Finite Group Databasebeta

← #46847965

Subgroups — #46847965

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×#22576184C212yesmaximal, complemented
2#31774551C211yesmaximal, characteristic, complemented
3D4×D1084C211yesmaximal, characteristic, complemented
4#31774270C211yesmaximal, characteristic, complemented
5#31774144C212yesmaximal, complemented
6#31774102C211yesmaximal, characteristic, complemented
7#31774092C211yesmaximal, characteristic, complemented
8#31774049C212yesmaximal, complemented
9#31774043C212yesmaximal, complemented
10#31773976C211yesmaximal, characteristic, complemented
11C4×D8×C271C211yesmaximal, characteristic, complemented
12C2×#16864653C2212yescomplemented
13C22×D1084—21no—
14C2×D4×D271C2212yes—
15#22576184—28no—
16C2×#16864643C2212yescomplemented
17C2×#16864643C2212yescomplemented
18C2×#16864642C2212yes—
19C2×#16864639C2212yescomplemented
20C2×#16864637C2212yescomplemented
21#22576160C2211yescharacteristic, complemented
22#22576157C2211yescharacteristic
23#22576144—22no—
24#22576140C2212yescomplemented
25#22576135C2212yes—
26#22576134—21no—
27#22576134—21no—
28#22576130—22no—
29C4×D1084C2211yescharacteristic
30C2×C4⋊Dic271—21no—
31C4⋊(C271⋊C8)C2211yescharacteristic, complemented
32#22576083C2212yescomplemented
33#22576080C2212yescomplemented
34(C2×D271)⋊C8C2212yescomplemented
35C2×D8×C271C2211yescharacteristic
36C4×D4×C271C2211yescharacteristic
37C4×D4×C271C2211yescharacteristic, complemented
38#107×C271C2211yescharacteristic, complemented
39#102×C271C2211yescharacteristic
40#102×C271C2211yescharacteristic, complemented
41C4×C8×C271C2211yescharacteristic, complemented
42C23×D271—22no—
43#16864653—24no—
44#16864653—42no—
45C2×Dic542C2312yes—
46C2×D1084—22no—
47C2×D1084—21no—
48C2×D1084—21no—
49C2×D1084—44no—
50C2×D1084D411yescharacteristic
51C2×D1084C2312yes—
52C2×C22⋊D271—22no—
53C2×C22⋊D271—22no—
54D4×D271—24no—
55D4×D271—42no—
56C22×Dic271—22no—
57C2×C4×D271C2312yes—
58#16864643—24no—
59#16864643—42no—
60#16864642—24no—
61#16864639—24no—
62#16864637—24no—
63C2×C271⋊C8C2312yes—
64C4⋊Dic271—22no—
65C4⋊Dic271—21no—
66C4⋊Dic271—21no—
67C4⋊Dic271D411yescharacteristic
68(C2×D271)⋊C4—22no—
69(C2×D271)⋊C4—22no—
70(C2×D271)⋊C4D412yes—
71C2×D4×C271C2311yescharacteristic
72C2×D4×C271C2311yescharacteristic
73C22×C4×C271—21no—
74C22×C4×C271—21no—
75D8×C271D414yescomplemented
76C2×C8×C271C2311yescharacteristic
77C2×C8×C271C2311yescharacteristic
78C4⋊C4×C271C2311yescharacteristic
79C4⋊C4×C271C2311yescharacteristic
80C22⋊C4×C271—21no—
81C22⋊C4×C271—21no—
82C42×C271C2311yescharacteristic
83C22×D271—22no—
84C22×D271—44no—
85C22×D271—44no—
86C22×D271—82no—
87C22×D271C2×D412yes—
88Dic542—22no—
89Dic542—22no—
90D1084—22no—
91D1084—22no—
92D1084—22no—
93D1084—44no—
94D1084—42no—
95D1084—81no—
96C22⋊D271—44no—
97C22⋊D271—44no—
98C2×Dic271—22no—
99C2×Dic271—44no—
100C2×Dic271C2×D412yes—
101C4×D271—22no—
102C4×D271—22no—
103C271⋊C8—22no—
104C23×C271—21no—
105C23×C271—21no—
106D4×C271—21no—
107D4×C271—21no—
108D4×C271C2×D412yes—
109D4×C271C2×D412yescomplemented
110C2×C4×C271—21no—
111C2×C4×C271—21no—
112C2×C4×C271—42no—
113C2×C4×C271—42no—
114C2×C4×C271C2×D411yescharacteristic
115C2×C4×C271C2×D411yescharacteristic
116C2×C4×C271C2411yescharacteristic
117C8×C271—21no—
118C8×C271C2×D412yescomplemented
119C22×C271—22no—
120C22×C271—22no—
121C22×C271—41no—
122C22×C271—41no—
123C22×C271—41no—
124C22×C271—41no—
125C22×C271C22×D411yescharacteristic
126C2×D271—22no—
127C2×D271—22no—
128C2×D271—22no—
129C2×D271—42no—
130C2×D271—42no—
131C2×D271—42no—
132C2×D271—88no—
133C4×C271—21no—
134C4×C271—41no—
135C4×C271—41no—
136C4×C271AGL(1,ℤ/8)12yescomplemented
137C4×C271C22×D411yescharacteristic
138C4×C271H(5,𝔽2)11yescharacteristic
139Dic271—22no—
140Dic271—42no—
141C2×C271—42no—
142C2×C271—42no—
143C2×C271D4211yescharacteristic
144C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C271#57711yescharacteristic
146D271—42no—
147D271—82no—
148C271#327211yescharacteristic, complemented, Sylow
149#3272—2711nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2712no—
151#547—2711no—
152D42—2711no—
153#492—2711no—
154#468—2712no—
155#461—2711no—
