Finite Group Databasebeta

← #46847963

Subgroups — #46847963

Order 34688.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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#31774550C211yesmaximal, characteristic, complemented
3#31774532C211yesmaximal, characteristic, complemented
4#31774140C212yesmaximal, complemented
5#31774099C211yesmaximal, characteristic, complemented
6#31774089C211yesmaximal, characteristic, complemented
7#31774046C212yesmaximal, complemented
8#31774041C212yesmaximal, complemented
9C4×D2168C211yesmaximal, characteristic, complemented
10#31773976C211yesmaximal, characteristic, complemented
11#492×C271C211yesmaximal, characteristic, complemented
12C2×#16864655C2212yescomplemented
13C2×D4×D271—21no—
14C2×D4×D271C2212yes—
15#22576187—28no—
16C2×#16864641C2212yes—
17C2×D2168C2211yescharacteristic
18C2×#16864639C2212yescomplemented
19C2×#16864637C2212yescomplemented
20#22576145—22no—
21#22576136C2212yescomplemented
22#22576135—21no—
23#22576131—21no—
24#22576131C2212yes—
25#22576128—22no—
26C4×D1084C2211yescharacteristic
27C4×D1084C2211yescharacteristic
28C4×D1084C2211yescharacteristic, complemented
29C4⋊C4×D271—21no—
30#22576094C2211yescharacteristic, complemented
31C4⋊(C271⋊C8)C2211yescharacteristic, complemented
32#22576084C2212yescomplemented
33#22576082C2211yescharacteristic
34#22576082C2211yescharacteristic, complemented
35#22576081C2212yescomplemented
36(C2×D271)⋊C8C2212yescomplemented
37C2×SD16×C271C2212yescomplemented
38C2×SD16×C271C2212yescomplemented
39C4⋊Q8×C271C2211yescharacteristic, complemented
40#127×C271C2211yescharacteristic
41C4×C8×C271C2211yescharacteristic, complemented
42C23×D271—22no—
43#16864655—24no—
44#16864655—42no—
45C2×D1084—22no—
46C2×D1084D411yescharacteristic
47C2×D1084C2311yescharacteristic
48C2×D1084C2311yescharacteristic
49C2×C22⋊D271—22no—
50C2×C22⋊D271—22no—
51D4×D271—24no—
52D4×D271—42no—
53D4×D271—44no—
54C2×C4×D271—22no—
55C2×C4×D271—21no—
56C2×C4×D271—21no—
57C2×C4×D271C2312yes—
58#16864641—24no—
59D2168D414yescomplemented
60#16864639—24no—
61#16864637—24no—
62C2×C271⋊C8C2312yes—
63C4⋊Dic271D411yescharacteristic
64C4⋊Dic271C2311yescharacteristic
65C4⋊Dic271C2311yescharacteristic
66Dic271⋊C4—22no—
67C22⋊Dic271—22no—
68(C2×D271)⋊C4—22no—
69(C2×D271)⋊C4—21no—
70(C2×D271)⋊C4—21no—
71(C2×D271)⋊C4D412yes—
72C2×Q8×C271C2312yes—
73C2×D4×C271—21no—
74C2×D4×C271—21no—
75C2×D4×C271C2312yes—
76SD16×C271—24no—
77SD16×C271—42no—
78C2×C8×C271C2311yescharacteristic
79C2×C8×C271C2311yescharacteristic
80C4⋊C4×C271—21no—
81C4⋊C4×C271—21no—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—21no—
85C22×D271—44no—
86C22×D271—44no—
87C22×D271—82no—
88C22×D271C2×D412yes—
89D1084—22no—
90D1084—21no—
91D1084—21no—
92D1084—44no—
93D1084C2×D412yes—
94D1084C2×D412yescomplemented
95C22⋊D271—44no—
96C22⋊D271—44no—
97C2×Dic271—21no—
98C2×Dic271—21no—
99C2×Dic271C2×D412yes—
100C4×D271—22no—
101C4×D271—22no—
102C4×D271—44no—
103C4×D271—42no—
104C4×D271—42no—
105C271⋊C8—22no—
106C23×C271—22no—
107Q8×C271—22no—
108Q8×C271—22no—
109D4×C271—22no—
110D4×C271—22no—
111D4×C271—42no—
112D4×C271—81no—
113C2×C4×C271—22no—
114C2×C4×C271C2×D411yescharacteristic
115C2×C4×C271C2×D411yescharacteristic
116C2×C4×C271C2411yescharacteristic
117C8×C271—21no—
118C8×C271C2×D412yescomplemented
119C22×C271—42no—
120C22×C271—42no—
121C22×C271—42no—
122C22×C271C22×D411yescharacteristic
123C2×D271—22no—
124C2×D271—22no—
125C2×D271—22no—
126C2×D271—22no—
127C2×D271—22no—
128C2×D271—41no—
129C2×D271—41no—
130C2×D271—41no—
131C2×D271—41no—
132C2×D271—88no—
133C4×C271—21no—
134C4×C271—42no—
135C4×C271AGL(1,ℤ/8)12yescomplemented
136C4×C271C22×D411yescharacteristic
137C4×C271H(5,𝔽2)11yescharacteristic
138Dic271—22no—
139Dic271—41no—
140Dic271—41no—
141C2×C271—82no—
142C2×C271D4211yescharacteristic
143C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C271#57711yescharacteristic
145D271—42no—
146D271—42no—
147D271—42no—
148C271#327211yescharacteristic, complemented, Sylow
