Finite Group Databasebeta

← #27181612

Subgroups — #27181612

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×#15692279C212yesmaximal, complemented
2C2×#15692274C211yesmaximal, characteristic, complemented
3C2×D8×D101C211yesmaximal, characteristic, complemented
4#20265740C212yesmaximal, complemented
5#20265565C212yesmaximal, complemented
6#20265564C212yesmaximal, complemented
7#20265561C211yesmaximal, characteristic, complemented
8#20265560C211yesmaximal, characteristic, complemented
9#20265552C211yesmaximal, characteristic, complemented
10#20265131C211yesmaximal, characteristic, complemented
11#493×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2212yes—
13C2×D4×D101C2212yes—
14#15692279—28no—
15#15692274—24no—
16C2×Dic404C2211yescharacteristic, complemented
17C2×#13108063C2212yescomplemented
18C2×#13108062C2212yescomplemented
19C2×#13108062C2212yescomplemented
20C2×D808C2211yescharacteristic
21C2×#13108059C2212yes—
22C2×#13108059C2212yescomplemented
23D8×D101—24no—
24C2×#13108057C2211yescharacteristic
25C2×C8×D101C2211yescharacteristic
26#15692251C2212yescomplemented
27#15692243C2212yescomplemented
28#15692242C2212yescomplemented
29#15692227—24no—
30#15692223—24no—
31#15692212C2211yescharacteristic
32#15692166C2211yescharacteristic, complemented
33#15692165C2211yescharacteristic, complemented
34C8×Dic101C2211yescharacteristic, complemented
35C2×D8×C101C2211yescharacteristic
36C2×D8×C101C2211yescharacteristic, complemented
37C2×D8×C101C2212yescomplemented
38#127×C101C2212yescomplemented
39C4×C8×C101C2211yescharacteristic, complemented
40C23×D101—22no—
41#13108073—24no—
42#13108073—24no—
43C2×Dic202C2312yes—
44C2×D404C2312yes—
45C2×C22⋊D101—22no—
46C2×C22⋊D101—22no—
47C2×C22⋊D101—24no—
48D4×D101—24no—
49D4×D101—24no—
50C22×Dic101—22no—
51C2×C4×D101C2311yescharacteristic
52Dic404—22no—
53#13108063—24no—
54#13108062—24no—
55#13108062—24no—
56D808—22no—
57#13108059—24no—
58#13108059—24no—
59#13108057D414yescomplemented
60C2×#11717579C2311yescharacteristic
61C2×#11717579C2311yescharacteristic
62C8×D101D414yescomplemented
63#13108053D412yes—
64C22⋊Dic101—24no—
65#13108051—24no—
66#13108051D412yes—
67C4×Dic101C2311yescharacteristic
68C2×D4×C101—24no—
69C2×D4×C101C2312yes—
70C2×D4×C101C2312yes—
71D8×C101—22no—
72D8×C101—22no—
73D8×C101—24no—
74C2×C8×C101C2311yescharacteristic
75C2×C8×C101C2311yescharacteristic
76C42×C101C2311yescharacteristic
77C23×C101—22no—
78C23×C101—22no—
79C22×D101—22no—
80C22×D101—44no—
81C22×D101—84no—
82C22×D101C2×D411yescharacteristic
83D4×C101—22no—
84D4×C101—22no—
85D4×C101—22no—
86D4×C101—22no—
87D4×C101—84no—
88C2×C4×C101C2×D412yes—
89C2×C4×C101C2411yescharacteristic
90C22⋊D101—44no—
91C22⋊D101—44no—
92C22⋊D101—84no—
93C2×Dic101—22no—
94C2×Dic101—44no—
95C2×Dic101C2×D412yes—
96C2×Dic101C2×D411yescharacteristic
97D404—22no—
98D404—22no—
99C4×D101C2×D412yes—
100C4×D101C2×D412yes—
101Dic202—22no—
102Dic202—22no—
103C8×C101C2×D412yes—
104C8×C101C2×D412yes—
105#11717579C2×D412yes—
106#11717579C2×D412yes—
107C22×C101—42no—
108C22×C101—42no—
109C22×C101—44no—
110C22×C101—44no—
111C22×C101C22×D411yescharacteristic
112C2×D101—22no—
113C2×D101—42no—
114C2×D101—44no—
115C2×D101—88no—
116C2×D101#12712yes—
117C4×C101—22no—
118C4×C101C22×D411yescharacteristic
119C4×C101C22×D411yescharacteristic
120Dic101—22no—
121Dic101—42no—
122Dic101#12712yes—
123C2×C101—82no—
124C2×C101—82no—
