Finite Group Databasebeta

← #46847986

Subgroups — #46847986

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×#22576188C212yesmaximal, complemented
2#31774568C211yesmaximal, characteristic, complemented
3D4×D1084C211yesmaximal, characteristic, complemented
4#31774274C211yesmaximal, characteristic, complemented
5#31774150C212yesmaximal, complemented
6#31774098C211yesmaximal, characteristic, complemented
7#31774093C211yesmaximal, characteristic, complemented
8#31774044C212yesmaximal, complemented
9#31774033C212yesmaximal, complemented
10C4×#16864643C211yesmaximal, characteristic, complemented
11#436×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2212yescomplemented
13C22×D1084C2212yes—
14C2×D4×D271—21no—
15#22576188—28no—
16C2×#16864643C2211yescharacteristic
17C2×#16864643C2212yescomplemented
18C2×D2168C2212yes—
19C2×D2168C2212yescomplemented
20C2×D2168C2212yescomplemented
21#22576157C2211yescharacteristic
22#22576157C2211yescharacteristic, complemented
23#22576156—21no—
24#22576135—21no—
25#22576134C2212yes—
26#22576134C2212yescomplemented
27#22576133—22no—
28#22576130—22no—
29C4×Dic542C2211yescharacteristic, complemented
30C4×D1084C2211yescharacteristic
31#22576122—21no—
32#22576092C2211yescharacteristic, complemented
33#22576085C2211yescharacteristic, complemented
34#22576082C2211yescharacteristic
35#22576082C2212yescomplemented
36#22576082C2212yescomplemented
37C2×M16×C271C2212yescomplemented
38C4×D4×C271C2211yescharacteristic
39C4⋊C8×C271C2211yescharacteristic, complemented
40C22⋊C8×C271C2212yescomplemented
41C4×C8×C271C2211yescharacteristic, complemented
42C23×D271—22no—
43#16864654—24no—
44#16864654—42no—
45C2×Dic542C2311yescharacteristic
46C2×D1084—24no—
47C2×D1084—22no—
48C2×D1084—21no—
49C2×D1084—21no—
50C2×D1084—21no—
51C2×D1084—21no—
52C2×D1084—42no—
53C2×D1084C2312yes—
54C2×D1084C2312yes—
55C2×D1084C2311yescharacteristic
56C2×C22⋊D271—22no—
57C2×C22⋊D271—22no—
58C2×C22⋊D271—22no—
59D4×D271—44no—
60C2×C4×D271—22no—
61C2×C4×D271—21no—
62#16864643—24no—
63#16864643D414yescomplemented
64D2168—24no—
65D2168—24no—
66D2168—42no—
67C4⋊Dic271C2311yescharacteristic
68C4⋊Dic271C2311yescharacteristic
69Dic271⋊C4—21no—
70(C2×D271)⋊C4—21no—
71(C2×D271)⋊C4—22no—
72(C2×D271)⋊C4—22no—
73C4×Dic271—21no—
74C2×D4×C271D411yescharacteristic
75C22×C4×C271C2312yes—
76M16×C271—24no—
77C2×C8×C271C2312yes—
78C2×C8×C271C2311yescharacteristic
79C2×C8×C271C2311yescharacteristic
80C4⋊C4×C271D411yescharacteristic
81C22⋊C4×C271D412yes—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—22no—
85C22×D271—22no—
86C22×D271—44no—
87C22×D271—44no—
88C22×D271—82no—
89Dic542—21no—
90Dic542C2×D412yescomplemented
91D1084—22no—
92D1084—22no—
93D1084—21no—
94D1084—22no—
95D1084—22no—
96D1084—44no—
97D1084—42no—
98D1084—42no—
99D1084—81no—
100D1084—81no—
101D1084C2×D412yes—
102C22⋊D271—44no—
103C22⋊D271—44no—
104C22⋊D271—82no—
105C2×Dic271—21no—
106C2×Dic271—21no—
107C2×Dic271—21no—
108C4×D271—44no—
109C4×D271—42no—
110C23×C271C2×D412yes—
111D4×C271—22no—
112C2×C4×C271—22no—
113C2×C4×C271—22no—
114C2×C4×C271C2×D411yescharacteristic
115C2×C4×C271C2×D412yes—
116C2×C4×C271C2×D411yescharacteristic
117C2×C4×C271C2411yescharacteristic
118C8×C271—22no—
119C8×C271—21no—
120C8×C271C2×D412yescomplemented
121C22×C271—22no—
122C22×C271—22no—
123C22×C271—22no—
124C22×C271C22×D411yescharacteristic
125C2×D271—22no—
126C2×D271—41no—
127C2×D271—41no—
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×C271AGL(1,ℤ/8)12yescomplemented
138C4×C271C22×D411yescharacteristic
139C4×C271H(5,𝔽2)11yescharacteristic
140Dic271—22no—
141Dic271—41no—
142Dic271—41no—
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×SD16—2711no—
162#436D27111yescharacteristic, complemented
163C2×#41—2712no—
164C22×D4—2712no—
165C22×D4—5421no—
166AGL(1,ℤ/8)—5428no—
167C2×SD16—2711no—
168C2×SD16—2712no—
169C2×D8—2712no—
