×C8×C271C2311yescharacteristic
76C2×C8×C271C2311yescharacteristic
77C22⋊C4×C271—22no—
78C22⋊C4×C271—22no—
79C42×C271C2311yescharacteristic
80C22×D271—22no—
81C22×D271—44no—
82C22×D271—44no—
83C22×D271—82no—
84C22×D271C2×D411yescharacteristic
85C22×D271C2×D411yescharacteristic
86D1084—22no—
87D1084—22no—
88D1084—44no—
89D1084C2×D414yes—
90C22⋊D271—44no—
91C22⋊D271—82no—
92C2×Dic271—22no—
93C2×Dic271C2×D411yescharacteristic
94C2×Dic271C2×D411yescharacteristic
95C4×D271—21no—
96C4×D271—21no—
97C4×D271—21no—
98C4×D271—21no—
99C4×D271—44no—
100C4×D271—82no—
101C271⋊C8—21no—
102C271⋊C8—21no—
103C23×C271—22no—
104Q8×C271—22no—
105Q8×C271—22no—
106D4×C271—22no—
107D4×C271—22no—
108C2×C4×C271—22no—
109C2×C4×C271C2×D411yescharacteristic
110C2×C4×C271C2×D411yescharacteristic
111C2×C4×C271C2411yescharacteristic
112C8×C271—21no—
113C8×C271C2×D412yes—
114C22×C271—42no—
115C22×C271—42no—
116C22×C271—42no—
117C22×C271C22×D411yescharacteristic
118C2×D271—21no—
119C2×D271—21no—
120C2×D271—21no—
121C2×D271—24no—
122C2×D271—21no—
123C2×D271—21no—
124C2×D271—42no—
125C2×D271—42no—
126C2×D271—88no—
127C2×D271#13512yescomplemented
128C4×C271—21no—
129C4×C271—21no—
130C4×C271—42no—
131C4×C271C22×D411yescharacteristic
132C4×C271H(5,𝔽2)11yescharacteristic
133Dic271—21no—
134Dic271—21no—
135Dic271—42no—
136C2×C271—82no—
137C2×C271D4211yescharacteristic
138C2×C271C2×#13511yescharacteristic
139C2×C271#57611yescharacteristic
140D271—22no—
141D271—41no—
142D271—44no—
143C271#328011yescharacteristic, complemented, Sylow
144#3280—2711nomaximal, Sylow
145C2×#135—2711no—
146C22×D8—2711no—
147#546—2712no—
148#495D27111yescharacteristic, complemented
149#467—2711no—
150#466—2711no—
151#463—2712no—
152#450—2712no—
153#449—2712no—
154C4×D8—2711no—
155C8×D4—2711no—
156C2×#41—2712no—
157C22×D4—2712no—
158#135—5424no—
159C2×Q16C2×D27111yescharacteristic, complemented
160C2×SD16—2712no—
161C2×SD16C2×D27112yescomplemented
162C2×D8—2712no—
163C2×D8—2711no—
164C2×D8—5424no—
165C2×D8C2×D27111yescharacteristic
166C22×C8—2711no—
167C22×C8—2711no—
168#124C2×D27112yescomplemented
169#123—5422no—
170C22⋊Q8—2712no—
171#121—2712no—
172#121—5422no—
173#120—5422no—
174C4×D4—2711no—
175C4×D4—2712no—
176C2×C22⋊C4—5422no—
177#107—2711no—
178C4⋊C8—2711no—
179#103—2712no—
180#102—2712no—
181#102—2712no—
182C22⋊C8—2711no—
183C22⋊C8—2711no—
184C4×C8C2×D27111yescharacteristic, complemented
185C24—5422no—
186#41—5424no—
187#41—10842no—
188C2×Q8C22×D27112yes—
189C2×D4—2711no—
190C2×D4—2712no—
191C2×D4—5422no—
192C2×D4—5422no—
193C2×D4—5424no—
194C2×D4—5422no—
195C2×D4—10842no—
196C2×D4C22×D27112yes—
197C22×C4—2711no—
198C22×C4—2711no—
199C22×C4—5422no—
200C22×C4—5422no—
201Q16—22no—
202SD16—42no—
203SD16—5424no—
204D8—22no—
