Finite Group Databasebeta

← #46847985

Subgroups — #46847985

Order 34688.

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

normal subgroups only

This group has 556 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×#22576187C212yesmaximal, complemented
2#31774569C211yesmaximal, characteristic, complemented
3D4×D1084C211yesmaximal, characteristic, complemented
4#31774274C211yesmaximal, characteristic, complemented
5#31774151C212yesmaximal, complemented
6#31774097C211yesmaximal, characteristic, complemented
7#31774092C211yesmaximal, characteristic, complemented
8#31774045C212yesmaximal, complemented
9#31774032C212yesmaximal, complemented
10#31773976C211yesmaximal, characteristic, complemented
11C4×SD16×C271C211yesmaximal, characteristic, complemented
12C2×#16864655C2212yescomplemented
13C22×D1084—21no—
14C2×D4×D271C2212yes—
15#22576187—28no—
16C2×#16864641C2212yescomplemented
17C2×D2168C2212yescomplemented
18C2×D2168C2212yescomplemented
19C2×#16864639C2212yes—
20C2×#16864637C2212yescomplemented
21#22576157—21no—
22#22576157C2211yescharacteristic
23#22576157C2211yescharacteristic, complemented
24#22576135—22no—
25#22576135C2212yes—
26#22576135C2212yescomplemented
27#22576134—21no—
28#22576130—22no—
29C4×D1084—21no—
30C4×D1084C2211yescharacteristic
31C4⋊(C271⋊C8)C2211yescharacteristic, complemented
32#22576080C2212yescomplemented
33#22576080C2212yescomplemented
34(C2×D271)⋊C8C2212yescomplemented
35C2×SD16×C271C2211yescharacteristic
36C4×Q8×C271C2211yescharacteristic, complemented
37C4×D4×C271C2211yescharacteristic
38#106×C271C2211yescharacteristic, complemented
39#103×C271C2211yescharacteristic, complemented
40#102×C271C2211yescharacteristic
41C4×C8×C271C2211yescharacteristic, complemented
42C23×D271—22no—
43#16864655—24no—
44#16864655—42no—
45C2×D1084—22no—
46C2×D1084—22no—
47C2×D1084—22no—
48C2×D1084—21no—
49C2×D1084—21no—
50C2×D1084—21no—
51C2×D1084—21no—
52C2×D1084—44no—
53C2×D1084D411yescharacteristic
54C2×D1084C2312yes—
55C2×D1084C2312yes—
56C2×C22⋊D271—22no—
57D4×D271—24no—
58D4×D271—42no—
59C2×C4×D271—22no—
60C2×C4×D271C2312yes—
61#16864641—24no—
62D2168—24no—
63D2168—42no—
64#16864639—24no—
65#16864637—24no—
66C2×C271⋊C8C2312yes—
67C4⋊Dic271D411yescharacteristic
68(C2×D271)⋊C4—22no—
69(C2×D271)⋊C4—22no—
70(C2×D271)⋊C4D412yes—
71C2×Q8×C271C2311yescharacteristic
72C2×D4×C271C2311yescharacteristic
73C22×C4×C271—21no—
74SD16×C271D414yescomplemented
75C2×C8×C271C2311yescharacteristic
76C2×C8×C271C2311yescharacteristic
77C4⋊C4×C271—21no—
78C4⋊C4×C271C2311yescharacteristic
79C4⋊C4×C271C2311yescharacteristic
80C22⋊C4×C271—21no—
81C42×C271—21no—
82C42×C271C2311yescharacteristic
83C22×D271—22no—
84C22×D271—22no—
85C22×D271—44no—
86C22×D271—44no—
87C22×D271—82no—
88C22×D271C2×D412yes—
89D1084—22no—
90D1084—22no—
91D1084—22no—
92D1084—22no—
93D1084—22no—
94D1084—44no—
95D1084—44no—
96D1084—42no—
97D1084—42no—
98D1084—81no—
99D1084—81no—
100D1084—82no—
101C22⋊D271—44no—
102C2×Dic271C2×D412yes—
103C4×D271—22no—
104C4×D271—22no—
105C4×D271—44no—
106C271⋊C8—22no—
107C23×C271—21no—
108Q8×C271—21no—
109Q8×C271C2×D412yescomplemented
110D4×C271—21no—
111D4×C271C2×D412yes—
112C2×C4×C271—21no—
113C2×C4×C271—21no—
114C2×C4×C271—21no—
115C2×C4×C271—42no—
116C2×C4×C271C2×D411yescharacteristic
117C2×C4×C271C2×D411yescharacteristic
118C2×C4×C271C2411yescharacteristic
119C8×C271—21no—
120C8×C271C2×D412yescomplemented
121C22×C271—22no—
122C22×C271—41no—
123C22×C271—41no—
124C22×C271C22×D411yescharacteristic
125C2×D271—22no—
126C2×D271—22no—
127C2×D271—22no—
128C2×D271—42no—
129C2×D271—42no—
130C2×D271—42no—
131C2×D271—42no—
132C2×D271—42no—
133C2×D271—42no—
134C2×D271—88no—
135C4×C271—22no—
136C4×C271—21no—
137C4×C271—41no—
138C4×C271—41no—
139C4×C271AGL(1,ℤ/8)12yescomplemented
140C4×C271C22×D411yescharacteristic
141C4×C271H(5,𝔽2)11yescharacteristic
142Dic271—22no—
143C2×C271—42no—
144C2×C271D4211yescharacteristic
145C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C271#57611yescharacteristic
147D271—42no—
148D271—82no—
149D271—82no—
150C271#327411yescharacteristic, complemented, Sylow
151#3274—2711nomaximal, Sylow
152C2×AGL(1,ℤ/8)—2712no—
153#550—2711no—
154D42—2711no—
155#493—2711no—
156#469—2712no—
157#460—2711no—
158#459—2711no—