149#3272—2711nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2712no—
151#547—2711no—
152D42—2711no—
153#492D27111yescharacteristic, complemented
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—2712no—
163C22×D4—5421no—
164AGL(1,ℤ/8)—5428no—
165C2×SD16—2712no—
166C2×SD16C2×D27112yescomplemented
167C2×SD16C2×D27112yescomplemented
168C2×D8—2712no—
169C2×D8—2711no—
170C2×M16—2712no—
171C4⋊Q8C2×D27111yescharacteristic, complemented
172#127C2×D27111yescharacteristic
173#123—5422no—
174C22⋊Q8—2712no—
175#121—2712no—
176#121—5421no—
177#121—5421no—
178#120—5422no—
179C4×D4—2711no—
180C4×D4—2711no—
181C4×D4—2711no—
182C2×C4⋊C4—5421no—
183#107—2711no—
184C4⋊C8—2711no—
185#103—2712no—
186#102—2712no—
187#102—2711no—
188#102—2711no—
189C22⋊C8—2712no—
190C4×C8C2×D27111yescharacteristic, complemented
191C24—5422no—
192#41—5424no—
193#41—10842no—
194C2×Q8C22×D27112yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—2711no—
198C2×D4—2711no—
199C2×D4—2711no—
200C2×D4—5422no—
201C2×D4—5422no—
202C2×D4—5424no—
203C2×D4—5422no—
204C2×D4—10842no—
205C2×D4—10844no—
206C2×D4C22×D27112yes—
207C22×C4—2712no—
208C22×C4—5422no—
209C22×C4—5421no—
210C22×C4—5421no—
211SD16—24no—
212SD16—42no—
213SD16—5424no—
214D8—2714no—
215D8—5424no—
216M16—5424no—
217C2×C8—2712no—
218C2×C8C22×D27111yescharacteristic
219C2×C8C22×D27111yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—2711no—
223C4⋊C4—2711no—
224C4⋊C4—2711no—
225C4⋊C4—5422no—
226C22⋊C4—2712no—
227C22⋊C4—5422no—
228C22⋊C4—5422no—
229C22⋊C4—5421no—
230C22⋊C4—5421no—
231C42C22×D27111yescharacteristic
232C23—22no—
233C23—2712no—
234C23—5421no—
235C23—5421no—
236C23—10844no—
237C23—10844no—
238C23—21682no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—2712no—
246D4—2712no—
247D4—5422no—
248D4—5421no—
249D4—5421no—
250D4—10844no—
251D4—10844no—
252D4—10844no—
253C2×C4—22no—
254C2×C4—2712no—
255C2×C4—5422no—
256C2×C4—5422no—
257C2×C4—5421no—
258C2×C4—5421no—
259C2×C4—10844no—
260C2×C4—10842no—
261C2×C4—10842no—
262C2×C4D4×D27111yescharacteristic
263C2×C4D4×D27111yescharacteristic
264C2×C4C23×D27111yescharacteristic
265C8—21no—
266C8—5422no—
267C8D4×D27112yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—5422no—
272C22—5422no—
273C22—5422no—
274C22—5422no—
275C22—5422no—
276C22—10841no—
277C22—10841no—
278C22—10841no—
279C22—10841no—
280C22—21688no—
281C22C2×D4×D27111yescharacteristic
282C4—21no—
283C4—42no—
284C4—5422no—
285C4—10841no—
286C4—10841no—
287C4#2257618712yescomplemented
288C4C2×D4×D27111yescharacteristic
289C4#2257620511yescharacteristic
290C2—82no—
291C2—10842no—
292C2—10842no—
293C2—10842no—
294C2#3177453211yescharacteristic
295C2C2×#2257618711yescharacteristic
296C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#492×C271C2
socleC22×C271C22×D4
hypercenter#492D271
hypocenterC271#3272
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. #31774550C2semidirect
3. #31774532C2semidirect
4. #31774140C2semidirect
5. #31774099C2semidirect
6. #31774089C2semidirect
7. #31774046C2semidirect
8. #31774041C2semidirect
9. C4×D2168C2semidirect
10. #31773976C2semidirect
11. #492×C271C2semidirect
12. C2×#16864655C22semidirect
18. C2×#16864639C22semidirect
19. C2×#16864637C22semidirect
21. #22576136C22semidirect
28. C4×D1084C22semidirect
30. #22576094C22semidirect
31. C4⋊(C271⋊C8)C22semidirect
32. #22576084C22semidirect
34. #22576082C22semidirect
35. #22576081C22semidirect
36. (C2×D271)⋊C8C22semidirect
37. C2×SD16×C271C22semidirect
38. C2×SD16×C271C22semidirect
39. C4⋊Q8×C271C22semidirect
41. C4×C8×C271C22semidirect
59. D2168D4semidirect
94. D1084C2×D4semidirect
118. C8×C271C2×D4semidirect
135. C4×C271AGL(1,ℤ/8)semidirect
148. C271#3272semidirect
153. #492D271semidirect
166. C2×SD16C2×D271semidirect
167. C2×SD16C2×D271semidirect
171. C4⋊Q8C2×D271semidirect
190. C4×C8C2×D271semidirect
267. C8D4×D271semidirect
287. C4#22576187semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
#317745502
#317745322
#317741402
#317740992
#317740892
#317740462
#317740412
C4×D21682
#317739762
#492×C2712
#3272271