205D8—2714no—
206D8—5424no—
207C2×C8—2711no—
208C2×C8—2711no—
209C2×C8—5422no—
210C2×C8—5422no—
211C2×C8C22×D27111yescharacteristic
212C2×C8C22×D27111yescharacteristic
213C4⋊C4—2711no—
214C4⋊C4—2712no—
215C4⋊C4—5422no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—2711no—
219C22⋊C4—2711no—
220C22⋊C4—5422no—
221C22⋊C4—5422no—
222C22⋊C4—5422no—
223C42C22×D27111yescharacteristic
224C23—22no—
225C23—2711no—
226C23—2711no—
227C23—5422no—
228C23—10844no—
229C23—10844no—
230C23—21682no—
231Q8—22no—
232Q8—22no—
233D4—22no—
234D4—22no—
235D4—2714no—
236D4—5422no—
237D4—5422no—
238D4—10844no—
239D4—10844no—
240D4—21682no—
241C2×C4—22no—
242C2×C4—2711no—
243C2×C4—2711no—
244C2×C4—5421no—
245C2×C4—5421no—
246C2×C4—5421no—
247C2×C4—5421no—
248C2×C4—5422no—
249C2×C4—10844no—
250C2×C4—21682no—
251C2×C4D4×D27111yescharacteristic
252C2×C4D4×D27111yescharacteristic
253C2×C4C23×D27111yescharacteristic
254C8—21no—
255C8—5421no—
256C8—5421no—
257C8D4×D27112yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—2712no—
262C22—5421no—
263C22—5421no—
264C22—5421no—
265C22—5424no—
266C22—5421no—
267C22—5421no—
268C22—10842no—
269C22—10842no—
270C22—21688no—
271C22C2×D4×D27111yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—5421no—
276C4—5421no—
277C4—10842no—
278C4C2×D4×D27111yescharacteristic
279C4#2257620511yescharacteristic
280C2—82no—
281C2—5422no—
282C2—10841no—
283C2—10844no—
284C2#3177453211yescharacteristic
285C2#135×D27111yescharacteristic
286C2#3177469911yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#495×C271C2
socleC22×C271C22×D4
hypercenter#495D271
hypocenterC271#3280
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#22576183C2semidirect
2. C2×D8×D271C2semidirect
3. #31774540C2semidirect
4. #31774135C2semidirect
5. #31774130C2semidirect
6. #31774109C2semidirect
7. #31774046C2semidirect
8. #31774041C2semidirect
9. C4×D2168C2semidirect
10. C8×D1084C2semidirect
11. #495×C271C2semidirect
12. C2×#16864655C22semidirect
15. C2×#16864641C22semidirect
20. C2×C8×D271C22semidirect
24. #22576131C22semidirect
27. C4×D1084C22semidirect
29. #22576094C22semidirect
30. C4⋊(C271⋊C8)C22semidirect
31. #22576084C22semidirect
32. #22576082C22semidirect
33. #22576081C22semidirect
35. (C2×D271)⋊C8C22semidirect
36. C2×Q16×C271C22semidirect
37. C2×SD16×C271C22semidirect
39. #124×C271C22semidirect
40. C4×C8×C271C22semidirect
56. D2168D4semidirect
127. C2×D271#135semidirect
143. C271#3280semidirect
148. #495D271semidirect
159. C2×Q16C2×D271semidirect
161. C2×SD16C2×D271semidirect
168. #124C2×D271semidirect
184. C4×C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761832
C2×D8×D2712
#317745402
#317741352
#317741302
#317741092
#317740462
#317740412
C4×D21682
C8×D10842
#495×C2712
#3280271