Finite Group Databasebeta

← #46847960

Subgroups — #46847960

Order 34688.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22576186C212yesmaximal, complemented
2#31774552C211yesmaximal, characteristic, complemented
3#31774269C211yesmaximal, characteristic, complemented
4#31774141C212yesmaximal, complemented
5#31774102C211yesmaximal, characteristic, complemented
6#31774089C211yesmaximal, characteristic, complemented
7#31774048C212yesmaximal, complemented
8#31774040C212yesmaximal, complemented
9C4×#16864639C211yesmaximal, characteristic, complemented
10#31773975C211yesmaximal, characteristic, complemented
11D42×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2212yescomplemented
13#22576186—28no—
14C2×#16864642C2212yes—
15C2×#16864642C2212yescomplemented
16C2×#16864642C2212yescomplemented
17C2×#16864639C2211yescharacteristic
18C2×#16864639C2212yescomplemented
19C2×#16864638C2212yescomplemented
20#22576160C2211yescharacteristic, complemented
21#22576142—22no—
22#22576139C2212yescomplemented
23#22576131—21no—
24C4×D1084C2211yescharacteristic, complemented
25C4⋊C4×D271—21no—
26#22576093C2211yescharacteristic, complemented
27C4⋊(C271⋊C8)C2211yescharacteristic, complemented
28#22576083C2212yescomplemented
29#22576081C2211yescharacteristic
30#22576081C2212yescomplemented
31#22576080C2211yescharacteristic, complemented
32C22⋊(C271⋊C8)C2212yescomplemented
33C4×C271⋊C8C2211yescharacteristic, complemented
34C22×D4×C271—21no—
35C22×D4×C271C2212yes—
36#127×C271C2211yescharacteristic
37#121×C271—21no—
38#121×C271C2212yes—
39#120×C271—22no—
40C4×D4×C271C2211yescharacteristic
41C4×D4×C271C2211yescharacteristic
42#16864654—24no—
43#16864654—42no—
44C2×Dic542C2312yes—
45C2×D1084C2311yescharacteristic
46C2×C22⋊D271—22no—
47C2×C4×D271—21no—
48C2×C4×D271—22no—
49#16864642—24no—
50#16864642—24no—
51#16864642—42no—
52#16864639—24no—
53#16864639D414yescomplemented
54#16864638—24no—
55C2×C271⋊C8C2312yes—
56C2×C271⋊C8C2311yescharacteristic
57C2×C271⋊C8C2311yescharacteristic
58C4⋊Dic271—21no—
59C4⋊Dic271—21no—
60C4⋊Dic271C2311yescharacteristic
61Dic271⋊C4—22no—
62C22⋊Dic271—22no—
63(C2×D271)⋊C4—21no—
64C24×C271—22no—
65C2×D4×C271—24no—
66C2×D4×C271—22no—
67C2×D4×C271—22no—
68C2×D4×C271—21no—
69C2×D4×C271—21no—
70C2×D4×C271—42no—
71C2×D4×C271—44no—
72C2×D4×C271D411yescharacteristic
73C2×D4×C271C2312yes—
74C2×D4×C271C2311yescharacteristic
75C22×C4×C271—21no—
76C22×C4×C271C2312yes—
77C4⋊C4×C271D411yescharacteristic
78C4⋊C4×C271C2311yescharacteristic
79C22⋊C4×C271—21no—
80C22⋊C4×C271—22no—
81C22⋊C4×C271D412yes—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84Dic542—22no—
85Dic542—22no—
86D1084—21no—
87D1084C2×D412yescomplemented
88C22⋊D271—44no—
89C2×Dic271—21no—
90C2×Dic271—22no—
91C4×D271—44no—
92C4×D271—42no—
93C271⋊C8—22no—
94C271⋊C8—21no—
95C271⋊C8C2×D412yescomplemented
96C23×C271—21no—
97C23×C271—22no—
98C23×C271—44no—
99C23×C271—44no—
100C23×C271—82no—
101C23×C271C2×D412yes—
102D4×C271—22no—
103D4×C271—22no—
104D4×C271—21no—
105D4×C271—22no—
106D4×C271—44no—
107D4×C271—44no—
108D4×C271—42no—
109D4×C271—81no—
110D4×C271C2×D412yes—
111C2×C4×C271—22no—
112C2×C4×C271—22no—
113C2×C4×C271—21no—
114C2×C4×C271—42no—
115C2×C4×C271C2×D411yescharacteristic
116C2×C4×C271C2×D412yes—
117C2×C4×C271C2×D411yescharacteristic
118C2×C4×C271C2411yescharacteristic
119C22×C271—22no—
120C22×C271—22no—
121C22×C271—22no—
122C22×C271—22no—
123C22×C271—41no—
124C22×C271—41no—
125C22×C271—42no—
126C22×C271—42no—
127C22×C271—42no—
128C22×C271—88no—
129C22×C271C22×D411yescharacteristic
130C2×D271—22no—
131C2×D271—41no—
132C2×D271—41no—
133C4×C271—22no—
134C4×C271—21no—
135C4×C271—41no—
136C4×C271AGL(1,ℤ/8)12yescomplemented
137C4×C271C22×D411yescharacteristic
138C4×C271H(5,𝔽2)11yescharacteristic
139Dic271—42no—
140Dic271—41no—
141C2×C271—42no—
142C2×C271—42no—
143C2×C271—82no—
144C2×C271D4211yescharacteristic
