Finite Group Databasebeta

← #46848171

Subgroups — #46848171

Order 34688.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D1084C211yesmaximal, characteristic, complemented
2#31774245C211yesmaximal, characteristic, complemented
3#31774210C212yesmaximal, complemented
4#31774174C211yesmaximal, characteristic, complemented
5#31774131C212yesmaximal, complemented
6#31774092C211yesmaximal, characteristic, complemented
7#31774030C212yesmaximal, complemented
8C4×#16864639C211yesmaximal, characteristic, complemented
9D4×C271⋊C8C211yesmaximal, characteristic, complemented
10C2×#22576080C212yesmaximal, complemented
11#547×C271C211yesmaximal, characteristic, complemented
12C22×D1084C2212yes—
13C2×D4×D271—21no—
14C2×#16864639C2211yescharacteristic
15C2×#16864639C2212yescomplemented
16C22×C271⋊C8C2212yescomplemented
17#22576157C2211yescharacteristic
18#22576135—21no—
19#22576134C2212yes—
20#22576130—22no—
21C4×D1084C2211yescharacteristic
22#22576093C2211yescharacteristic, complemented
23#22576093C2212yescomplemented
24C4⋊(C271⋊C8)C2211yescharacteristic, complemented
25#22576081C2211yescharacteristic, complemented
26#22576080C2211yescharacteristic
27#22576080C2212yes—
28#22576080C2212yescomplemented
29#22576080C2212yescomplemented
30#22576080C2212yescomplemented
31C22⋊(C271⋊C8)C2212yescomplemented
32C4×C271⋊C8C2211yescharacteristic, complemented
33C2×#41×C271—21no—
34C4⋊Q8×C271C2211yescharacteristic, complemented
35#123×C271—22no—
36C22⋊Q8×C271—21no—
37#121×C271C2212yescomplemented
38C4×D4×C271C2211yescharacteristic
39C4×D4×C271C2211yescharacteristic, complemented
40C2×C4⋊C4×C271C2212yescomplemented
41C23×D271—22no—
42C2×D1084—24no—
43C2×D1084—22no—
44C2×D1084—21no—
45C2×D1084—21no—
46C2×D1084—42no—
47C2×D1084C2312yes—
48C2×D1084C2311yescharacteristic
49C2×C22⋊D271—22no—
50D4×D271—44no—
51C2×C4×D271—21no—
52#16864639—22no—
53#16864639—42no—
54C2×C271⋊C8—42no—
55C2×C271⋊C8C2311yescharacteristic
56C2×C271⋊C8C2312yes—
57C2×C271⋊C8C2311yescharacteristic
58C4⋊Dic271C2311yescharacteristic
59(C2×D271)⋊C4—21no—
60(C2×D271)⋊C4—22no—
61#41×C271—44no—
62C2×Q8×C271—21no—
63C2×D4×C271—22no—
64C2×D4×C271D411yescharacteristic
65C2×D4×C271C2311yescharacteristic
66C22×C4×C271—22no—
67C22×C4×C271—21no—
68C22×C4×C271C2312yes—
69C4⋊C4×C271—22no—
70C4⋊C4×C271—21no—
71C4⋊C4×C271D411yescharacteristic
72C4⋊C4×C271C2311yescharacteristic
73C4⋊C4×C271C2312yes—
74C22⋊C4×C271—21no—
75C22⋊C4×C271—22no—
76C22⋊C4×C271D412yes—
77C42×C271C2311yescharacteristic
78C22×D271—21no—
79C22×D271—22no—
80C22×D271—44no—
81C22×D271—44no—
82C22×D271—82no—
83D1084—22no—
84D1084—21no—
85D1084—22no—
86D1084—44no—
87D1084—42no—
88D1084—81no—
89D1084#4112yescomplemented
90C22⋊D271—44no—
91C2×Dic271—21no—
92C4×D271—42no—
93C271⋊C8—21no—
94C271⋊C8—21no—
95C271⋊C8—22no—
96C23×C271—21no—
97C23×C271C2×D412yes—
98Q8×C271—42no—
99D4×C271—21no—
100D4×C271—21no—
101D4×C271—44no—
102D4×C271D814yescomplemented
103C2×C4×C271—22no—
104C2×C4×C271—22no—
105C2×C4×C271—21no—
106C2×C4×C271—22no—
107C2×C4×C271—44no—
108C2×C4×C271—42no—
109C2×C4×C271C2×D411yescharacteristic
110C2×C4×C271C2×D412yes—
111C2×C4×C271C2×D411yescharacteristic
112C2×C4×C271C2411yescharacteristic
113C22×C271—22no—
114C22×C271—22no—
115C22×C271—41no—
116C22×C271—41no—
117C22×C271—41no—
118C22×C271C2×D814yes—
119C22×C271C22×D411yescharacteristic
120C2×D271—22no—
121C2×D271—41no—
122C2×D271—41no—
123C2×D271—42no—
124C2×D271—42no—
125C2×D271—42no—
126C2×D271—88no—
127C4×C271—22no—
128C4×C271—21no—
129C4×C271—41no—
130C4×C271—42no—
131C4×C271C2×D812yes—
132C4×C271C2×#4111yescharacteristic
133C4×C271H(5,𝔽2)11yescharacteristic
134Dic271—41no—
135C2×C271—24no—
136C2×C271—81no—
137C2×C271#54611yescharacteristic
138C2×C271C22×D811yescharacteristic
139C2×C271#57711yescharacteristic
140D271—42no—
141D271—82no—
142C271#329411yescharacteristic, complemented, Sylow
143#3294—2711nomaximal, Sylow
144#547D27111yescharacteristic, complemented
145D42—2711no—