156#459—2711no—
157#450—2712no—
158#449—2712no—
159C4×D8D27111yescharacteristic, complemented
160#436—2711no—
161C2×#41—2712no—
162C22×D4—2712no—
163C22×D4—5421no—
164AGL(1,ℤ/8)—5428no—
165C2×SD16—2712no—
166C2×SD16—2712no—
167C2×SD16—2712no—
168C2×D8—2712no—
169C2×D8C2×D27111yescharacteristic
170C2×M16—2712no—
171C4⋊Q8—2711no—
172#127—2711no—
173#123—5422no—
174C22⋊Q8—2712no—
175#121—2712no—
176#121—5421no—
177#121—5421no—
178#120—5422no—
179C4×D4—2711no—
180C4×D4C2×D27111yescharacteristic
181C4×D4C2×D27111yescharacteristic, complemented
182C2×C4⋊C4—5421no—
183#107C2×D27111yescharacteristic, complemented
184C4⋊C8—2711no—
185#103—2712no—
186#102—2712no—
187#102C2×D27111yescharacteristic
188#102C2×D27111yescharacteristic, complemented
189C22⋊C8—2712no—
190C4×C8C2×D27111yescharacteristic, complemented
191C24—5422no—
192#41—5424no—
193#41—10842no—
194C2×Q8—2712no—
195C2×D4—2711no—
196C2×D4—2712no—
197C2×D4—5422no—
198C2×D4—5422no—
199C2×D4—5424no—
200C2×D4—5422no—
201C2×D4—5421no—
202C2×D4—5421no—
203C2×D4—10844no—
204C2×D4—10842no—
205C2×D4C22×D27111yescharacteristic
206C2×D4C22×D27111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—2712no—
210C22×C4—5422no—
211SD16—5424no—
212SD16—5424no—
213SD16—10842no—
214D8—5424no—
215D8D108414yescomplemented
216M16—5424no—
217C2×C8—2712no—
218C2×C8C22×D27111yescharacteristic
219C2×C8C22×D27111yescharacteristic
220C4⋊C4—2711no—
221C4⋊C4—5422no—
222C4⋊C4—5421no—
223C4⋊C4—5421no—
224C4⋊C4C22×D27111yescharacteristic
225C4⋊C4C22×D27111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—2712no—
229C22⋊C4—5422no—
230C22⋊C4—5422no—
231C42C22×D27111yescharacteristic
232C23—21no—
233C23—21no—
234C23—2712no—
235C23—5422no—
236C23—10844no—
237C23—10844no—
238C23—21682no—
239Q8—5422no—
240Q8—5422no—
241D4—21no—
242D4—21no—
243D4—5422no—
244D4—5422no—
245D4—5422no—
246D4—10844no—
247D4—10844no—
248D4—10844no—
249D4—10842no—
250D4—21681no—
251D4C2×D108412yes—
252D4C2×D108412yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—2712no—
258C2×C4—5422no—
259C2×C4—5422no—
260C2×C4—5422no—
261C2×C4—10844no—
262C2×C4D4×D27111yescharacteristic
263C2×C4D4×D27111yescharacteristic
264C2×C4C23×D27111yescharacteristic
265C8—21no—
266C8—5422no—
267C8C2×D108412yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—5422no—
275C22—5422no—
276C22—5422no—
277C22—10842no—
278C22—10842no—
279C22—10842no—
280C22—21688no—
281C22C2×D4×D27111yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—5422no—
286C4—10842no—
287C4#2257618412yescomplemented
288C4C22×D108411yescharacteristic
289C4#2257620611yescharacteristic
290C2—42no—
291C2—42no—
292C2—10842no—
293C2—21682no—
294C2D4×D108411yescharacteristic
295C2C2×#2257618411yescharacteristic
296C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC4×D8×C271C2
socleC22×C271C22×D4
hypercenterC4×D8D271
hypocenterC271#3272
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. C2×#22576184C2semidirect
2. #31774551C2semidirect
3. D4×D1084C2semidirect
4. #31774270C2semidirect
5. #31774144C2semidirect
6. #31774102C2semidirect
7. #31774092C2semidirect
8. #31774049C2semidirect
9. #31774043C2semidirect
10. #31773976C2semidirect
11. C4×D8×C271C2semidirect
12. C2×#16864653C22semidirect
16. C2×#16864643C22semidirect
17. C2×#16864643C22semidirect
19. C2×#16864639C22semidirect
20. C2×#16864637C22semidirect
21. #22576160C22semidirect
24. #22576140C22semidirect
31. C4⋊(C271⋊C8)C22semidirect
32. #22576083C22semidirect
33. #22576080C22semidirect
34. (C2×D271)⋊C8C22semidirect
37. C4×D4×C271C22semidirect
38. #107×C271C22semidirect
40. #102×C271C22semidirect
41. C4×C8×C271C22semidirect
75. D8×C271D4semidirect
109. D4×C271C2×D4semidirect
118. C8×C271C2×D4semidirect
136. C4×C271AGL(1,ℤ/8)semidirect
148. C271#3272semidirect
159. C4×D8D271semidirect
181. C4×D4C2×D271semidirect
183. #107C2×D271semidirect
188. #102C2×D271semidirect
190. C4×C8C2×D271semidirect
215. D8D1084semidirect
252. D4C2×D1084semidirect
267. C8C2×D1084semidirect
287. C4#22576184semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
#317745512
D4×D10842
#317742702
#317741442
#317741022
#317740922
#317740492
#317740432
#317739762
C4×D8×C2712
#3272271