Finite Group Databasebeta

← #46848046

Subgroups — #46848046

Order 34688.

526 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 125 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
2C22×D2168C211yesmaximal, characteristic, complemented
3#31774538C212yesmaximal, complemented
4#31774282C211yesmaximal, characteristic, complemented
5#31774134C211yesmaximal, characteristic, complemented
6#31774115C212yesmaximal, complemented
7#31774047C212yesmaximal, complemented
8#31774045C212yesmaximal, complemented
9#31773979C211yesmaximal, characteristic, complemented
10C8×C22⋊D271C211yesmaximal, characteristic, complemented
11#467×C271C211yesmaximal, characteristic, complemented
12C2×#16864655C2212yescomplemented
13C22×D1084C2212yes—
14#22576183—24no—
15C2×#16864641C2212yescomplemented
16C2×#16864641C2212yescomplemented
17C2×D2168—24no—
18C2×D2168C2211yescharacteristic
19C2×D2168C2212yes—
20C2×D2168C2211yescharacteristic
21C2×C8×D271C2211yescharacteristic, complemented
22#22576149C2212yescomplemented
23#22576145—22no—
24#22576135—22no—
25#22576135C2212yescomplemented
26#22576130—22no—
27#22576122C2212yescomplemented
28C4×C22⋊D271C2211yescharacteristic
29C22⋊C4×D271—22no—
30Dic271⋊C8C2211yescharacteristic, complemented
31#22576080C2212yescomplemented
32#22576080C2212yescomplemented
33C22⋊(C271⋊C8)C2211yescharacteristic
34(C2×D271)⋊C8C2211yescharacteristic, complemented
35C8×Dic271C2211yescharacteristic, complemented
36C2×Q16×C271C2211yescharacteristic, complemented
37C22×C8×C271C2211yescharacteristic
38C22⋊Q8×C271C2212yes—
39#107×C271C2211yescharacteristic, complemented
40#103×C271C2212yescomplemented
41C23×D271—22no—
42#16864655—24no—
43#16864655—42no—
44C2×D1084—24no—
45C2×D1084—22no—
46C2×D1084—22no—
47C2×D1084—22no—
48C2×D1084—42no—
49C2×D1084C2312yes—
50C2×D1084C2312yes—
51C2×C22⋊D271D411yescharacteristic
52C2×C4×D271—22no—
53C2×C4×D271—22no—
54C2×C4×D271C2311yescharacteristic
55#16864641—24no—
56#16864641—42no—
57D2168—24no—
58D2168—22no—
59D2168D414yescomplemented
60C2×C271⋊C8C2311yescharacteristic
61C2×C271⋊C8C2311yescharacteristic
62C8×D271—22no—
63Dic271⋊C4D411yescharacteristic
64C22⋊Dic271D411yescharacteristic
65(C2×D271)⋊C4—22no—
66(C2×D271)⋊C4—22no—
67(C2×D271)⋊C4—22no—
68(C2×D271)⋊C4—22no—
69(C2×D271)⋊C4D411yescharacteristic
70C4×Dic271C2311yescharacteristic
71C2×Q8×C271C2312yes—
72C22×C4×C271C2311yescharacteristic
73Q16×C271—22no—
74C2×C8×C271—22no—
75C2×C8×C271C2311yescharacteristic
76C2×C8×C271C2311yescharacteristic
77C4⋊C4×C271—22no—
78C4⋊C4×C271C2312yes—
79C22⋊C4×C271—22no—
80C22×D271—22no—
81C22×D271—22no—
82C22×D271—44no—
83C22×D271—44no—
84C22×D271—82no—
85C22×D271C2×D411yescharacteristic
86D1084—22no—
87D1084—22no—
88D1084—22no—
89D1084—44no—
90D1084—44no—
91D1084—82no—
92D1084C2×D414yes—
93C22⋊D271—22no—
94C2×Dic271C2×D411yescharacteristic
95C2×Dic271C2×D411yescharacteristic
96C2×Dic271C2×D411yescharacteristic
97C4×D271—21no—
98C4×D271—21no—
99C4×D271—44no—
100C4×D271—82no—
101C271⋊C8—21no—
102C271⋊C8—21no—
103C23×C271C2×D411yescharacteristic
104Q8×C271—22no—
105Q8×C271—22no—
106C2×C4×C271—21no—
107C2×C4×C271—21no—
108C2×C4×C271—22no—
109C2×C4×C271—22no—
110C2×C4×C271C2×D411yescharacteristic
111C2×C4×C271C2411yescharacteristic
112C8×C271—21no—
113C8×C271C2×D412yes—
114C22×C271—21no—
115C22×C271—21no—
116C22×C271#13512yescomplemented
117C22×C271C22×D411yescharacteristic
118C2×D271—21no—
119C2×D271—24no—
120C2×D271—21no—
121C2×D271—21no—
122C2×D271—42no—
123C2×D271—42no—
124C2×D271—42no—
125C2×D271—42no—
126C2×D271—42no—
127C2×D271—88no—
128C4×C271—21no—
129C4×C271—42no—
130C4×C271—42no—
131C4×C271C22×D411yescharacteristic
132C4×C271H(5,𝔽2)11yescharacteristic
133Dic271—21no—
134Dic271—21no—
135Dic271—21no—
136C2×C271—22no—
137C2×C271D4211yescharacteristic
138C2×C271C2×#13511yescharacteristic
139C2×C271#57611yescharacteristic
140D271—41no—
141D271—44no—
142D271—82no—
