Finite Group Databasebeta

← #46848179

Subgroups — #46848179

Order 34688.

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

normal subgroups only

This group has 494 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
1#31774552C211yesmaximal, characteristic, complemented
2D4×D1084C211yesmaximal, characteristic, complemented
3#31774244C211yesmaximal, characteristic, complemented
4#31774212C212yesmaximal, complemented
5#31774177C211yesmaximal, characteristic, complemented
6#31774132C212yesmaximal, complemented
7#31774092C211yesmaximal, characteristic, complemented
8#31774029C212yesmaximal, complemented
9C8×D1084C211yesmaximal, characteristic, complemented
10#102×D271C212yesmaximal, complemented
11C4×D8×C271C211yesmaximal, characteristic, complemented
12C2×#16864654—21no—
13C22×D1084—21no—
14C2×D4×D271C2212yes—
15C2×#16864639C2212yescomplemented
16C2×C8×D271C2212yescomplemented
17#22576160C2211yescharacteristic, complemented
18#22576157C2211yescharacteristic
19#22576142—22no—
20#22576139—21no—
21#22576135C2212yes—
22#22576134—21no—
23#22576131C2212yescomplemented
24#22576130—22no—
25C4×D1084C2211yescharacteristic
26C4⋊C4×D271C2212yescomplemented
27#22576093C2212yescomplemented
28C4⋊(C271⋊C8)C2211yescharacteristic, complemented
29#22576082C2212yescomplemented
30#22576082C2212yescomplemented
31#22576081C2212yes—
32#22576080C2212yescomplemented
33(C2×D271)⋊C8C2212yescomplemented
34C2×D8×C271C2211yescharacteristic
35C4×D4×C271C2211yescharacteristic
36C4×D4×C271C2211yescharacteristic, complemented
37#107×C271C2211yescharacteristic, complemented
38#102×C271C2211yescharacteristic
39#102×C271C2211yescharacteristic, complemented
40C4×C8×C271C2211yescharacteristic, complemented
41C23×D271—22no—
42#16864654—44no—
43C2×Dic542—21no—
44C2×D1084—22no—
45C2×D1084—21no—
46C2×D1084—21no—
47C2×D1084—44no—
48C2×D1084D411yescharacteristic
49C2×D1084C2312yes—
50C2×C22⋊D271—22no—
51C2×C22⋊D271—22no—
52D4×D271—24no—
53D4×D271—42no—
54C2×C4×D271—22no—
55C2×C4×D271C2312yes—
56#16864639—42no—
57C2×C271⋊C8C2312yes—
58C8×D271—42no—
59C4⋊Dic271—21no—
60C4⋊Dic271D411yescharacteristic
61C4⋊Dic271C2312yes—
62Dic271⋊C4—22no—
63C22⋊Dic271—22no—
64(C2×D271)⋊C4—22no—
65(C2×D271)⋊C4D412yes—
66C2×D4×C271C2311yescharacteristic
67C2×D4×C271C2311yescharacteristic
68C22×C4×C271—21no—
69C22×C4×C271—21no—
70D8×C271—22no—
71C2×C8×C271C2311yescharacteristic
72C2×C8×C271C2311yescharacteristic
73C4⋊C4×C271C2311yescharacteristic
74C4⋊C4×C271C2311yescharacteristic
75C22⋊C4×C271—21no—
76C22⋊C4×C271—21no—
77C42×C271C2311yescharacteristic
78C22×D271—22no—
79C22×D271—44no—
80C22×D271—44no—
81C22×D271—82no—
82C22×D271C2×D412yes—
83Dic542—42no—
84D1084—22no—
85D1084—22no—
86D1084—44no—
87D1084—42no—
88D1084—81no—
89D1084D814yescomplemented
90C22⋊D271—44no—
91C22⋊D271—44no—
92C2×Dic271—22no—
93C2×Dic271C2×D412yes—
94C4×D271—22no—
95C4×D271—22no—
96C4×D271—44no—
97C271⋊C8—22no—
98C23×C271—21no—
99C23×C271—21no—
100D4×C271—21no—
101D4×C271—21no—
102D4×C271—21no—
103D4×C271#4112yescomplemented
104C2×C4×C271—21no—
105C2×C4×C271—21no—
106C2×C4×C271—42no—
107C2×C4×C271—42no—
108C2×C4×C271C2×D411yescharacteristic
109C2×C4×C271C2×D411yescharacteristic
110C2×C4×C271C2411yescharacteristic
111C8×C271—21no—
112C8×C271—21no—
113C22×C271—22no—
114C22×C271—41no—
115C22×C271—41no—
116C22×C271—41no—
117C22×C271—41no—
118C22×C271—41no—
119C22×C271C22×D411yescharacteristic
120C2×D271—22no—
121C2×D271—22no—
122C2×D271—42no—
123C2×D271—42no—
124C2×D271—42no—
125C2×D271—88no—
126C2×D271C2×D814yes—
127C4×C271—21no—
128C4×C271—41no—
129C4×C271—41no—
130C4×C271C2×D812yes—
131C4×C271C2×#4111yescharacteristic
132C4×C271H(5,𝔽2)11yescharacteristic
133Dic271—22no—
134Dic271—42no—
135C2×C271—42no—
136C2×C271—81no—
137C2×C271#54611yescharacteristic
138C2×C271C22×D811yescharacteristic
139C2×C271#57711yescharacteristic
140D271—24no—
141D271—82no—
142C271#329411yescharacteristic, complemented, Sylow
143#3294—2711nomaximal, Sylow
144#547—2711no—
145D42—2711no—