145C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C271#57711yescharacteristic
147D271—42no—
148C271#327211yescharacteristic, complemented, Sylow
149#3272—2711nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2712no—
151#547—2711no—
152D42D27111yescharacteristic, complemented
153#492—2711no—
154#468—2712no—
155#461—2711no—
156#459—2711no—
157#450—2712no—
158#449—2712no—
159C4×D8—2711no—
160#436—2711no—
161C2×#41—2712no—
162C22×D4—21no—
163C22×D4C2×D27112yes—
164AGL(1,ℤ/8)—5428no—
165C2×SD16—2712no—
166C2×SD16—2712no—
167C2×SD16—2712no—
168C2×D8—2711no—
169C2×D8—2712no—
170C2×M16—2712no—
171C4⋊Q8—2711no—
172#127C2×D27111yescharacteristic
173#123—5422no—
174C22⋊Q8—2712no—
175#121—21no—
176#121—5421no—
177#121C2×D27112yes—
178#120—22no—
179C4×D4—2711no—
180C4×D4C2×D27111yescharacteristic
181C4×D4C2×D27111yescharacteristic
182C2×C4⋊C4—5421no—
183#107—2711no—
184C4⋊C8—2711no—
185#103—2712no—
186#102—2711no—
187#102—2712no—
188#102—2711no—
189C22⋊C8—2712no—
190C4×C8—2711no—
191C24—22no—
192#41—5424no—
193#41—10842no—
194C2×Q8—2712no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—2711no—
203C2×D4—5422no—
204C2×D4C22⋊D27111yescharacteristic
205C2×D4C22×D27112yes—
206C2×D4C22×D27111yescharacteristic
207C22×C4—21no—
208C22×C4—5421no—
209C22×C4—5422no—
210C22×C4C22×D27112yes—
211SD16—5424no—
212SD16—5424no—
213SD16—10842no—
214D8—2714no—
215D8—5424no—
216M16—5424no—
217C2×C8—2711no—
218C2×C8—2711no—
219C2×C8—2712no—
220C4⋊C4—2711no—
221C4⋊C4—5421no—
222C4⋊C4—5422no—
223C4⋊C4—5421no—
224C4⋊C4C22⋊D27111yescharacteristic
225C4⋊C4C22×D27111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—5422no—
229C22⋊C4—5421no—
230C22⋊C4C22⋊D27112yes—
231C42C22×D27111yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—5421no—
238C23C2×C22⋊D27112yes—
239Q8—5422no—
240Q8—5422no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—2712no—
250D4—5421no—
251D4—10844no—
252D4D4×D27112yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—5421no—
258C2×C4—5422no—
259C2×C4—10844no—
260C2×C4—10842no—
261C2×C4C2×C22⋊D27111yescharacteristic
262C2×C4C2×C22⋊D27112yes—
263C2×C4C2×C22⋊D27111yescharacteristic
264C2×C4C23×D27111yescharacteristic
265C8—2712no—
266C8—5422no—
267C8—5421no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—5422no—
279C22—10841no—
280C22—10841no—
281C22C22×C22⋊D27111yescharacteristic
282C4—22no—
283C4—21no—
284C4—41no—
285C4—10842no—
286C4—10841no—
287C4#2257618612yescomplemented
288C4C2×D4×D27111yescharacteristic
289C4#2257620511yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—10842no—
294C2D4×C22⋊D27111yescharacteristic
295C2C2×#2257618611yescharacteristic
296C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingD42×C271C2
socleC22×C271C22×D4
hypercenterD42D271
hypocenterC271#3272
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#22576186C2semidirect
2. #31774552C2semidirect
3. #31774269C2semidirect
4. #31774141C2semidirect
5. #31774102C2semidirect
6. #31774089C2semidirect
7. #31774048C2semidirect
8. #31774040C2semidirect
9. C4×#16864639C2semidirect
10. #31773975C2semidirect
11. D42×C271C2semidirect
12. C2×#16864654C22semidirect
15. C2×#16864642C22semidirect
16. C2×#16864642C22semidirect
18. C2×#16864639C22semidirect
19. C2×#16864638C22semidirect
20. #22576160C22semidirect
22. #22576139C22semidirect
24. C4×D1084C22semidirect
26. #22576093C22semidirect
27. C4⋊(C271⋊C8)C22semidirect
28. #22576083C22semidirect
30. #22576081C22semidirect
31. #22576080C22semidirect
32. C22⋊(C271⋊C8)C22semidirect
33. C4×C271⋊C8C22semidirect
53. #16864639D4semidirect
87. D1084C2×D4semidirect
95. C271⋊C8C2×D4semidirect
136. C4×C271AGL(1,ℤ/8)semidirect
148. C271#3272semidirect
152. D42D271semidirect
287. C4#22576186semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
#317745522
#317742692
#317741412
#317741022
#317740892
#317740482
#317740402
C4×#168646392
#317739752
D42×C2712
#3272271