159#449—2712no—
160#447—2712no—
161C4×SD16D27111yescharacteristic, complemented
162#436—2711no—
163C2×#41—2712no—
164C22×D4—2712no—
165C22×D4—5421no—
166AGL(1,ℤ/8)—5428no—
167C2×SD16—2712no—
168C2×SD16C2×D27111yescharacteristic
169C2×D8—2712no—
170C2×D8—2712no—
171C2×D8—2712no—
172C2×M16—2712no—
173#127—2711no—
174#127—2711no—
175#127—5421no—
176#121—2712no—
177#121—2712no—
178#121—5422no—
179#121—5421no—
180#120—5422no—
181C4×Q8C2×D27111yescharacteristic, complemented
182C4×D4—2711no—
183C4×D4—5421no—
184C4×D4C2×D27111yescharacteristic
185#106C2×D27111yescharacteristic, complemented
186C4⋊C8—2711no—
187#103C2×D27111yescharacteristic, complemented
188#102—2712no—
189#102—2712no—
190#102C2×D27111yescharacteristic
191C22⋊C8—2712no—
192C4×C8C2×D27111yescharacteristic, complemented
193C24—5422no—
194#41—5424no—
195#41—10842no—
196C2×Q8C22×D27111yescharacteristic
197C2×D4—2711no—
198C2×D4—2712no—
199C2×D4—2712no—
200C2×D4—5422no—
201C2×D4—5422no—
202C2×D4—5422no—
203C2×D4—5422no—
204C2×D4—5424no—
205C2×D4—5421no—
206C2×D4—5421no—
207C2×D4—5421no—
208C2×D4—5421no—
209C2×D4—10844no—
210C2×D4—10842no—
211C2×D4C22×D27111yescharacteristic
212C22×C4—21no—
213C22×C4—2712no—
214C22×C4—5422no—
215SD16—5424no—
216SD16D108414yescomplemented
217D8—5424no—
218D8—5424no—
219D8—10842no—
220M16—5424no—
221C2×C8—2712no—
222C2×C8C22×D27111yescharacteristic
223C2×C8C22×D27111yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—2711no—
226C4⋊C4C22×D27111yescharacteristic
227C4⋊C4C22×D27111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2712no—
230C22⋊C4—5422no—
231C22⋊C4—5422no—
232C42—21no—
233C42C22×D27111yescharacteristic
234C23—21no—
235C23—2712no—
236C23—5422no—
237C23—5422no—
238C23—10844no—
239C23—10844no—
240C23—21682no—
241Q8—21no—
242Q8C2×D108412yescomplemented
243D4—21no—
244D4—5422no—
245D4—5422no—
246D4—5422no—
247D4—5422no—
248D4—5422no—
249D4—10844no—
250D4—10844no—
251D4—10844no—
252D4—10842no—
253D4—10842no—
254D4—21681no—
255D4—21681no—
256D4—21682no—
257D4C2×D108412yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—2712no—
263C2×C4—5422no—
264C2×C4—5422no—
265C2×C4—10844no—
266C2×C4D4×D27111yescharacteristic
267C2×C4D4×D27111yescharacteristic
268C2×C4C23×D27111yescharacteristic
269C8—21no—
270C8—5422no—
271C8C2×D108412yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—5422no—
276C22—5422no—
277C22—5422no—
278C22—10842no—
279C22—10842no—
280C22—10842no—
281C22—10842no—
282C22—10842no—
283C22—10842no—
284C22—21688no—
285C22C2×D4×D27111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—5422no—
291C4#2257618712yescomplemented
292C4C22×D108411yescharacteristic
293C4#2257620611yescharacteristic
294C2—42no—
295C2—10842no—
296C2—21682no—
297C2—21682no—
298C2D4×D108411yescharacteristic
299C2C2×#2257618711yescharacteristic
300C2#3177469911yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC4×SD16×C271C2
socleC22×C271C22×D4
hypercenterC4×SD16D271
hypocenterC271#3274
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. #31774569C2semidirect
3. D4×D1084C2semidirect
4. #31774274C2semidirect
5. #31774151C2semidirect
6. #31774097C2semidirect
7. #31774092C2semidirect
8. #31774045C2semidirect
9. #31774032C2semidirect
10. #31773976C2semidirect
11. C4×SD16×C271C2semidirect
12. C2×#16864655C22semidirect
16. C2×#16864641C22semidirect
17. C2×D2168C22semidirect
18. C2×D2168C22semidirect
20. C2×#16864637C22semidirect
23. #22576157C22semidirect
26. #22576135C22semidirect
31. C4⋊(C271⋊C8)C22semidirect
32. #22576080C22semidirect
33. #22576080C22semidirect
34. (C2×D271)⋊C8C22semidirect
36. C4×Q8×C271C22semidirect
38. #106×C271C22semidirect
39. #103×C271C22semidirect
41. C4×C8×C271C22semidirect
74. SD16×C271D4semidirect
109. Q8×C271C2×D4semidirect
120. C8×C271C2×D4semidirect
139. C4×C271AGL(1,ℤ/8)semidirect
150. C271#3274semidirect
161. C4×SD16D271semidirect
181. C4×Q8C2×D271semidirect
185. #106C2×D271semidirect
187. #103C2×D271semidirect
192. C4×C8C2×D271semidirect
216. SD16D1084semidirect
242. Q8C2×D1084semidirect
271. C8C2×D1084semidirect
291. C4#22576187semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
#317745692
D4×D10842
#317742742
#317741512
#317740972
#317740922
#317740452
#317740322
#317739762
C4×SD16×C2712
#3274271