Finite Group Databasebeta

← #27182929

Subgroups — #27182929

Order 12928.

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

normal subgroups only

This group has 476 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
1#20265980C212yesmaximal, complemented
2#20265979C211yesmaximal, characteristic, complemented
3#20265552C211yesmaximal, characteristic, complemented
4#20265447C211yesmaximal, characteristic, complemented
5#20265445C212yesmaximal, complemented
6#20265444C211yesmaximal, characteristic, complemented
7#20265345C212yesmaximal, complemented
8#20265343C212yesmaximal, complemented
9#20265294C211yesmaximal, characteristic, complemented
10#20265292C211yesmaximal, characteristic, complemented
11#576×C101C211yesmaximal, characteristic, complemented
12#15692301C2212yescomplemented
13#15692282—22no—
14#15692281—22no—
15#15692281C2212yescomplemented
16#15692280—22no—
17#15692280C2212yescomplemented
18#15692280C2212yescomplemented
19#15692278—21no—
20#15692275—22no—
21#15692275C2211yescharacteristic
22C2×D808C2211yescharacteristic, complemented
23C2×#13108059—22no—
24C2×#13108059C2212yescomplemented
25D8×D101—21no—
26#15692262C2212yescomplemented
27#15692261C2211yescharacteristic, complemented
28#15692251C2212yescomplemented
29#15692191C2211yescharacteristic, complemented
30#15692190C2211yescharacteristic, complemented
31#15692182C2211yescharacteristic, complemented
32#15692182C2212yescomplemented
33#15692181C2211yescharacteristic, complemented
34#15692172C2212yescomplemented
35#15692171C2212yescomplemented
36C8×Dic101C2211yescharacteristic, complemented
37H(5,𝔽2)×C101C2212yescomplemented
38AGL(1,ℤ/8)×C101—22no—
39AGL(1,ℤ/8)×C101C2212yescomplemented
40#135×C101—21no—
41#135×C101C2211yescharacteristic
42C2×D8×C101—21no—
43C2×D8×C101C2211yescharacteristic, complemented
44#131×C101C2211yescharacteristic, complemented
45#13108075—22no—
46#13108074—22no—
47#13108074C2311yescharacteristic
48C2×D404—22no—
49C2×D404C2312yes—
50C2×C22⋊D101—42no—
51D4×D101—22no—
52D4×D101—42no—
53#13108064D411yescharacteristic, complemented
54#13108062—22no—
55#13108062D411yescharacteristic, complemented
56#13108061—22no—
57#13108061—22no—
58#13108061D411yescharacteristic, complemented
59D808—21no—
60D808—21no—
61#13108059—22no—
62#13108059—22no—
63#13108059—22no—
64#13108059—22no—
65#13108059—22no—
66#13108059D411yescharacteristic, complemented
67#13108058—22no—
68#13108058C2312yes—
69#13108058C2311yescharacteristic
70#13108057—21no—
71C2×#11717579—22no—
72C2×#11717579C2311yescharacteristic
73C8×D101—21no—
74C4×Dic101C2311yescharacteristic
75#41×C101—22no—
76#41×C101—22no—
77#41×C101C2312yes—
78#41×C101C2311yescharacteristic
79C2×D4×C101—22no—
80C2×D4×C101—22no—
81C2×D4×C101—42no—
82C2×D4×C101C2312yes—
83Q16×C101D411yescharacteristic, complemented
84SD16×C101—22no—
85SD16×C101D412yescomplemented
86D8×C101—22no—
87D8×C101—21no—
88D8×C101—21no—
89D8×C101D411yescharacteristic, complemented
90M16×C101—21no—
91M16×C101C2311yescharacteristic
92C2×C8×C101—21no—
93C2×C8×C101C2311yescharacteristic
94C23×C101—22no—
95C23×C101—42no—
96C22×D101—22no—
97C22×D101—42no—
98Q8×C101C2×D411yescharacteristic
99Q8×C101C2×D412yes—
100D4×C101—21no—
101D4×C101—22no—
102D4×C101—22no—
103D4×C101—22no—
104D4×C101—42no—
105D4×C101—42no—
106D4×C101C2×D411yescharacteristic
107D4×C101C2×D412yes—
108C2×C4×C101—21no—
109C2×C4×C101—22no—
110C2×C4×C101—42no—
111C2×C4×C101C2411yescharacteristic
112C22⋊D101—21no—
113C22⋊D101—42no—
114C22⋊D101—84no—
115C2×Dic101—21no—
116D404—22no—
117D404—22no—
118D404—42no—
119D404C2×D411yescharacteristic
120C4×D101—21no—
121C4×D101—42no—
122Dic202C2×D411yescharacteristic
123C8×C101—21no—
124C8×C101C2×D411yescharacteristic
125C8×C101C2×D411yescharacteristic
126#11717579—21no—
127#11717579—22no—
128#11717579C2×D411yescharacteristic
129#11717579C2×D411yescharacteristic
130C22×C101—22no—
131C22×C101—21no—
132C22×C101—42no—
133C22×C101—82no—
134C22×C101—162no—
135C22×C101H(5,𝔽2)11yescharacteristic
136C2×D101—21no—
137C2×D101—42no—
138C2×D101—82no—
139C2×D101—162no—
140C4×C101—22no—
141C4×C101—21no—
142C4×C101C22×D411yescharacteristic
143C4×C101C22×D411yescharacteristic
144Dic101—22no—
145Dic101—21no—
146C2×C101—21no—
147C2×C101—42no—
148C2×C101—41no—
149C2×C101—82no—
150C2×C101D4211yescharacteristic
151D101—41no—
152D101—82no—
153#3292—1011nomaximal, Sylow
154C101#329211yescharacteristic, complemented, Sylow