143C271#328011yescharacteristic, complemented, Sylow
144#3280—2711nomaximal, Sylow
145C2×#135—2711no—
146C22×D8—2711no—
147#546—2712no—
148#495—2711no—
149#467D27111yescharacteristic, complemented
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×SD16—2712no—
162C2×D8—2711no—
163C2×D8—2712no—
164C2×D8—2711no—
165C2×D8—5424no—
166C22×C8—2711no—
167C22×C8C2×D27111yescharacteristic
168#124—2712no—
169#123—5422no—
170C22⋊Q8C2×D27112yes—
171#121—2712no—
172#121—5422no—
173#120—5422no—
174C4×D4—2711no—
175C4×D4—2712no—
176C2×C22⋊C4—5422no—
177#107C2×D27111yescharacteristic, complemented
178C4⋊C8—2711no—
179#103C2×D27112yescomplemented
180#102—2712no—
181#102—2712no—
182C22⋊C8—2711no—
183C22⋊C8—2711no—
184C4×C8—2711no—
185C24—5422no—
186#41—5424no—
187#41—10842no—
188C2×Q8C22×D27112yes—
189C2×D4—2711no—
190C2×D4—2712no—
191C2×D4—2712no—
192C2×D4—5424no—
193C2×D4—5422no—
194C2×D4—5422no—
195C2×D4—5422no—
196C2×D4—10842no—
197C22×C4—2711no—
198C22×C4—5422no—
199C22×C4—5422no—
200C22×C4C22×D27111yescharacteristic
201Q16—22no—
202SD16—5424no—
203SD16—10842no—
204D8—2714no—
205D8—5424no—
206D8—5422no—
207C2×C8—22no—
208C2×C8—2711no—
209C2×C8—2711no—
210C2×C8—5422no—
211C2×C8C22×D27111yescharacteristic
212C2×C8C22×D27111yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—2711no—
215C4⋊C4C22×D27112yes—
216C22⋊C4—22no—
217C22⋊C4—2711no—
218C22⋊C4—2711no—
219C22⋊C4—5422no—
220C22⋊C4—5422no—
221C22⋊C4—5422no—
222C22⋊C4—5422no—
223C42—2711no—
224C23—2711no—
225C23—5422no—
226C23—5422no—
227C23—10844no—
228C23—10844no—
229C23—21682no—
230C23D4×D27111yescharacteristic
231Q8—22no—
232Q8—22no—
233D4—2714no—
234D4—5422no—
235D4—5422no—
236D4—5422no—
237D4—5422no—
238D4—10844no—
239D4—10844no—
240D4—21682no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—2711no—
246C2×C4—2711no—
247C2×C4—2711no—
248C2×C4—5421no—
249C2×C4—5421no—
250C2×C4—10844no—
251C2×C4—21682no—
252C2×C4D4×D27111yescharacteristic
253C2×C4C23×D27111yescharacteristic
254C8—21no—
255C8—5421no—
256C8—5421no—
257C8D4×D27112yes—
258C22—21no—
259C22—21no—
260C22—5421no—
261C22—5424no—
262C22—5421no—
263C22—5421no—
264C22—10842no—
265C22—10842no—
266C22—10842no—
267C22—10842no—
268C22—10842no—
269C22—21688no—
270C22#2257618312yescomplemented
271C22C2×D4×D27111yescharacteristic
272C4—21no—
273C4—42no—
274C4—42no—
275C4—5421no—
276C4—5421no—
277C4—5421no—
278C4C2×D4×D27111yescharacteristic
279C4#2257620611yescharacteristic
280C2—22no—
281C2—10841no—
282C2—10844no—
283C2—21682no—
284C2#3177453111yescharacteristic
285C2C2×#2257618311yescharacteristic
286C2#3177469911yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#467×C271C2
socleC22×C271C22×D4
hypercenter#467D271
hypocenterC271#3280
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#22576183C2semidirect
2. C22×D2168C2semidirect
3. #31774538C2semidirect
4. #31774282C2semidirect
5. #31774134C2semidirect
6. #31774115C2semidirect
7. #31774047C2semidirect
8. #31774045C2semidirect
9. #31773979C2semidirect
10. C8×C22⋊D271C2semidirect
11. #467×C271C2semidirect
12. C2×#16864655C22semidirect
15. C2×#16864641C22semidirect
16. C2×#16864641C22semidirect
21. C2×C8×D271C22semidirect
22. #22576149C22semidirect
25. #22576135C22semidirect
27. #22576122C22semidirect
30. Dic271⋊C8C22semidirect
31. #22576080C22semidirect
32. #22576080C22semidirect
34. (C2×D271)⋊C8C22semidirect
35. C8×Dic271C22semidirect
36. C2×Q16×C271C22semidirect
39. #107×C271C22semidirect
40. #103×C271C22semidirect
59. D2168D4semidirect
116. C22×C271#135semidirect
143. C271#3280semidirect
149. #467D271semidirect
159. C2×Q16C2×D271semidirect
177. #107C2×D271semidirect
179. #103C2×D271semidirect
270. C22#22576183semidirect

Maximal subgroups

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

subgroupindex
C2×#225761832
C22×D21682
#317745382
#317742822
#317741342
#317741152
#317740472
#317740452
#317739792
C8×C22⋊D2712
#467×C2712
#3280271