170C2×D8—2712no—
171C2×D8—2712no—
172C2×M16C2×D27112yescomplemented
173#127—2711no—
174#127—2711no—
175#127—5421no—
176#121—2712no—
177#121—2712no—
178#121—5421no—
179#121—5422no—
180#120—5422no—
181C4×Q8—2711no—
182C4×D4—2711no—
183C4×D4—5421no—
184C4×D4C2×D27111yescharacteristic
185#106—2711no—
186C4⋊C8C2×D27111yescharacteristic, complemented
187#103—2711no—
188#102—2711no—
189#102—2712no—
190#102—2712no—
191C22⋊C8C2×D27112yescomplemented
192C4×C8C2×D27111yescharacteristic, complemented
193C24—5422no—
194#41—5424no—
195#41—10842no—
196C2×Q8—2711no—
197C2×D4—2711no—
198C2×D4—2712no—
199C2×D4—2712no—
200C2×D4—5424no—
201C2×D4—5422no—
202C2×D4—5422no—
203C2×D4—5422no—
204C2×D4—5421no—
205C2×D4—5422no—
206C2×D4—5421no—
207C2×D4—5421no—
208C2×D4—5421no—
209C2×D4—10842no—
210C2×D4—10844no—
211C2×D4D108411yescharacteristic
212C22×C4—5421no—
213C22×C4—5422no—
214C22×C4C22×D27112yes—
215SD16—2714no—
216SD16—5424no—
217D8—5424no—
218D8—5424no—
219D8—10842no—
220M16—24no—
221C2×C8C22×D27111yescharacteristic
222C2×C8C22×D27112yes—
223C2×C8C22×D27111yescharacteristic
224C4⋊C4—2711no—
225C4⋊C4—2711no—
226C4⋊C4—5421no—
227C4⋊C4D108411yescharacteristic
228C22⋊C4—5421no—
229C22⋊C4—5422no—
230C22⋊C4—5422no—
231C22⋊C4D108412yes—
232C42—5421no—
233C42C22×D27111yescharacteristic
234C23—5421no—
235C23—5422no—
236C23—5422no—
237C23—10844no—
238C23—10844no—
239C23—21682no—
240C23C2×D108412yes—
241Q8—2712no—
242Q8—5421no—
243D4—22no—
244D4—2712no—
245D4—5422no—
246D4—5422no—
247D4—5421no—
248D4—5422no—
249D4—5422no—
250D4—10844no—
251D4—10844no—
252D4—10844no—
253D4—10842no—
254D4—10842no—
255D4—21681no—
256D4—21681no—
257D4—21682no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—5421no—
261C2×C4—5421no—
262C2×C4—5421no—
263C2×C4—10844no—
264C2×C4—10842no—
265C2×C4C2×D108411yescharacteristic
266C2×C4C2×D108412yes—
267C2×C4C2×D108411yescharacteristic
268C2×C4C23×D27111yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D27112yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—5422no—
276C22—10841no—
277C22—10841no—
278C22—10842no—
279C22—10842no—
280C22—10842no—
281C22—10842no—
282C22—10842no—
283C22—10842no—
284C22—21688no—
285C22C22×D108411yescharacteristic
286C4—22no—
287C4—21no—
288C4—5422no—
289C4—10841no—
290C4—10841no—
291C4#2257618812yescomplemented
292C4C2×D4×D27111yescharacteristic
293C4#2257620611yescharacteristic
294C2—42no—
295C2—10842no—
296C2—21682no—
297C2—21682no—
298C2D4×D108411yescharacteristic
299C2C2×#2257618811yescharacteristic
300C2#3177470211yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#436×C271C2
socleC22×C271C22×D4
hypercenter#436D271
hypocenterC271#3274
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. #31774568C2semidirect
3. D4×D1084C2semidirect
4. #31774274C2semidirect
5. #31774150C2semidirect
6. #31774098C2semidirect
7. #31774093C2semidirect
8. #31774044C2semidirect
9. #31774033C2semidirect
10. C4×#16864643C2semidirect
11. #436×C271C2semidirect
12. C2×#16864654C22semidirect
17. C2×#16864643C22semidirect
19. C2×D2168C22semidirect
20. C2×D2168C22semidirect
22. #22576157C22semidirect
26. #22576134C22semidirect
29. C4×Dic542C22semidirect
32. #22576092C22semidirect
33. #22576085C22semidirect
35. #22576082C22semidirect
36. #22576082C22semidirect
37. C2×M16×C271C22semidirect
39. C4⋊C8×C271C22semidirect
40. C22⋊C8×C271C22semidirect
41. C4×C8×C271C22semidirect
63. #16864643D4semidirect
90. Dic542C2×D4semidirect
120. C8×C271C2×D4semidirect
137. C4×C271AGL(1,ℤ/8)semidirect
150. C271#3274semidirect
162. #436D271semidirect
172. C2×M16C2×D271semidirect
186. C4⋊C8C2×D271semidirect
191. C22⋊C8C2×D271semidirect
192. C4×C8C2×D271semidirect
271. C8D4×D271semidirect
291. C4#22576188semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
#317745682
D4×D10842
#317742742
#317741502
#317740982
#317740932
#317740442
#317740332
C4×#168646432
#436×C2712
#3274271