Finite Group Databasebeta

← #27181581

Subgroups — #27181581

Order 12928.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15692279C212yesmaximal, complemented
3#20265726C211yesmaximal, characteristic, complemented
4#127×D101C211yesmaximal, characteristic, complemented
5#20265567C211yesmaximal, characteristic, complemented
6#20265566C211yesmaximal, characteristic, complemented
7#20265564C211yesmaximal, characteristic, complemented
8#20265552C212yesmaximal, complemented
9#20265548C212yesmaximal, complemented
10#20265129C211yesmaximal, characteristic, complemented
11#496×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2211yescharacteristic
13C2×#13108073C2211yescharacteristic
14C2×D4×D101—22no—
15C2×D4×D101C2212yes—
16#15692282—28no—
17#15692279—28no—
18C2×#13108063C2212yescomplemented
19C2×#13108062C2212yescomplemented
20C2×#13108061C2212yescomplemented
21C2×D808C2212yescomplemented
22C2×#13108059C2212yescomplemented
23C2×#13108059C2212yescomplemented
24C2×#13108057C2212yes—
25#15692253C2211yescharacteristic, complemented
26#15692252C2211yescharacteristic
27#15692251—22no—
28#15692251C2212yes—
29#15692251C2211yescharacteristic, complemented
30#15692243C2211yescharacteristic, complemented
31#15692228—22no—
32#15692215—22no—
33C42×D101C2211yescharacteristic
34#15692166C2211yescharacteristic, complemented
35C8×Dic101C2212yescomplemented
36C2×SD16×C101C2212yescomplemented
37C2×D8×C101C2212yescomplemented
38#127×C101C2211yescharacteristic
39#124×C101C2211yescharacteristic, complemented
40#97×C101C2211yescharacteristic, complemented
41C23×D101—22no—
42#13108075—24no—
43#13108073—24no—
44C2×Dic202C2311yescharacteristic
45C2×D404—21no—
46C2×D404—22no—
47C2×D404C2312yes—
48C2×D404C2311yescharacteristic
49C2×C22⋊D101—21no—
50C2×C22⋊D101—24no—
51C2×C22⋊D101—22no—
52C2×C22⋊D101—22no—
53D4×D101—24no—
54D4×D101—24no—
55D4×D101—48no—
56C22×Dic101—21no—
57C2×C4×D101—21no—
58C2×C4×D101D412yes—
59C2×C4×D101C2311yescharacteristic
60#13108063—24no—
61#13108062—24no—
62#13108061—24no—
63D808—24no—
64#13108059—24no—
65#13108059—24no—
66#13108057D418yescomplemented
67C2×#11717579C2312yes—
68#13108053—22no—
69#13108051—22no—
70C4×Dic101D412yes—
71C4×Dic101C2311yescharacteristic
72C2×Q8×C101C2311yescharacteristic
73C2×D4×C101—22no—
74C2×D4×C101C2312yes—
75C2×D4×C101C2311yescharacteristic
76SD16×C101—24no—
77D8×C101—24no—
78C2×C8×C101C2312yes—
79C22⋊C4×C101—22no—
80C42×C101C2311yescharacteristic
81C23×C101—21no—
82C23×C101—22no—
83C22×D101—21no—
84C22×D101—22no—
85C22×D101—48no—
86C22×D101—44no—
87C22×D101C2×D411yescharacteristic
88Q8×C101—22no—
89D4×C101—22no—
90D4×C101—22no—
91D4×C101—22no—
92D4×C101—44no—
93C2×C4×C101—21no—
94C2×C4×C101C2×D412yes—
95C2×C4×C101C2411yescharacteristic
96C22⋊D101—42no—
97C22⋊D101—44no—
98C22⋊D101—48no—
99C22⋊D101—44no—
100C2×Dic101—21no—
101C2×Dic101—42no—
102C2×Dic101C2×D411yescharacteristic
103C2×Dic101C2×D412yes—
104D404—22no—
105D404—22no—
106D404—22no—
107D404—44no—
108D404—42no—
109C4×D101—24no—
110C4×D101—42no—
111C4×D101C2×D414yes—
112Dic202—22no—
113C8×C101C2×D414yescomplemented
114#11717579C2×D414yescomplemented
115C22×C101—41no—
116C22×C101—42no—
117C22×C101—44no—
118C22×C101—42no—
119C22×C101C22×D411yescharacteristic
120C2×D101—22no—
121C2×D101—41no—
122C2×D101—42no—
123C2×D101—44no—
124C2×D101—42no—
125C2×D101—88no—
126C2×D101#12712yes—
127C4×C101—22no—
128C4×C101—41no—
129C4×C101C22×D412yes—
130Dic101—41no—
131Dic101#12712yes—
132Dic101AGL(1,ℤ/8)14yescomplemented
133C2×C101—82no—
134C2×C101—81no—
135C2×C101C2×#12711yescharacteristic
136C2×C101C2×AGL(1,ℤ/8)12yes—
137D101—22no—
138D101—81no—
139D101—82no—
140#3151—1011nomaximal, Sylow
141C101#315111yescharacteristic, complemented, Sylow
142C2×AGL(1,ℤ/8)—1012no—
