Finite Group Databasebeta

← #46848050

Subgroups — #46848050

Order 34688.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×#22576179C211yesmaximal, characteristic, complemented
2C2×D8×D271C211yesmaximal, characteristic, complemented
3#31774542C212yesmaximal, complemented
4#31774281C211yesmaximal, characteristic, complemented
5#31774139C211yesmaximal, characteristic, complemented
6#31774134C211yesmaximal, characteristic, complemented
7#31774112C212yesmaximal, complemented
8#31774049C212yesmaximal, complemented
9#31774043C212yesmaximal, complemented
10C8×D1084C211yesmaximal, characteristic, complemented
11C4×D8×C271C211yesmaximal, characteristic, complemented
12C2×#16864653C2212yescomplemented
13C2×D4×D271C2212yes—
14#22576179—24no—
15C2×Dic1084C2211yescharacteristic, complemented
16C2×#16864643C2212yescomplemented
17C2×#16864642C2212yescomplemented
18C2×D2168C2211yescharacteristic
19C2×#16864639C2212yes—
20D8×D271—24no—
21C2×C8×D271C2211yescharacteristic
22C2×C8×D271C2211yescharacteristic, complemented
23#22576148C2212yescomplemented
24#22576144—22no—
25#22576140C2212yes—
26#22576135C2212yescomplemented
27#22576134—22no—
28#22576130—22no—
29C4×D1084C2211yescharacteristic
30C2×(C2×D271)⋊C4—22no—
31C4⋊(C271⋊C8)C2211yescharacteristic, complemented
32#22576083C2212yescomplemented
33#22576080C2212yescomplemented
34(C2×D271)⋊C8C2211yescharacteristic
35(C2×D271)⋊C8C2211yescharacteristic, complemented
36C2×D8×C271C2211yescharacteristic
37C4×D4×C271C2212yescomplemented
38#107×C271C2211yescharacteristic, complemented
39#102×C271C2212yescomplemented
40C4×C8×C271C2211yescharacteristic, complemented
41C23×D271—22no—
42#16864653—24no—
43#16864653—42no—
44C2×Dic542C2312yes—
45C2×D1084—22no—
46C2×D1084D411yescharacteristic
47C2×D1084C2312yes—
48C2×C22⋊D271—22no—
49C2×C22⋊D271—22no—
50D4×D271—24no—
51D4×D271—42no—
52C22×Dic271—22no—
53C2×C4×D271C2311yescharacteristic
54C2×C4×D271C2311yescharacteristic
55Dic1084—22no—
56#16864643—42no—
57#16864642—24no—
58D2168—22no—
59#16864639—24no—
60C2×C271⋊C8C2311yescharacteristic
61C2×C271⋊C8C2311yescharacteristic
62C8×D271—22no—
63C8×D271—22no—
64C4⋊Dic271—22no—
65C4⋊Dic271D411yescharacteristic
66(C2×D271)⋊C4—22no—
67(C2×D271)⋊C4—22no—
68(C2×D271)⋊C4—22no—
69(C2×D271)⋊C4—22no—
70(C2×D271)⋊C4D411yescharacteristic
71(C2×D271)⋊C4D411yescharacteristic
72C2×D4×C271C2312yes—
73C22×C4×C271—22no—
74D8×C271D414yescomplemented
75C2×C8×C271C2311yescharacteristic
76C2×C8×C271C2311yescharacteristic
77C4⋊C4×C271C2312yes—
78C22⋊C4×C271—22no—
79C42×C271C2311yescharacteristic
80C22×D271—22no—
81C22×D271—44no—
82C22×D271—44no—
83C22×D271—82no—
84C22×D271C2×D411yescharacteristic
85C22×D271C2×D411yescharacteristic
86Dic542—22no—
87Dic542—22no—
88D1084—22no—
89D1084—22no—
90D1084—22no—
91D1084—82no—
92C22⋊D271—44no—
93C22⋊D271—44no—
94C2×Dic271—22no—
95C2×Dic271—44no—
96C2×Dic271C2×D411yescharacteristic
97C2×Dic271C2×D411yescharacteristic
98C4×D271—21no—
99C4×D271—21no—
100C4×D271—21no—
101C4×D271—21no—
102C271⋊C8—21no—
103C271⋊C8—21no—
104C23×C271—22no—
105D4×C271—22no—
106D4×C271C2×D414yes—
107C2×C4×C271—22no—
108C2×C4×C271—82no—
109C2×C4×C271C2×D411yescharacteristic
110C2×C4×C271C2×D411yescharacteristic
111C2×C4×C271C2411yescharacteristic
112C8×C271—21no—
113C8×C271C2×D412yes—
114C22×C271—24no—
115C22×C271—42no—
116C22×C271—42no—
117C22×C271C22×D411yescharacteristic
118C2×D271—21no—
119C2×D271—21no—
120C2×D271—21no—
121C2×D271—21no—
122C2×D271—21no—
123C2×D271—42no—
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—44no—
137C2×C271D4211yescharacteristic
138C2×C271C2×#13511yescharacteristic
139C2×C271#57611yescharacteristic
140D271—22no—
141D271—41no—
142D271—82no—
143C271#328011yescharacteristic, complemented, Sylow
144#3280—2711nomaximal, Sylow
