Finite Group Databasebeta

← #46848162

Subgroups — #46848162

Order 34688.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#31774702C211yesmaximal, characteristic, complemented
2#31774701C212yesmaximal, complemented
3#31774698C211yesmaximal, characteristic, complemented
4#31774272C211yesmaximal, characteristic, complemented
5#31774168C211yesmaximal, characteristic, complemented
6#31774166C212yesmaximal, complemented
7#31774067C212yesmaximal, complemented
8#31774063C212yesmaximal, complemented
9#31774013C211yesmaximal, characteristic, complemented
10#31774012C211yesmaximal, characteristic, complemented
11#472×C271C211yesmaximal, characteristic, complemented
12#22576206C2212yescomplemented
13#22576205C2212yescomplemented
14#22576188—22no—
15#22576188C2212yescomplemented
16#22576188C2212yescomplemented
17#22576187—22no—
18#22576186C2212yescomplemented
19#22576184—22no—
20#22576184—22no—
21#22576182—21no—
22#22576182C2211yescharacteristic
23#22576181—22no—
24#22576180C2211yescharacteristic
25#22576179—21no—
26C2×D2168—21no—
27C2×D2168C2211yescharacteristic, complemented
28C2×#16864639C2212yescomplemented
29D8×D271—21no—
30D8×D271—22no—
31#22576168C2212yescomplemented
32#22576166C2211yescharacteristic, complemented
33#22576156C2212yescomplemented
34#22576095C2211yescharacteristic, complemented
35#22576087C2211yescharacteristic, complemented
36#22576086C2211yescharacteristic, complemented
37#22576086C2212yescomplemented
38#22576076C2212yescomplemented
39C8×Dic271C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C271C2212yescomplemented
41C2×D8×C271C2211yescharacteristic, complemented
42#131×C271C2211yescharacteristic, complemented
43#108×C271C2211yescharacteristic, complemented
44#100×C271C2212yescomplemented
45#16864655—22no—
46#16864654—22no—
47#16864654C2311yescharacteristic
48#16864654C2312yes—
49#16864653—22no—
50#16864653—22no—
51C2×D1084—22no—
52C2×D1084C2312yes—
53C2×C22⋊D271—42no—
54C2×C22⋊D271—42no—
55D4×D271—22no—
56D4×D271—22no—
57D4×D271—22no—
58D4×D271—42no—
59Dic1084D411yescharacteristic, complemented
60#16864644D411yescharacteristic, complemented
61#16864643—22no—
62#16864643—22no—
63#16864643D412yescomplemented
64#16864642—22no—
65#16864642D411yescharacteristic, complemented
66#16864641D411yescharacteristic, complemented
67D2168—22no—
68D2168—22no—
69D2168—21no—
70D2168—21no—
71D2168D411yescharacteristic, complemented
72#16864639—22no—
73#16864639—22no—
74#16864639—22no—
75#16864639D411yescharacteristic, complemented
76#16864638C2311yescharacteristic
77#16864637—22no—
78#16864637—21no—
79C2×C271⋊C8C2311yescharacteristic
80C8×D271—22no—
81C8×D271—21no—
82C4×Dic271C2311yescharacteristic
83#41×C271C2311yescharacteristic
84C2×D4×C271C2312yes—
85SD16×C271—22no—
86D8×C271—22no—
87D8×C271—21no—
88D8×C271—21no—
89M16×C271—21no—
90M16×C271C2312yes—
91M16×C271C2311yescharacteristic
92C2×C8×C271—21no—
93C2×C8×C271C2311yescharacteristic
94C22×D271—22no—
95C22×D271—42no—
96C22×D271—42no—
97Dic542C2×D412yes—
98Dic542C2×D411yescharacteristic
99D1084—22no—
100D1084—22no—
101D1084—42no—
102D1084C2×D412yes—
103D1084C2×D411yescharacteristic
104C22⋊D271—22no—
105C22⋊D271—21no—
106C22⋊D271—42no—
107C22⋊D271—42no—
108C22⋊D271—42no—
109C22⋊D271—84no—
110C2×Dic271—21no—
111C2×Dic271—42no—
112C4×D271—21no—
113C4×D271—22no—
114C4×D271—42no—
115C271⋊C8—21no—
116C271⋊C8C2×D411yescharacteristic
117C271⋊C8C2×D411yescharacteristic
118C23×C271—22no—
119Q8×C271C2×D411yescharacteristic
120D4×C271—21no—
121D4×C271—22no—
122D4×C271—22no—
123D4×C271C2×D411yescharacteristic
124C2×C4×C271—21no—
125C2×C4×C271C2411yescharacteristic
126C8×C271—21no—
127C8×C271—22no—
128C8×C271C2×D411yescharacteristic
129C8×C271C2×D411yescharacteristic
130C22×C271—21no—
131C22×C271—42no—
132C22×C271—82no—
133C22×C271H(5,𝔽2)11yescharacteristic
134C2×D271—21no—
135C2×D271—22no—
136C2×D271—42no—
137C2×D271—82no—
138C2×D271—162no—
139C2×D271—162no—
140C4×C271—21no—
141C4×C271C22×D411yescharacteristic
142C4×C271C22×D411yescharacteristic
143Dic271—22no—
144Dic271—22no—
145Dic271—21no—
146C2×C271—21no—
147C2×C271—41no—
148C2×C271—82no—
149C2×C271D4211yescharacteristic
150D271—41no—
151D271—42no—
152D271—82no—
153C271#329211yescharacteristic, complemented, Sylow
154#3292—2711nomaximal, Sylow
155#576—2712no—
156#576—2711no—
157#576—2711no—