125C2×C101C2×#12711yescharacteristic
126C2×C101#57612yes—
127D101—22no—
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#495—1011no—
139#493—1011no—
140#493D10111yescharacteristic, complemented
141#405—1011no—
142C2×#41—1012no—
143C22×D4—1012no—
144AGL(1,ℤ/8)—2028no—
145#135—2024no—
146C2×Q16—1011no—
147C2×SD16—1012no—
148C2×SD16—1012no—
149C2×SD16—1012no—
150C2×D8—1012no—
151C2×D8—1012no—
152C2×D8—1011no—
153C2×D8—2024no—
154C2×D8C2×D10111yescharacteristic
155C2×D8C2×D10111yescharacteristic, complemented
156C2×D8C2×D10112yescomplemented
157C2×M16—1011no—
158C22×C8—1011no—
159#127—1012no—
160#127C2×D10112yescomplemented
161#124—1012no—
162#124—1012no—
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×Q8—1012no—
174C2×D4—24no—
175C2×D4—1012no—
176C2×D4—2024no—
177C2×D4—2024no—
178C2×D4—2024no—
179C2×D4—2022no—
180C2×D4—2022no—
181C2×D4C22×D10112yes—
182C2×D4C22×D10112yes—
183C22×C4—1011no—
184C22×C4—2022no—
185Q16—2022no—
186SD16—2024no—
187SD16—2024no—
188SD16—2024no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—2024no—
193D8—2024no—
194D8—2022no—
195M16—1014no—
196C2×C8—1011no—
197C2×C8—1011no—
198C2×C8—1014no—
199C2×C8C22×D10111yescharacteristic
200C2×C8C22×D10111yescharacteristic
201C4⋊C4—1012no—
202C22⋊C4—1012no—
203C22⋊C4—2024no—
204C22⋊C4—2024no—
205C42—1011no—
206C42C22×D10111yescharacteristic
207C23—22no—
208C23—22no—
209C23—8084no—
210C23—1011no—
211C23—4044no—
212C23—2022no—
213Q8—2022no—
214Q8—2022no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—8084no—
221D4—4044no—
222D4—4044no—
223D4—2022no—
224D4—2022no—
225C2×C4—1011no—
226C2×C4—1012no—
227C2×C4—1012no—
228C2×C4—1012no—
229C2×C4—4044no—
230C2×C4—2022no—
231C2×C4D4×D10112yes—
232C2×C4C23×D10111yescharacteristic
233C8—1012no—
234C8—1012no—
235C8D4×D10112yes—
236C8D4×D10112yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—8088no—
242C22—1012no—
243C22—4042no—
244C22—4044no—
245C22—2022no—
246C22C2×D4×D10111yescharacteristic
247C4—22no—
248C4—1012no—
249C4—4042no—
250C4—2022no—
251C4C2×D4×D10111yescharacteristic
252C4C2×D4×D10111yescharacteristic
253C2—82no—
254C2—82no—
255C2—8082no—
256C2—2022no—
257C2#127×D10111yescharacteristic
258C2#2026597812yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#493×C101C2
socleC22×C101C22×D4
hypercenter#493D101
hypocenterC101#3155
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. C2×#15692274C2semidirect
3. C2×D8×D101C2semidirect
4. #20265740C2semidirect
5. #20265565C2semidirect
6. #20265564C2semidirect
7. #20265561C2semidirect
8. #20265560C2semidirect
9. #20265552C2semidirect
10. #20265131C2semidirect
11. #493×C101C2semidirect
16. C2×Dic404C22semidirect
17. C2×#13108063C22semidirect
18. C2×#13108062C22semidirect
19. C2×#13108062C22semidirect
22. C2×#13108059C22semidirect
26. #15692251C22semidirect
27. #15692243C22semidirect
28. #15692242C22semidirect
32. #15692166C22semidirect
33. #15692165C22semidirect
34. C8×Dic101C22semidirect
36. C2×D8×C101C22semidirect
37. C2×D8×C101C22semidirect
38. #127×C101C22semidirect
39. C4×C8×C101C22semidirect
59. #13108057D4semidirect
62. C8×D101D4semidirect
130. C101#3155semidirect
140. #493D101semidirect
155. C2×D8C2×D101semidirect
156. C2×D8C2×D101semidirect
160. #127C2×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×#156922792
C2×#156922742
C2×D8×D1012
#202657402
#202655652
#202655642
#202655612
#202655602
#202655522
#202651312
#493×C1012
#3155101