Finite Group Databasebeta

← #46846414

Subgroups — #46846414

Order 34688.

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

normal subgroups only

This group has 454 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#31774433C211yesmaximal, characteristic, complemented
2C2×#22576122C211yesmaximal, characteristic, complemented
3#31774215C211yesmaximal, characteristic, complemented
4#31774200C212yesmaximal, complemented
5#31774115C212yesmaximal, complemented
6#31774047C211yesmaximal, characteristic, complemented
7#31774033C211yesmaximal, characteristic
8#31773982C212yesmaximal, complemented
9#31773979C212yesmaximal, complemented
10#31773947C211yesmaximal, characteristic, complemented
11#484×C271C211yesmaximal, characteristic, complemented
12C22×D1084C2211yescharacteristic
13C22×C4×D271—21no—
14C2×#16864641C2212yes—
15C2×D2168C2212yes—
16#22576155—21no—
17#22576151C2211yescharacteristic, complemented
18#22576149C2211yescharacteristic, complemented
19#22576144—21no—
20#22576134C2211yescharacteristic
21Q8×Dic271C2211yescharacteristic, complemented
22#22576122—21no—
23#22576122—21no—
24#22576122C2212yes—
25#22576122C2212yes—
26#22576122C2211yescharacteristic, complemented
27(C22⋊D271)⋊C4—21no—
28#22576118—21no—
29C2×(C2×D271)⋊C4—21no—
30C2×C4×Dic271C2211yescharacteristic
31#22576091C2212yescomplemented
32Dic271⋊C8C2212yescomplemented
33#22576082C2212yes—
34#22576080C2212yes—
35#22576080C2212yescomplemented
36C22⋊(C271⋊C8)C2211yescharacteristic
37C8×Dic271C2212yescomplemented
38C22⋊Q8×C271C2211yescharacteristic
39C2×C4⋊C4×C271C2211yescharacteristic
40#107×C271C2212yescomplemented
41#103×C271C2212yescomplemented
42C22⋊C8×C271C2211yescharacteristic
43C23×D271—21no—
44C2×D1084—24no—
45C2×D1084—21no—
46C2×D1084C2311yescharacteristic
47C2×D1084C2312yes—
48C2×C22⋊D271—21no—
49C22×Dic271D411yescharacteristic
50C22×Dic271D411yescharacteristic
51C2×C4×D271—22no—
52C2×C4×D271—22no—
53C2×C4×D271—21no—
54C2×C4×D271—44no—
55#16864641—24no—
56D2168—24no—
57C2×C271⋊C8C2312yes—
58C4⋊Dic271—21no—
59C4⋊Dic271—21no—
60C4⋊Dic271C2311yescharacteristic
61Dic271⋊C4—21no—
62(C2×D271)⋊C4—21no—
63(C2×D271)⋊C4—21no—
64(C2×D271)⋊C4—21no—
65(C2×D271)⋊C4—22no—
66(C2×D271)⋊C4—22no—
67(C2×D271)⋊C4—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—21no—
85C22×D271—42no—
86C22×D271—42no—
87D1084—24no—
88D1084—22no—
89D1084—22no—
90D1084#4114yes—
91C22⋊D271—42no—
92C2×Dic271—22no—
93C2×Dic271—22no—
94C2×Dic271—21no—
95C2×Dic271C2×D411yescharacteristic
96C2×Dic271C2×D411yescharacteristic
97C2×Dic271C2×D411yescharacteristic
98C2×Dic271C2×D411yescharacteristic
99C4×D271—44no—
100C4×D271—42no—
101C4×D271—44no—
102C271⋊C8—22no—
103C23×C271C2×D411yescharacteristic
104Q8×C271—22no—
105C2×C4×C271—22no—
106C2×C4×C271—21no—
107C2×C4×C271—21no—
108C2×C4×C271—22no—
109C2×C4×C271—42no—
110C2×C4×C271C2×D411yescharacteristic
111C2×C4×C271C2411yescharacteristic
112C8×C271—22no—
113C22×C271—21no—
114C22×C271—21no—
115C22×C271#13512yes—
116C22×C271C22×D411yescharacteristic
117C2×D271—24no—
118C2×D271—48no—
119C2×D271—42no—
120C2×D271—42no—
121C2×D271—41no—
122C2×D271—41no—
123C2×D271—41no—
124C4×C271—21no—
125C4×C271—41no—
126C4×C271—42no—
127C4×C271—41no—
128C4×C271C2×#4112yes—
129Dic271—21no—
130Dic271—21no—
131Dic271—21no—
132Dic271—41no—
133Dic271AGL(1,ℤ/8)12yes—
134C2×C271—22no—
135C2×C271#52511yescharacteristic
136C2×C271C2×#13511yescharacteristic
137C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
138D271—44no—
139D271—81no—
140C271#310511yescharacteristic, complemented, Sylow
141#3105—2711nomaximal, Sylow