146#486—2711no—
147#480—2712no—
148#474—2711no—
149#466—2712no—
150#459—2711no—
151#447—2712no—
152C4×D8D27111yescharacteristic, complemented
153C8×D4—2711no—
154C2×#102—2712no—
155C2×#41—5421no—
156C22×D4—2712no—
157C22×D4—5421no—
158C2×D8—2712no—
159C2×D8C2×D27111yescharacteristic
160C22×C8—2712no—
161C4⋊Q8—2711no—
162#127—2711no—
163#123—5422no—
164C22⋊Q8—5421no—
165#121—2712no—
166#121—2712no—
167#121—5421no—
168#120—5422no—
169C4×D4—2711no—
170C4×D4C2×D27111yescharacteristic
171C4×D4C2×D27111yescharacteristic, complemented
172C2×C4⋊C4—2712no—
173#107—2712no—
174#107C2×D27111yescharacteristic, complemented
175C4⋊C8—2711no—
176#102—2712no—
177#102—2712no—
178#102—2712no—
179#102—2712no—
180#102C2×D27111yescharacteristic
181#102C2×D27111yescharacteristic, complemented
182C22⋊C8—2712no—
183C4×C8C2×D27111yescharacteristic, complemented
184C24—5422no—
185#41—10844no—
186C2×Q8—5421no—
187C2×D4—2711no—
188C2×D4—2712no—
189C2×D4—5422no—
190C2×D4—5422no—
191C2×D4—5424no—
192C2×D4—5422no—
193C2×D4—5421no—
194C2×D4—5421no—
195C2×D4—10844no—
196C2×D4—10842no—
197C2×D4C22×D27111yescharacteristic
198C2×D4C22×D27111yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—2712no—
202C22×C4—5422no—
203D8—22no—
204D8—10842no—
205C2×C8—2712no—
206C2×C8—10842no—
207C2×C8C22×D27111yescharacteristic
208C2×C8C22×D27111yescharacteristic
209C4⋊C4—2711no—
210C4⋊C4—2712no—
211C4⋊C4—5422no—
212C4⋊C4—5421no—
213C4⋊C4C22×D27111yescharacteristic
214C4⋊C4C22×D27111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—2712no—
218C22⋊C4—5422no—
219C22⋊C4—5422no—
220C42C22×D27111yescharacteristic
221C23—21no—
222C23—21no—
223C23—2712no—
224C23—5422no—
225C23—10844no—
226C23—10844no—
227C23—21682no—
228Q8—10842no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—2714no—
233D4—5422no—
234D4—5422no—
235D4—10844no—
236D4—10844no—
237D4—10844no—
238D4—10842no—
239D4—21681no—
240D4#1686465412yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—2712no—
246C2×C4—5422no—
247C2×C4—5422no—
248C2×C4—5422no—
249C2×C4—10844no—
250C2×C4D4×D27111yescharacteristic
251C2×C4D4×D27111yescharacteristic
252C2×C4C23×D27111yescharacteristic
253C8—21no—
254C8—21no—
255C8—5422no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—2714no—
263C22—5422no—
264C22—5422no—
265C22—10842no—
266C22—10842no—
267C22—10842no—
268C22—21688no—
269C22C2×D4×D27111yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—5422no—
274C4—10842no—
275C4D8×D27112yes—
276C4C2×#1686465411yescharacteristic
277C4#2257620611yescharacteristic
278C2—42no—
279C2—81no—
280C2—5424no—
281C2—21682no—
282C2#3177454311yescharacteristic
283C2C2×D8×D27111yescharacteristic
284C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC4×D8×C271C2
socleC22×C271C22×D4
hypercenterC4×D8D271
hypocenterC271#3294
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. #31774552C2semidirect
2. D4×D1084C2semidirect
3. #31774244C2semidirect
4. #31774212C2semidirect
5. #31774177C2semidirect
6. #31774132C2semidirect
7. #31774092C2semidirect
8. #31774029C2semidirect
9. C8×D1084C2semidirect
10. #102×D271C2semidirect
11. C4×D8×C271C2semidirect
15. C2×#16864639C22semidirect
16. C2×C8×D271C22semidirect
17. #22576160C22semidirect
23. #22576131C22semidirect
26. C4⋊C4×D271C22semidirect
27. #22576093C22semidirect
28. C4⋊(C271⋊C8)C22semidirect
29. #22576082C22semidirect
30. #22576082C22semidirect
32. #22576080C22semidirect
33. (C2×D271)⋊C8C22semidirect
36. C4×D4×C271C22semidirect
37. #107×C271C22semidirect
39. #102×C271C22semidirect
40. C4×C8×C271C22semidirect
89. D1084D8semidirect
103. D4×C271#41semidirect
142. C271#3294semidirect
152. C4×D8D271semidirect
171. C4×D4C2×D271semidirect
174. #107C2×D271semidirect
181. #102C2×D271semidirect
183. C4×C8C2×D271semidirect
240. D4#16864654semidirect

Maximal subgroups

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

subgroupindex
#317745522
D4×D10842
#317742442
#317742122
#317741772
#317741322
#317740922
#317740292
C8×D10842
#102×D2712
C4×D8×C2712
#3294271