Finite Group Databasebeta

← #46846460

Subgroups — #46846460

Order 34688.

444 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 119 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#31774449C211yesmaximal, characteristic, complemented
2C2×#22576122C211yesmaximal, characteristic
3#31774232C211yesmaximal, characteristic, complemented
4#31774176C212yesmaximal, complemented
5#31774101C212yesmaximal, complemented
6#31774050C211yesmaximal, characteristic
7#31774047C211yesmaximal, characteristic
8#31774000C212yesmaximal, complemented
9#31773996C212yesmaximal, complemented
10#31773951C211yesmaximal, characteristic, complemented
11#484×C271C211yesmaximal, characteristic, complemented
12C22×D1084C2211yescharacteristic
13C22×C4×D271—21no—
14C2×#16864643C2212yes—
15C2×#16864641C2212yes—
16#22576159C2211yescharacteristic, complemented
17#22576158C2211yescharacteristic, complemented
18#22576150—21no—
19#22576139C2211yescharacteristic
20#22576137—21no—
21Q8×Dic271C2211yescharacteristic, complemented
22#22576125—21no—
23#22576125C2211yescharacteristic, complemented
24#22576122—21no—
25#22576122—21no—
26#22576122C2212yes—
27#22576122C2212yes—
28#22576116—21no—
29C2×(C2×D271)⋊C4—21no—
30C2×C4×Dic271C2211yescharacteristic
31#22576093C2212yescomplemented
32Dic271⋊C8C2212yescomplemented
33#22576083C2212yescomplemented
34#22576082C2212yes—
35#22576080C2212yes—
36C22⋊(C271⋊C8)C2211yescharacteristic
37#22576070C2212yescomplemented
38C22⋊Q8×C271C2211yescharacteristic
39C2×C4⋊C4×C271C2211yescharacteristic
40#107×C271C2212yescomplemented
41#103×C271C2212yescomplemented
42C22⋊C8×C271C2211yescharacteristic
43C23×D271—21no—
44C2×Dic542—21no—
45C2×Dic542C2311yescharacteristic
46C2×D1084—24no—
47C2×D1084—21no—
48C2×D1084C2312yes—
49C22×Dic271D411yescharacteristic
50C22×Dic271D411yescharacteristic
51C2×C4×D271—22no—
52C2×C4×D271—22no—
53C2×C4×D271—44no—
54#16864643—24no—
55#16864641—24no—
56C2×C271⋊C8C2312yes—
57C4⋊Dic271—21no—
58C4⋊Dic271C2311yescharacteristic
59Dic271⋊C4—21no—
60Dic271⋊C4—21no—
61Dic271⋊C4—21no—
62Dic271⋊C4—21no—
63Dic271⋊C4—21no—
64C22⋊Dic271—21no—
65(C2×D271)⋊C4—22no—
66(C2×D271)⋊C4—22no—
67C4×Dic271—21no—
68C4×Dic271—21no—
69C4×Dic271D411yescharacteristic
70C4×Dic271D411yescharacteristic
71C4×Dic271C2311yescharacteristic
72C4×Dic271C2311yescharacteristic
73C2×Q8×C271C2311yescharacteristic
74C22×C4×C271—21no—
75C22×C4×C271C2311yescharacteristic
76C2×C8×C271C2312yes—
77C4⋊C4×C271—21no—
78C4⋊C4×C271—21no—
79C4⋊C4×C271C2311yescharacteristic
80C4⋊C4×C271C2312yes—
81C22⋊C4×C271—21no—
82C22×D271—24no—
83C22×D271—22no—
84C22×D271—42no—
85C22×D271—42no—
86Dic542—22no—
87Dic542—42no—
88D1084—24no—
89D1084—22no—
90D1084#4114yes—
91C2×Dic271—22no—
92C2×Dic271—22no—
93C2×Dic271—21no—
94C2×Dic271—21no—
95C2×Dic271C2×D411yescharacteristic
96C2×Dic271C2×D411yescharacteristic
97C2×Dic271C2×D411yescharacteristic
98C2×Dic271C2×D411yescharacteristic
99C4×D271—44no—
100C4×D271—44no—
101C271⋊C8—22no—
102C23×C271C2×D411yescharacteristic
103Q8×C271—22no—
104C2×C4×C271—22no—
105C2×C4×C271—21no—
106C2×C4×C271—21no—
107C2×C4×C271—22no—
108C2×C4×C271—42no—
109C2×C4×C271C2×D411yescharacteristic
110C2×C4×C271C2411yescharacteristic
111C8×C271—22no—
112C22×C271—21no—
113C22×C271—21no—
114C22×C271#13712yes—
115C22×C271C22×D411yescharacteristic
116C2×D271—24no—
117C2×D271—48no—
118C2×D271—42no—
119C2×D271—42no—
120C4×C271—21no—
121C4×C271—41no—
122C4×C271—42no—
123C4×C271—41no—
124C4×C271C2×#4112yes—
125Dic271—21no—
126Dic271—21no—
127Dic271—21no—
128Dic271—41no—
129Dic271—41no—
130Dic271AGL(1,ℤ/8)12yes—
131C2×C271—22no—
132C2×C271#52511yescharacteristic
