Finite Group Databasebeta

← #27181614

Subgroups — #27181614

Order 12928.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15692282C212yesmaximal, complemented
2C2×#15692278C211yesmaximal, characteristic, complemented
3C2×D8×D101C211yesmaximal, characteristic, complemented
4#20265745C212yesmaximal, complemented
5#20265567C212yesmaximal, complemented
6#20265563C212yesmaximal, complemented
7#20265562C211yesmaximal, characteristic, complemented
8#20265554C211yesmaximal, characteristic, complemented
9#20265552C211yesmaximal, characteristic, complemented
10#20265131C211yesmaximal, characteristic, complemented
11#495×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2212yes—
13C2×D4×D101C2212yes—
14#15692282—28no—
15#15692278—24no—
16C2×#13108061C2212yescomplemented
17C2×#13108061C2212yescomplemented
18C2×D808C2211yescharacteristic
19C2×D808C2211yescharacteristic, complemented
20C2×D808C2212yescomplemented
21C2×#13108059C2212yes—
22C2×#13108059C2212yescomplemented
23D8×D101—24no—
24C2×#13108057C2211yescharacteristic
25C2×C8×D101C2211yescharacteristic
26#15692252C2212yescomplemented
27#15692251C2212yescomplemented
28#15692244C2212yescomplemented
29#15692228—24no—
30#15692225—24no—
31#15692212C2211yescharacteristic
32#15692166C2211yescharacteristic, complemented
33#15692165C2211yescharacteristic, complemented
34C8×Dic101C2211yescharacteristic, complemented
35C2×Q16×C101C2211yescharacteristic, complemented
36C2×SD16×C101C2212yescomplemented
37C2×D8×C101C2211yescharacteristic
38#124×C101C2212yescomplemented
39C4×C8×C101C2211yescharacteristic, complemented
40C23×D101—22no—
41#13108075—24no—
42#13108075—24no—
43C2×D404—22no—
44C2×D404—24no—
45C2×D404C2312yes—
46C2×D404C2312yes—
47C2×C22⋊D101—22no—
48C2×C22⋊D101—24no—
49D4×D101—24no—
50D4×D101—24no—
51C2×C4×D101—22no—
52C2×C4×D101C2311yescharacteristic
53#13108061—24no—
54#13108061—24no—
55D808—22no—
56D808—22no—
57D808—24no—
58#13108059—24no—
59#13108059—24no—
60#13108057D414yescomplemented
61C2×#11717579C2311yescharacteristic
62C2×#11717579C2311yescharacteristic
63C8×D101D414yescomplemented
64#13108053D412yes—
65#13108051—24no—
66#13108051D412yes—
67C4×Dic101C2311yescharacteristic
68C2×Q8×C101C2312yes—
69C2×D4×C101C2312yes—
70Q16×C101—22no—
71SD16×C101—24no—
72D8×C101—22no—
73C2×C8×C101C2311yescharacteristic
74C2×C8×C101C2311yescharacteristic
75C22⋊C4×C101—24no—
76C42×C101C2311yescharacteristic
77C23×C101—22no—
78C22×D101—22no—
79C22×D101—22no—
80C22×D101—44no—
81C22×D101—84no—
82C22×D101C2×D411yescharacteristic
83Q8×C101—22no—
84Q8×C101—22no—
85D4×C101—22no—
86D4×C101—22no—
87C2×C4×C101—22no—
88C2×C4×C101C2×D412yes—
89C2×C4×C101C2411yescharacteristic
90C22⋊D101—44no—
91C22⋊D101—84no—
92C2×Dic101C2×D412yes—
93C2×Dic101C2×D411yescharacteristic
94D404—22no—
95D404—22no—
96D404—22no—
97D404—22no—
98D404—44no—
99D404—84no—
100C4×D101—44no—
101C4×D101C2×D412yes—
102C4×D101C2×D412yes—
103C8×C101C2×D412yes—
104C8×C101C2×D412yes—
105#11717579C2×D412yes—
106#11717579C2×D412yes—
107C22×C101—42no—
108C22×C101—44no—
109C22×C101C22×D411yescharacteristic
110C2×D101—22no—
111C2×D101—42no—
112C2×D101—42no—
113C2×D101—44no—
114C2×D101—44no—
115C2×D101—88no—
116C2×D101#12712yes—
117C4×C101—22no—
118C4×C101—42no—
119C4×C101C22×D411yescharacteristic
120C4×C101C22×D411yescharacteristic
121Dic101—22no—
122Dic101#12712yes—
123C2×C101—82no—
124C2×C101C2×#12711yescharacteristic
125C2×C101#57612yes—
126D101—22no—
127D101—82no—
128D101—82no—
129#3155—1011nomaximal, Sylow
130C101#315511yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—1012no—
