Finite Group Databasebeta

← #46847984

Subgroups — #46847984

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×#22576185C212yesmaximal, complemented
2D4×D1084C211yesmaximal, characteristic, complemented
3#31774273C211yesmaximal, characteristic, complemented
4#31774148C212yesmaximal, complemented
5#31774097C211yesmaximal, characteristic, complemented
6#31774089C211yesmaximal, characteristic, complemented
7#31774042C212yesmaximal, complemented
8#31774030C212yesmaximal, complemented
9C4×#16864641C211yesmaximal, characteristic, complemented
10#31773975C211yesmaximal, characteristic, complemented
11#550×C271C211yesmaximal, characteristic, complemented
12C22×D1084C2212yes—
13C2×D4×D271—21no—
14#22576185—28no—
15C2×#16864641C2211yescharacteristic
16C2×#16864641C2212yescomplemented
17C2×#16864639C2212yes—
18C2×#16864639C2212yescomplemented
19C2×#16864639C2212yescomplemented
20C2×#16864638C2212yescomplemented
21#22576157C2211yescharacteristic
22#22576135—21no—
23#22576134C2212yes—
24#22576130—22no—
25C4×D1084C2211yescharacteristic
26#22576091C2211yescharacteristic, complemented
27C4⋊(C271⋊C8)C2211yescharacteristic, complemented
28#22576084C2211yescharacteristic, complemented
29#22576081C2212yescomplemented
30#22576080C2211yescharacteristic
31#22576080C2212yescomplemented
32C22⋊(C271⋊C8)C2212yescomplemented
33C4×C271⋊C8C2211yescharacteristic, complemented
34C2×#41×C271C2212yescomplemented
35#127×C271—21no—
36#127×C271C2211yescharacteristic, complemented
37#121×C271—22no—
38#121×C271C2212yescomplemented
39C4×Q8×C271C2211yescharacteristic, complemented
40C4×D4×C271—21no—
41C4×D4×C271C2211yescharacteristic
42C23×D271—22no—
43C2×D1084—24no—
44C2×D1084—22no—
45C2×D1084—21no—
46C2×D1084—21no—
47C2×D1084—42no—
48C2×D1084C2312yes—
49C2×D1084C2311yescharacteristic
50C2×C22⋊D271—22no—
51D4×D271—44no—
52C2×C4×D271—21no—
53#16864641—24no—
54#16864641D414yescomplemented
55#16864639—24no—
56#16864639—24no—
57#16864639—42no—
58#16864638—24no—
59C2×C271⋊C8C2312yes—
60C2×C271⋊C8C2311yescharacteristic
61C2×C271⋊C8C2311yescharacteristic
62C4⋊Dic271C2311yescharacteristic
63(C2×D271)⋊C4—21no—
64(C2×D271)⋊C4—22no—
65#41×C271—24no—
66#41×C271—42no—
67C2×Q8×C271C2311yescharacteristic
68C2×D4×C271—22no—
69C2×D4×C271—22no—
70C2×D4×C271—21no—
71C2×D4×C271—21no—
72C2×D4×C271D411yescharacteristic
73C2×D4×C271C2312yes—
74C22×C4×C271—22no—
75C22×C4×C271C2312yes—
76C4⋊C4×C271—21no—
77C4⋊C4×C271D411yescharacteristic
78C4⋊C4×C271C2311yescharacteristic
79C22⋊C4×C271—22no—
80C22⋊C4×C271D412yes—
81C42×C271—21no—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—22no—
85C22×D271—44no—
86C22×D271—44no—
87C22×D271—82no—
88D1084—22no—
89D1084—21no—
90D1084—22no—
91D1084—44no—
92D1084—42no—
93D1084—81no—
94D1084C2×D412yes—
95C22⋊D271—44no—
96C2×Dic271—21no—
97C4×D271—42no—
98C271⋊C8—22no—
99C271⋊C8—21no—
100C271⋊C8C2×D412yescomplemented
101C23×C271—22no—
102C23×C271C2×D412yes—
103Q8×C271—21no—
104Q8×C271C2×D412yescomplemented
105D4×C271—22no—
106D4×C271—22no—
107D4×C271—22no—
108D4×C271—44no—
109D4×C271—42no—
110D4×C271—81no—
111D4×C271—82no—
112C2×C4×C271—21no—
113C2×C4×C271—22no—
114C2×C4×C271—22no—
115C2×C4×C271—21no—
116C2×C4×C271—44no—
117C2×C4×C271C2×D411yescharacteristic
118C2×C4×C271C2×D412yes—
119C2×C4×C271C2×D411yescharacteristic
120C2×C4×C271C2411yescharacteristic
121C22×C271—22no—
122C22×C271—22no—
123C22×C271—22no—
124C22×C271—42no—
125C22×C271—42no—
126C22×C271—42no—
127C22×C271C22×D411yescharacteristic
128C2×D271—22no—
129C2×D271—41no—
130C2×D271—41no—
131C2×D271—42no—
132C2×D271—42no—
133C2×D271—42no—
134C2×D271—88no—
135C4×C271—22no—
136C4×C271—22no—
137C4×C271—21no—
138C4×C271—41no—
139C4×C271AGL(1,ℤ/8)12yescomplemented
140C4×C271C22×D411yescharacteristic
141C4×C271H(5,𝔽2)11yescharacteristic
142Dic271—41no—
143C2×C271—42no—
144C2×C271—82no—
145C2×C271D4211yescharacteristic
146C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C271#57611yescharacteristic
148D271—42no—
149D271—82no—
150C271#327411yescharacteristic, complemented, Sylow
151#3274—2711nomaximal, Sylow