158#493—2711no—
159#472—2711no—
160#472—2712no—
161#472D27111yescharacteristic, complemented
162#453—2712no—
163#453—2712no—
164#443—2711no—
165#443—2711no—
166H(5,𝔽2)—2712no—
167H(5,𝔽2)—2712no—
168AGL(1,ℤ/8)—2712no—
169AGL(1,ℤ/8)—2712no—
170AGL(1,ℤ/8)—2712no—
171AGL(1,ℤ/8)—5422no—
172AGL(1,ℤ/8)—5422no—
173AGL(1,ℤ/8)—5422no—
174AGL(1,ℤ/8)—5422no—
175AGL(1,ℤ/8)C2×D27112yescomplemented
176#135—2711no—
177#135—2711no—
178#135—5421no—
179#135—5421no—
180#135—5422no—
181C2×D8—2712no—
182C2×D8—2711no—
183C2×D8—5421no—
184C2×D8—5422no—
185C2×D8—5421no—
186C2×D8C2×D27111yescharacteristic, complemented
187#131—2712no—
188#131—2711no—
189#131C2×D27111yescharacteristic, complemented
190#127—2712no—
191#108—2711no—
192#108C2×D27111yescharacteristic, complemented
193#104—2711no—
194#104—2712no—
195#104—2711no—
196#100—2712no—
197#100C2×D27112yescomplemented
198C4×C8—2711no—
199#41—2712no—
200#41—2711no—
201#41—5422no—
202#41—5422no—
203#41—5422no—
204#41—5422no—
205#41C22×D27111yescharacteristic
206C2×D4—2712no—
207C2×D4—5422no—
208C2×D4—5422no—
209C2×D4—5422no—
210C2×D4—5422no—
211C2×D4—10842no—
212C2×D4—10842no—
213C2×D4—10842no—
214C2×D4C22×D27112yes—
215Q16—2711no—
216Q16—2711no—
217SD16—22no—
218SD16—2711no—
219SD16—2711no—
220SD16—2712no—
221SD16—5422no—
222SD16—5422no—
223SD16—5422no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2711no—
228D8—2711no—
229D8—5422no—
230D8—5422no—
231D8—5422no—
232D8—5422no—
233D8—5422no—
234D8—5421no—
235D8—5421no—
236M16—21no—
237M16—2711no—
238M16—5422no—
239M16—5421no—
240M16C22×D27112yes—
241M16C22×D27111yescharacteristic
242C2×C8—21no—
243C2×C8—2711no—
244C2×C8—5422no—
245C2×C8—5421no—
246C2×C8C22×D27111yescharacteristic
247C42—2711no—
248C23—22no—
249C23—5422no—
250C23—10842no—
251C23—10842no—
252Q8—2711no—
253Q8—2712no—
254Q8D4×D27111yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—2711no—
259D4—2712no—
260D4—5422no—
261D4—5421no—
262D4—5422no—
263D4—5422no—
264D4—10842no—
265D4—10842no—
266D4—10842no—
267D4—10842no—
268D4—21684no—
269D4D4×D27111yescharacteristic
270C2×C4—21no—
271C2×C4—5421no—
272C2×C4—5422no—
273C2×C4—5421no—
274C2×C4—10842no—
275C2×C4—10842no—
276C2×C4C23×D27111yescharacteristic
277C8—21no—
278C8—22no—
279C8—2711no—
280C8—2711no—
281C8—5421no—
282C8D4×D27111yescharacteristic
283C8D4×D27111yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—5422no—
288C22—5421no—
289C22—10842no—
290C22—21682no—
291C22—43362no—
292C22—43362no—
293C22#2257620611yescharacteristic
294C4—21no—
295C4—5422no—
296C4—5422no—
297C4—5421no—
298C4C2×D4×D27111yescharacteristic
299C4C2×D4×D27111yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—10842no—
304C2—10841no—
305C2—21682no—
306C2#3177453111yescharacteristic
subgroupG/H
centerC2#31774531
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#472×C271C2
socleC2×C271D42
hypercenter#472D271
hypocenterC271#3292
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #31774702C2semidirect
2. #31774701C2semidirect
3. #31774698C2semidirect
4. #31774272C2semidirect
5. #31774168C2semidirect
6. #31774166C2semidirect
7. #31774067C2semidirect
8. #31774063C2semidirect
9. #31774013C2semidirect
10. #31774012C2semidirect
11. #472×C271C2semidirect
12. #22576206C22semidirect
13. #22576205C22semidirect
15. #22576188C22semidirect
16. #22576188C22semidirect
18. #22576186C22semidirect
27. C2×D2168C22semidirect
28. C2×#16864639C22semidirect
31. #22576168C22semidirect
32. #22576166C22semidirect
33. #22576156C22semidirect
34. #22576095C22semidirect
35. #22576087C22semidirect
36. #22576086C22semidirect
37. #22576086C22semidirect
38. #22576076C22semidirect
39. C8×Dic271C22semidirect
40. AGL(1,ℤ/8)×C271C22semidirect
41. C2×D8×C271C22semidirect
42. #131×C271C22semidirect
43. #108×C271C22semidirect
44. #100×C271C22semidirect
59. Dic1084D4semidirect
60. #16864644D4semidirect
63. #16864643D4semidirect
65. #16864642D4semidirect
66. #16864641D4semidirect
71. D2168D4semidirect
75. #16864639D4semidirect
153. C271#3292semidirect
161. #472D271semidirect
175. AGL(1,ℤ/8)C2×D271semidirect
186. C2×D8C2×D271semidirect
189. #131C2×D271semidirect
192. #108C2×D271semidirect
197. #100C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747022
#317747012
#317746982
#317742722
#317741682
#317741662
#317740672
#317740632
#317740132
#317740122
#472×C2712
#3292271