142#529—2711no—
143C2×C4×D4—2711no—
144#484D27111yescharacteristic, complemented
145#481—2711no—
146#478—2712no—
147#463—2712no—
148#450—2711no—
149#447—2711no—
150C4×SD16—2712no—
151C4×D8—2712no—
152#431—2711no—
153C22×D4—2711no—
154C23×C4—5421no—
155C2×SD16—2712no—
156C2×D8—2712no—
157#126—5421no—
158#125—2711no—
159#124—2711no—
160#123—5421no—
161C22⋊Q8C2×D27111yescharacteristic
162#121—2711no—
163C4×Q8—2711no—
164C4×D4—2712no—
165C4×D4—2712no—
166C4×D4—2711no—
167C4×D4—5421no—
168C4×D4—5421no—
169C4×D4—5421no—
170#117—5421no—
171C2×C4⋊C4C2×D27111yescharacteristic
172C2×C22⋊C4—5421no—
173C2×C42—2711no—
174#107C2×D27112yescomplemented
175#106—2712no—
176C4⋊C8—2712no—
177#103C2×D27112yescomplemented
178#102—2712no—
179#102—2712no—
180#102—2712no—
181C22⋊C8—2711no—
182C22⋊C8C2×D27111yescharacteristic
183C4×C8—2712no—
184C24—5421no—
185C2×Q8C22×D27111yescharacteristic
186C2×D4—2712no—
187C2×D4—2711no—
188C2×D4—5424no—
189C2×D4—5421no—
190C2×D4—5421no—
191C22×C4—21no—
192C22×C4—2711no—
193C22×C4—2711no—
194C22×C4—5421no—
195C22×C4—5422no—
196C22×C4—5422no—
197C22×C4—10844no—
198C22×C4C22×D27111yescharacteristic
199SD16—5424no—
200D8—5424no—
201C2×C8—2712no—
202C2×C8C22×D27112yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—2711no—
206C4⋊C4—5421no—
207C4⋊C4—5421no—
208C4⋊C4—5421no—
209C4⋊C4C22×D27111yescharacteristic
210C4⋊C4C22×D27112yes—
211C22⋊C4—21no—
212C22⋊C4—5422no—
213C22⋊C4—5422no—
214C22⋊C4—5421no—
215C22⋊C4—5421no—
216C22⋊C4—5421no—
217C22⋊C4—5421no—
218C42—2711no—
219C42—2711no—
220C42—2711no—
221C42—2711no—
222C42—5421no—
223C23—5424no—
224C23—5422no—
225C23—5421no—
226C23—10842no—
227C23—10842no—
228C23D4×D27111yescharacteristic
229Q8—22no—
230D4—2714no—
231D4—5424no—
232D4—5422no—
233D4—5422no—
234D4—10842no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—2711no—
241C2×C4—2711no—
242C2×C4—2711no—
243C2×C4—2711no—
244C2×C4—5422no—
245C2×C4—5422no—
246C2×C4—5421no—
247C2×C4—10844no—
248C2×C4—10842no—
249C2×C4—10844no—
250C2×C4D4×D27111yescharacteristic
251C2×C4C23×D27111yescharacteristic
252C8—22no—
253C8—5422no—
254C22—21no—
255C22—21no—
256C22—5424no—
257C22—10848no—
258C22—10842no—
259C22—10842no—
260C22—10841no—
261C22—10841no—
262C22—10841no—
263C22#2257618312yes—
264C22C2×D4×D27111yescharacteristic
265C4—21no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—2712no—
270C4—5421no—
271C4—5421no—
272C4—5421no—
273C4—10841no—
274C4#41×D27112yes—
275C2—22no—
276C2—10844no—
277C2—21681no—
278C2#3177440511yescharacteristic
279C2C2×#2257618311yescharacteristic
280C2AGL(1,ℤ/8)×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#484×C271C2
socleC22×C271C22×D4
prime radical#31774033C2
hypercenter#484D271
hypocenterC271#3105
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. #31774433C2semidirect
2. C2×#22576122C2semidirect
3. #31774215C2semidirect
4. #31774200C2semidirect
5. #31774115C2semidirect
6. #31774047C2semidirect
8. #31773982C2semidirect
9. #31773979C2semidirect
10. #31773947C2semidirect
11. #484×C271C2semidirect
17. #22576151C22semidirect
18. #22576149C22semidirect
21. Q8×Dic271C22semidirect
26. #22576122C22semidirect
31. #22576091C22semidirect
32. Dic271⋊C8C22semidirect
35. #22576080C22semidirect
37. C8×Dic271C22semidirect
40. #107×C271C22semidirect
41. #103×C271C22semidirect
140. C271#3105semidirect
144. #484D271semidirect
174. #107C2×D271semidirect
177. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744332
C2×#225761222
#317742152
#317742002
#317741152
#317740472
#317740332
#317739822
#317739792
#317739472
#484×C2712
#3105271