155#576—1012no—
156#576—1011no—
157#576D10111yescharacteristic, complemented
158#493—1011no—
159#472—1012no—
160#472—1011no—
161#472—1011no—
162#453—1012no—
163#453—1012no—
164#443—1011no—
165#443—1011no—
166H(5,𝔽2)—1012no—
167H(5,𝔽2)C2×D10112yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—1012no—
170AGL(1,ℤ/8)—1012no—
171AGL(1,ℤ/8)—1012no—
172AGL(1,ℤ/8)—2022no—
173AGL(1,ℤ/8)—2022no—
174AGL(1,ℤ/8)—2022no—
175AGL(1,ℤ/8)C2×D10112yescomplemented
176#135—21no—
177#135—1011no—
178#135—2022no—
179#135—2021no—
180#135C2×D10111yescharacteristic
181C2×D8—21no—
182C2×D8—1012no—
183C2×D8—1011no—
184C2×D8—2022no—
185C2×D8—2021no—
186C2×D8C2×D10111yescharacteristic, complemented
187#131—1012no—
188#131—1011no—
189#131C2×D10111yescharacteristic, complemented
190#127—1012no—
191#108—1011no—
192#108—1011no—
193#104—1011no—
194#104—1012no—
195#104—1011no—
196#100—1012no—
197#100—1012no—
198C4×C8—1011no—
199#41—22no—
200#41—22no—
201#41—1011no—
202#41—2022no—
203#41—2022no—
204#41C22×D10111yescharacteristic
205#41C22×D10112yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—1012no—
210C2×D4—4042no—
211C2×D4—4042no—
212C2×D4—2022no—
213C2×D4—2022no—
214C2×D4C22×D10112yes—
215Q16—1011no—
216Q16C22⋊D10111yescharacteristic, complemented
217SD16—22no—
218SD16—1011no—
219SD16—1011no—
220SD16—2022no—
221SD16—2022no—
222SD16—2022no—
223SD16C22⋊D10112yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—1011no—
228D8—2022no—
229D8—2022no—
230D8—2022no—
231D8—2022no—
232D8—2022no—
233D8—2021no—
234D8—2021no—
235D8C22⋊D10111yescharacteristic, complemented
236M16—21no—
237M16—1012no—
238M16—1011no—
239M16—2022no—
240M16—2021no—
241M16C22×D10111yescharacteristic
242C2×C8—21no—
243C2×C8—1011no—
244C2×C8—2022no—
245C2×C8—2021no—
246C2×C8C22×D10111yescharacteristic
247C42—1011no—
248C23—22no—
249C23—42no—
250C23—4042no—
251C23—2022no—
252Q8—1011no—
253Q8D4×D10111yescharacteristic
254Q8C2×C22⋊D10112yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—8084no—
262D4—1011no—
263D4—4042no—
264D4—4042no—
265D4—2021no—
266D4—2022no—
267D4—2022no—
268D4D4×D10111yescharacteristic
269D4C2×C22⋊D10112yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—4042no—
274C2×C4—2021no—
275C2×C4—2021no—
276C2×C4C23×D10111yescharacteristic
277C8—21no—
278C8—1011no—
279C8—1011no—
280C8—2022no—
281C8—2021no—
282C8C2×C22⋊D10111yescharacteristic
283C8C2×C22⋊D10111yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—8082no—
290C22—16162no—
291C22—4042no—
292C22—2021no—
293C22#1569230011yescharacteristic
294C4—21no—
295C4—22no—
296C4—2022no—
297C4—2021no—
298C4C2×D4×D10111yescharacteristic
299C4C22×C22⋊D10111yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—8082no—
305C2—4041no—
306C2D4×C22⋊D10111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#576×C101C2
socleC2×C101D42
hypercenter#576D101
hypocenterC101#3292
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20265980C2semidirect
2. #20265979C2semidirect
3. #20265552C2semidirect
4. #20265447C2semidirect
5. #20265445C2semidirect
6. #20265444C2semidirect
7. #20265345C2semidirect
8. #20265343C2semidirect
9. #20265294C2semidirect
10. #20265292C2semidirect
11. #576×C101C2semidirect
12. #15692301C22semidirect
15. #15692281C22semidirect
17. #15692280C22semidirect
18. #15692280C22semidirect
22. C2×D808C22semidirect
24. C2×#13108059C22semidirect
26. #15692262C22semidirect
27. #15692261C22semidirect
28. #15692251C22semidirect
29. #15692191C22semidirect
30. #15692190C22semidirect
31. #15692182C22semidirect
32. #15692182C22semidirect
33. #15692181C22semidirect
34. #15692172C22semidirect
35. #15692171C22semidirect
36. C8×Dic101C22semidirect
37. H(5,𝔽2)×C101C22semidirect
39. AGL(1,ℤ/8)×C101C22semidirect
43. C2×D8×C101C22semidirect
44. #131×C101C22semidirect
53. #13108064D4semidirect
55. #13108062D4semidirect
58. #13108061D4semidirect
66. #13108059D4semidirect
83. Q16×C101D4semidirect
85. SD16×C101D4semidirect
89. D8×C101D4semidirect
154. C101#3292semidirect
157. #576D101semidirect
167. H(5,𝔽2)C2×D101semidirect
175. AGL(1,ℤ/8)C2×D101semidirect
186. C2×D8C2×D101semidirect
189. #131C2×D101semidirect
216. Q16C22⋊D101semidirect
223. SD16C22⋊D101semidirect
235. D8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202659802
#202659792
#202655522
#202654472
#202654452
#202654442
#202653452
#202653432
#202652942
#202652922
#576×C1012
#3292101