145C2×#135—2711no—
146C22×D8—2711no—
147#546—2712no—
148#495—2711no—
149#467—2711no—
150#466—2711no—
151#463—2712no—
152#450—2712no—
153#449—2712no—
154C4×D8D27111yescharacteristic, complemented
155C8×D4—2711no—
156C2×#41—2712no—
157C22×D4—2712no—
158#135—5424no—
159C2×Q16—2711no—
160C2×SD16—2712no—
161C2×SD16—2712no—
162C2×D8—2712no—
163C2×D8—2711no—
164C2×D8—5424no—
165C2×D8C2×D27111yescharacteristic
166C22×C8—2711no—
167C22×C8—2711no—
168#124—2712no—
169#123—5422no—
170C22⋊Q8—2712no—
171#121—2712no—
172#121—5422no—
173#120—5422no—
174C4×D4—2711no—
175C4×D4C2×D27112yescomplemented
176C2×C22⋊C4—5422no—
177#107C2×D27111yescharacteristic, complemented
178C4⋊C8—2711no—
179#103—2712no—
180#102—2712no—
181#102C2×D27112yescomplemented
182C22⋊C8—2711no—
183C22⋊C8—2711no—
184C4×C8C2×D27111yescharacteristic, complemented
185C24—5422no—
186#41—5424no—
187#41—10842no—
188C2×Q8—2712no—
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—22no—
198C22×C4—2711no—
199C22×C4—2711no—
200C22×C4—5422no—
201Q16—5422no—
202SD16—5424no—
203SD16—10842no—
204D8—5424no—
205D8—5422no—
206D8D108414yescomplemented
207C2×C8—2711no—
208C2×C8—2711no—
209C2×C8—5422no—
210C2×C8—5422no—
211C2×C8C22×D27111yescharacteristic
212C2×C8C22×D27111yescharacteristic
213C4⋊C4—2711no—
214C4⋊C4—5422no—
215C4⋊C4C22×D27112yes—
216C22⋊C4—22no—
217C22⋊C4—2711no—
218C22⋊C4—2711no—
219C22⋊C4—5422no—
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—5422no—
232Q8—5422no—
233D4—22no—
234D4—5422no—
235D4—5422no—
236D4—5422no—
237D4—10844no—
238D4—10844no—
239D4—21682no—
240D4C2×D108414yes—
241C2×C4—22no—
242C2×C4—82no—
243C2×C4—2711no—
244C2×C4—2711no—
245C2×C4—5421no—
246C2×C4—5421no—
247C2×C4—5421no—
248C2×C4—5421no—
249C2×C4—5422no—
250C2×C4—10844no—
251C2×C4D4×D27111yescharacteristic
252C2×C4D4×D27111yescharacteristic
253C2×C4C23×D27111yescharacteristic
254C8—21no—
255C8—5421no—
256C8—5421no—
257C8C2×D108412yes—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—2712no—
262C22—5421no—
263C22—5421no—
264C22—5421no—
265C22—5421no—
266C22—5421no—
267C22—10842no—
268C22—10842no—
269C22—10842no—
270C22—21688no—
271C22C2×D4×D27111yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—5421no—
276C4—5421no—
277C4—10842no—
278C4C22×D108411yescharacteristic
279C4#2257620611yescharacteristic
280C2—44no—
281C2—5422no—
282C2—10841no—
283C2—21682no—
284C2D4×D108411yescharacteristic
285C2#135×D27111yescharacteristic
286C2#3177469811yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC4×D8×C271C2
socleC22×C271C22×D4
hypercenterC4×D8D271
hypocenterC271#3280
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#22576179C2semidirect
2. C2×D8×D271C2semidirect
3. #31774542C2semidirect
4. #31774281C2semidirect
5. #31774139C2semidirect
6. #31774134C2semidirect
7. #31774112C2semidirect
8. #31774049C2semidirect
9. #31774043C2semidirect
10. C8×D1084C2semidirect
11. C4×D8×C271C2semidirect
12. C2×#16864653C22semidirect
15. C2×Dic1084C22semidirect
16. C2×#16864643C22semidirect
17. C2×#16864642C22semidirect
22. C2×C8×D271C22semidirect
23. #22576148C22semidirect
26. #22576135C22semidirect
31. C4⋊(C271⋊C8)C22semidirect
32. #22576083C22semidirect
33. #22576080C22semidirect
35. (C2×D271)⋊C8C22semidirect
37. C4×D4×C271C22semidirect
38. #107×C271C22semidirect
39. #102×C271C22semidirect
40. C4×C8×C271C22semidirect
74. D8×C271D4semidirect
127. C2×D271#135semidirect
143. C271#3280semidirect
154. C4×D8D271semidirect
175. C4×D4C2×D271semidirect
177. #107C2×D271semidirect
181. #102C2×D271semidirect
184. C4×C8C2×D271semidirect
206. D8D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225761792
C2×D8×D2712
#317745422
#317742812
#317741392
#317741342
#317741122
#317740492
#317740432
C8×D10842
C4×D8×C2712
#3280271