132C2×#135—1011no—
133C22×D8—1011no—
134#535—1012no—
135#496—1012no—
136#496—1012no—
137#495—1011no—
138#495D10111yescharacteristic, complemented
139#493—1011no—
140#493—1011no—
141#405—1011no—
142C2×#41—1012no—
143C22×D4—1012no—
144AGL(1,ℤ/8)—2028no—
145#135—2024no—
146C2×Q16C2×D10111yescharacteristic, complemented
147C2×SD16—1012no—
148C2×SD16—1012no—
149C2×SD16C2×D10112yescomplemented
150C2×D8—1012no—
151C2×D8—1012no—
152C2×D8—1011no—
153C2×D8—1011no—
154C2×D8—1012no—
155C2×D8—2024no—
156C2×D8C2×D10111yescharacteristic
157C2×M16—1011no—
158C22×C8—1011no—
159#127—1012no—
160#127—1012no—
161#124—1012no—
162#124C2×D10112yescomplemented
163#121—2024no—
164#120—2024no—
165#117—1011no—
166#97—1011no—
167#97—1011no—
168C4×C8—1011no—
169C4×C8C2×D10111yescharacteristic, complemented
170C24—2022no—
171#41—2024no—
172#41—2024no—
173C2×Q8C22×D10112yes—
174C2×D4—1012no—
175C2×D4—1012no—
176C2×D4—2022no—
177C2×D4—2024no—
178C2×D4—2024no—
179C2×D4—2024no—
180C2×D4—2022no—
181C2×D4—2024no—
182C2×D4C22×D10112yes—
183C22×C4—1011no—
184C22×C4—2022no—
185Q16—22no—
186SD16—24no—
187SD16—2024no—
188SD16—2024no—
189D8—22no—
190D8—2024no—
191D8—2024no—
192D8—2022no—
193D8—2022no—
194D8—2024no—
195M16—1014no—
196C2×C8—1011no—
197C2×C8—1011no—
198C2×C8—1014no—
199C2×C8C22×D10111yescharacteristic
200C2×C8C22×D10111yescharacteristic
201C4⋊C4—1012no—
202C22⋊C4—24no—
203C22⋊C4—1012no—
204C22⋊C4—2024no—
205C42—1011no—
206C42C22×D10111yescharacteristic
207C23—22no—
208C23—8084no—
209C23—1011no—
210C23—4044no—
211C23—2022no—
212C23—2022no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—8084no—
218D4—8084no—
219D4—4044no—
220D4—4044no—
221D4—2022no—
222D4—2022no—
223D4—2022no—
224D4—2022no—
225C2×C4—22no—
226C2×C4—1011no—
227C2×C4—1012no—
228C2×C4—1012no—
229C2×C4—1012no—
230C2×C4—4044no—
231C2×C4D4×D10112yes—
232C2×C4C23×D10111yescharacteristic
233C8—1012no—
234C8—1012no—
235C8D4×D10112yes—
236C8D4×D10112yes—
237C22—42no—
238C22—44no—
239C22—8088no—
240C22—1012no—
241C22—4042no—
242C22—4042no—
243C22—4044no—
244C22—4044no—
245C22—2022no—
246C22C2×D4×D10111yescharacteristic
247C4—22no—
248C4—42no—
249C4—1012no—
250C4—2022no—
251C4C2×D4×D10111yescharacteristic
252C4C2×D4×D10111yescharacteristic
253C2—82no—
254C2—8082no—
255C2—8082no—
256C2—2022no—
257C2#127×D10111yescharacteristic
258C2#2026597912yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#495×C101C2
socleC22×C101C22×D4
hypercenter#495D101
hypocenterC101#3155
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. C2×#15692278C2semidirect
3. C2×D8×D101C2semidirect
4. #20265745C2semidirect
5. #20265567C2semidirect
6. #20265563C2semidirect
7. #20265562C2semidirect
8. #20265554C2semidirect
9. #20265552C2semidirect
10. #20265131C2semidirect
11. #495×C101C2semidirect
16. C2×#13108061C22semidirect
17. C2×#13108061C22semidirect
19. C2×D808C22semidirect
20. C2×D808C22semidirect
22. C2×#13108059C22semidirect
26. #15692252C22semidirect
27. #15692251C22semidirect
28. #15692244C22semidirect
32. #15692166C22semidirect
33. #15692165C22semidirect
34. C8×Dic101C22semidirect
35. C2×Q16×C101C22semidirect
36. C2×SD16×C101C22semidirect
38. #124×C101C22semidirect
39. C4×C8×C101C22semidirect
60. #13108057D4semidirect
63. C8×D101D4semidirect
130. C101#3155semidirect
138. #495D101semidirect
146. C2×Q16C2×D101semidirect
149. C2×SD16C2×D101semidirect
162. #124C2×D101semidirect
169. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C2×#156922782
C2×D8×D1012
#202657452
#202655672
#202655632
#202655622
#202655542
#202655522
#202651312
#495×C1012
#3155101