152C2×AGL(1,ℤ/8)—2712no—
153#550D27111yescharacteristic, complemented
154D42—2711no—
155#493—2711no—
156#469—2712no—
157#460—2711no—
158#459—2711no—
159#449—2712no—
160#447—2712no—
161C4×SD16—2711no—
162#436—2711no—
163C2×#41C2×D27112yescomplemented
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×M16—2712no—
173#127—21no—
174#127—2711no—
175#127C2×D27111yescharacteristic, complemented
176#121—22no—
177#121—2712no—
178#121—5421no—
179#121C2×D27112yescomplemented
180#120—5422no—
181C4×Q8C2×D27111yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—2711no—
184C4×D4C2×D27111yescharacteristic
185#106—2711no—
186C4⋊C8—2711no—
187#103—2711no—
188#102—2712no—
189#102—2711no—
190#102—2712no—
191C22⋊C8—2712no—
192C4×C8—2711no—
193C24—5422no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D27111yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2711no—
202C2×D4—2712no—
203C2×D4—5424no—
204C2×D4—5422no—
205C2×D4—5422no—
206C2×D4—5421no—
207C2×D4—5421no—
208C2×D4—10842no—
209C2×D4—10844no—
210C2×D4C22⋊D27111yescharacteristic
211C2×D4C22×D27112yes—
212C22×C4—22no—
213C22×C4—5421no—
214C22×C4C22×D27112yes—
215SD16—2714no—
216SD16—5424no—
217D8—5424no—
218D8—5424no—
219D8—10842no—
220M16—5424no—
221C2×C8—2711no—
222C2×C8—2711no—
223C2×C8—2712no—
224C4⋊C4—21no—
225C4⋊C4—2711no—
226C4⋊C4C22⋊D27111yescharacteristic
227C4⋊C4C22×D27111yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—5421no—
230C22⋊C4—5422no—
231C22⋊C4C22⋊D27112yes—
232C42—21no—
233C42C22×D27111yescharacteristic
234C23—22no—
235C23—5421no—
236C23—5422no—
237C23—10844no—
238C23—10844no—
239C23—21682no—
240C23C2×C22⋊D27112yes—
241Q8—21no—
242Q8D4×D27112yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—2712no—
251D4—5422no—
252D4—5421no—
253D4—5422no—
254D4—10844no—
255D4—10844no—
256D4—10842no—
257D4—21681no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—5421no—
264C2×C4—10842no—
265C2×C4C2×C22⋊D27111yescharacteristic
266C2×C4C2×C22⋊D27112yes—
267C2×C4C2×C22⋊D27111yescharacteristic
268C2×C4C23×D27111yescharacteristic
269C8—2712no—
270C8—5422no—
271C8—5421no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—5422no—
279C22—10841no—
280C22—10841no—
281C22—10842no—
282C22—10842no—
283C22—10842no—
284C22—21688no—
285C22C22×C22⋊D27111yescharacteristic
286C4—22no—
287C4—22no—
288C4—21no—
289C4—41no—
290C4—10841no—
291C4#2257618512yescomplemented
292C4C2×D4×D27111yescharacteristic
293C4#2257620511yescharacteristic
294C2—42no—
295C2—82no—
296C2—10842no—
297C2—21682no—
298C2D4×C22⋊D27111yescharacteristic
299C2C2×#2257618511yescharacteristic
300C2#3177470011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#550×C271C2
socleC22×C271C22×D4
hypercenter#550D271
hypocenterC271#3274
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#22576185C2semidirect
2. D4×D1084C2semidirect
3. #31774273C2semidirect
4. #31774148C2semidirect
5. #31774097C2semidirect
6. #31774089C2semidirect
7. #31774042C2semidirect
8. #31774030C2semidirect
9. C4×#16864641C2semidirect
10. #31773975C2semidirect
11. #550×C271C2semidirect
16. C2×#16864641C22semidirect
18. C2×#16864639C22semidirect
19. C2×#16864639C22semidirect
20. C2×#16864638C22semidirect
26. #22576091C22semidirect
27. C4⋊(C271⋊C8)C22semidirect
28. #22576084C22semidirect
29. #22576081C22semidirect
31. #22576080C22semidirect
32. C22⋊(C271⋊C8)C22semidirect
33. C4×C271⋊C8C22semidirect
34. C2×#41×C271C22semidirect
36. #127×C271C22semidirect
38. #121×C271C22semidirect
39. C4×Q8×C271C22semidirect
54. #16864641D4semidirect
100. C271⋊C8C2×D4semidirect
104. Q8×C271C2×D4semidirect
139. C4×C271AGL(1,ℤ/8)semidirect
150. C271#3274semidirect
153. #550D271semidirect
163. C2×#41C2×D271semidirect
175. #127C2×D271semidirect
179. #121C2×D271semidirect
181. C4×Q8C2×D271semidirect
242. Q8D4×D271semidirect
291. C4#22576185semidirect

Maximal subgroups

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

subgroupindex
C2×#225761852
D4×D10842
#317742732
#317741482
#317740972
#317740892
#317740422
#317740302
C4×#168646412
#317739752
#550×C2712
#3274271