146#486—2711no—
147#480—2712no—
148#474—2711no—
149#466—2712no—
150#459—2711no—
151#447—2712no—
152C4×D8—2711no—
153C8×D4—2711no—
154C2×#102—2712no—
155C2×#41—21no—
156C22×D4—2712no—
157C22×D4—5421no—
158C2×D8—2711no—
159C2×D8—2712no—
160C22×C8—2712no—
161C4⋊Q8C2×D27111yescharacteristic, complemented
162#127—2711no—
163#123—22no—
164C22⋊Q8—21no—
165#121—2712no—
166#121—5421no—
167#121C2×D27112yescomplemented
168#120—5422no—
169C4×D4—2711no—
170C4×D4C2×D27111yescharacteristic
171C4×D4C2×D27111yescharacteristic, complemented
172C2×C4⋊C4C2×D27112yescomplemented
173#107—2711no—
174#107—2712no—
175C4⋊C8—2711no—
176#102—2711no—
177#102—2711no—
178#102—2712no—
179#102—2712no—
180#102—2712no—
181#102—2712no—
182C22⋊C8—2712no—
183C4×C8—2711no—
184C24—5422no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—2711no—
189C2×D4—2712no—
190C2×D4—5424no—
191C2×D4—5422no—
192C2×D4—5422no—
193C2×D4—5421no—
194C2×D4—5421no—
195C2×D4—10842no—
196C2×D4—10844no—
197C2×D4C22⋊D27111yescharacteristic
198C2×D4C22×D27111yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—5421no—
202C22×C4C22×D27112yes—
203D8—5422no—
204D8—10842no—
205C2×C8—2711no—
206C2×C8—2711no—
207C2×C8—2712no—
208C2×C8—10842no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—2711no—
212C4⋊C4C22⋊D27111yescharacteristic
213C4⋊C4C22×D27111yescharacteristic
214C4⋊C4C22×D27112yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—5421no—
218C22⋊C4—5422no—
219C22⋊C4C22⋊D27112yes—
220C42C22×D27111yescharacteristic
221C23—21no—
222C23—5421no—
223C23—5422no—
224C23—10844no—
225C23—10844no—
226C23—21682no—
227C23C2×C22⋊D27112yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—2712no—
233D4—5422no—
234D4—5421no—
235D4—5422no—
236D4—10844no—
237D4—10844no—
238D4—10842no—
239D4—21681no—
240D4#1686463914yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—5421no—
248C2×C4—10842no—
249C2×C4C2×C22⋊D27111yescharacteristic
250C2×C4C2×C22⋊D27112yes—
251C2×C4C2×C22⋊D27111yescharacteristic
252C2×C4C23×D27111yescharacteristic
253C8—5421no—
254C8—5421no—
255C8—5422no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—5422no—
262C22—10841no—
263C22—10841no—
264C22—10842no—
265C22—10842no—
266C22—10842no—
267C22—21688no—
268C22C2×#1686463914yes—
269C22C22×C22⋊D27111yescharacteristic
270C4—22no—
271C4—21no—
272C4—41no—
273C4—42no—
274C4—10841no—
275C4C2×#1686463912yes—
276C4#41×D27111yescharacteristic
277C4#2257620611yescharacteristic
278C2—24no—
279C2—81no—
280C2—10842no—
281C2—21682no—
282C2#3177453511yescharacteristic
283C2C22×#1686463911yescharacteristic
284C2#3177470511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#547×C271C2
socleC22×C271C22×D4
hypercenter#547D271
hypocenterC271#3294
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D1084C2semidirect
2. #31774245C2semidirect
3. #31774210C2semidirect
4. #31774174C2semidirect
5. #31774131C2semidirect
6. #31774092C2semidirect
7. #31774030C2semidirect
8. C4×#16864639C2semidirect
9. D4×C271⋊C8C2semidirect
10. C2×#22576080C2semidirect
11. #547×C271C2semidirect
15. C2×#16864639C22semidirect
16. C22×C271⋊C8C22semidirect
22. #22576093C22semidirect
23. #22576093C22semidirect
24. C4⋊(C271⋊C8)C22semidirect
25. #22576081C22semidirect
28. #22576080C22semidirect
29. #22576080C22semidirect
30. #22576080C22semidirect
31. C22⋊(C271⋊C8)C22semidirect
32. C4×C271⋊C8C22semidirect
34. C4⋊Q8×C271C22semidirect
37. #121×C271C22semidirect
39. C4×D4×C271C22semidirect
40. C2×C4⋊C4×C271C22semidirect
89. D1084#41semidirect
102. D4×C271D8semidirect
142. C271#3294semidirect
144. #547D271semidirect
161. C4⋊Q8C2×D271semidirect
167. #121C2×D271semidirect
171. C4×D4C2×D271semidirect
172. C2×C4⋊C4C2×D271semidirect
240. D4#16864639semidirect

Maximal subgroups

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

subgroupindex
D4×D10842
#317742452
#317742102
#317741742
#317741312
#317740922
#317740302
C4×#168646392
D4×C271⋊C82
C2×#225760802
#547×C2712
#3294271