133C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
134C2×C271C2×#13711yescharacteristic
135D271—44no—
136C271#311011yescharacteristic, complemented, Sylow
137#3110—2711nomaximal, Sylow
138#533—2711no—
139C2×C4×D4—2711no—
140#484—2711no—
141#484D27111yescharacteristic, complemented
142#474—2712no—
143#461—2712no—
144#450—2711no—
145#450—2711no—
146#440—2712no—
147#440—2712no—
148#432—2711no—
149C22×D4—2711no—
150C23×C4—5421no—
151C2×SD16—2712no—
152C2×SD16—2712no—
153C4⋊Q8—2711no—
154C4⋊Q8—2711no—
155#125—5421no—
156C22⋊Q8—2711no—
157C22⋊Q8—5421no—
158C22⋊Q8C2×D27111yescharacteristic
159C4×Q8—2711no—
160C4×Q8—2711no—
161C4×Q8—5421no—
162C4×D4—2712no—
163C4×D4—2712no—
164C4×D4—5421no—
165C4×D4—5421no—
166#117—5421no—
167C2×C4⋊C4C2×D27111yescharacteristic
168C2×C22⋊C4—5421no—
169C2×C42—2711no—
170#107—2712no—
171#107C2×D27112yescomplemented
172C4⋊C8—2712no—
173#103—2712no—
174#103C2×D27112yescomplemented
175#102—2712no—
176#102—2712no—
177C22⋊C8—2711no—
178C22⋊C8C2×D27111yescharacteristic
179#97—2712no—
180C24—5421no—
181C2×Q8—2711no—
182C2×Q8—5421no—
183C2×Q8C22×D27111yescharacteristic
184C2×D4—2712no—
185C2×D4—5424no—
186C2×D4—5421no—
187C22×C4—21no—
188C22×C4—2711no—
189C22×C4—2711no—
190C22×C4—5422no—
191C22×C4—5422no—
192C22×C4—10844no—
193C22×C4C22×D27111yescharacteristic
194SD16—5424no—
195SD16—5424no—
196C2×C8—2712no—
197C2×C8C22×D27112yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—2711no—
201C4⋊C4—5421no—
202C4⋊C4—5421no—
203C4⋊C4—5421no—
204C4⋊C4—5421no—
205C4⋊C4—5421no—
206C4⋊C4—5421no—
207C4⋊C4C22×D27111yescharacteristic
208C4⋊C4C22×D27112yes—
209C22⋊C4—21no—
210C22⋊C4—5422no—
211C22⋊C4—5422no—
212C22⋊C4—5421no—
213C42—2711no—
214C42—2711no—
215C42—2711no—
216C42—2711no—
217C42—5421no—
218C42—5421no—
219C23—5424no—
220C23—5422no—
221C23—10842no—
222C23—10842no—
223C23D4×D27111yescharacteristic
224Q8—22no—
225Q8—5422no—
226Q8—10842no—
227D4—2714no—
228D4—5424no—
229D4—5422no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—2711no—
236C2×C4—2711no—
237C2×C4—2711no—
238C2×C4—2711no—
239C2×C4—5422no—
240C2×C4—5422no—
241C2×C4—5421no—
242C2×C4—5421no—
243C2×C4—10844no—
244C2×C4—10844no—
245C2×C4D4×D27111yescharacteristic
246C2×C4C23×D27111yescharacteristic
247C8—22no—
248C8—5422no—
249C22—21no—
250C22—21no—
251C22—5424no—
252C22—10848no—
253C22—10842no—
254C22—10842no—
255C22#2257619212yes—
256C22C2×D4×D27111yescharacteristic
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—2712no—
262C4—5421no—
263C4—5421no—
264C4—5421no—
265C4—10841no—
266C4—10841no—
267C4#41×D27112yes—
268C2—22no—
269C2—10844no—
270C2#3177440511yescharacteristic
271C2AGL(1,ℤ/8)×D27111yescharacteristic
272C2C2×#2257619211yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#484×C271C2
socleC22×C271C22×D4
prime radicalC22×D1084C22
hypercenter#484D271
hypocenterC271#3110
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #31774449C2semidirect
3. #31774232C2semidirect
4. #31774176C2semidirect
5. #31774101C2semidirect
8. #31774000C2semidirect
9. #31773996C2semidirect
10. #31773951C2semidirect
11. #484×C271C2semidirect
16. #22576159C22semidirect
17. #22576158C22semidirect
21. Q8×Dic271C22semidirect
23. #22576125C22semidirect
31. #22576093C22semidirect
32. Dic271⋊C8C22semidirect
33. #22576083C22semidirect
37. #22576070C22semidirect
40. #107×C271C22semidirect
41. #103×C271C22semidirect
136. C271#3110semidirect
141. #484D271semidirect
171. #107C2×D271semidirect
174. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744492
C2×#225761222
#317742322
#317741762
#317741012
#317740502
#317740472
#317740002
#317739962
#317739512
#484×C2712
#3110271