143C2×AGL(1,ℤ/8)—1012no—
144#532—1011no—
145C2×#127—1011no—
146#496—1011no—
147#496—1011no—
148#496—1011no—
149#496D10111yescharacteristic, complemented
150#493—1012no—
151#492—1012no—
152C4×M16—1011no—
153C2×#41—1011no—
154C2×#41—1011no—
155C22×D4—1012no—
156C22×D4—2022no—
157AGL(1,ℤ/8)—2028no—
158AGL(1,ℤ/8)—2028no—
159C2×SD16—1012no—
160C2×SD16—1012no—
161C2×SD16—1012no—
162C2×SD16C2×D10112yescomplemented
163C2×D8—1012no—
164C2×D8—1012no—
165C2×D8—1012no—
166C2×D8C2×D10112yescomplemented
167C2×M16—1012no—
168C4⋊Q8—1011no—
169#127—1012no—
170#127—1011no—
171#127—1011no—
172#127—2022no—
173#127C2×D10111yescharacteristic
174#124—1011no—
175#124C2×D10111yescharacteristic, complemented
176#121—2022no—
177C4×D4—2022no—
178C2×C42—1011no—
179#97—1011no—
180#97C2×D10111yescharacteristic, complemented
181C4×C8—1012no—
182C24—2022no—
183#41—2024no—
184#41—2024no—
185C2×Q8—1011no—
186C2×Q8C22×D10111yescharacteristic
187C2×D4—22no—
188C2×D4—1011no—
189C2×D4—1012no—
190C2×D4—4048no—
191C2×D4—2021no—
192C2×D4—2024no—
193C2×D4—2024no—
194C2×D4—2022no—
195C2×D4—2024no—
196C2×D4—2022no—
197C2×D4—2021no—
198C2×D4—2022no—
199C2×D4C22×D10111yescharacteristic
200C2×D4C22×D10112yes—
201C22×C4—1011no—
202C22×C4—1012no—
203C22×C4—2021no—
204C22×C4—2021no—
205SD16—24no—
206SD16—2024no—
207SD16—2024no—
208SD16—2024no—
209D8—24no—
210D8—2024no—
211D8—2024no—
212D8—2024no—
213M16—1018no—
214C2×C8—1012no—
215C2×C8C22×D10112yes—
216C4⋊C4—2022no—
217C22⋊C4—22no—
218C22⋊C4—2022no—
219C42—1012no—
220C42—1011no—
221C42C22×D10111yescharacteristic
222C23—21no—
223C23—22no—
224C23—1011no—
225C23—4044no—
226C23—4048no—
227C23—2021no—
228C23—2022no—
229Q8—22no—
230Q8—2022no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—4044no—
236D4—4042no—
237D4—4048no—
238D4—4044no—
239D4—4044no—
240D4—4042no—
241D4—2022no—
242D4—2022no—
243D4—2022no—
244C2×C4—21no—
245C2×C4—1012no—
246C2×C4—1011no—
247C2×C4—1014no—
248C2×C4—4042no—
249C2×C4—4042no—
250C2×C4—2024no—
251C2×C4—2021no—
252C2×C4D4×D10112yes—
253C2×C4C23×D10111yescharacteristic
254C8—1014no—
255C8D4×D10114yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—8088no—
261C22—1012no—
262C22—4044no—
263C22—4042no—
264C22—4041no—
265C22—4042no—
266C22—2022no—
267C22C2×D4×D10111yescharacteristic
268C4—22no—
269C4—41no—
270C4—1014no—
271C4—1012no—
272C4—4041no—
273C4C2×D4×D10112yes—
274C2—82no—
275C2—81no—
276C2—8082no—
277C2—8081no—
278C2—2022no—
279C2#127×D10111yescharacteristic
280C2AGL(1,ℤ/8)×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#496×C101C2
socleC22×C101C22×D4
hypercenter#496D101
hypocenterC101#3151
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. C2×#15692279C2semidirect
3. #20265726C2semidirect
4. #127×D101C2semidirect
5. #20265567C2semidirect
6. #20265566C2semidirect
7. #20265564C2semidirect
8. #20265552C2semidirect
9. #20265548C2semidirect
10. #20265129C2semidirect
11. #496×C101C2semidirect
18. C2×#13108063C22semidirect
19. C2×#13108062C22semidirect
20. C2×#13108061C22semidirect
21. C2×D808C22semidirect
22. C2×#13108059C22semidirect
23. C2×#13108059C22semidirect
25. #15692253C22semidirect
29. #15692251C22semidirect
30. #15692243C22semidirect
34. #15692166C22semidirect
35. C8×Dic101C22semidirect
36. C2×SD16×C101C22semidirect
37. C2×D8×C101C22semidirect
39. #124×C101C22semidirect
40. #97×C101C22semidirect
66. #13108057D4semidirect
113. C8×C101C2×D4semidirect
114. #11717579C2×D4semidirect
132. Dic101AGL(1,ℤ/8)semidirect
141. C101#3151semidirect
149. #496D101semidirect
162. C2×SD16C2×D101semidirect
166. C2×D8C2×D101semidirect
175. #124C2×D101semidirect
180. #97C2×D101semidirect
255. C8D4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C2×#156922792
#202657262
#127×D1012
#202655672
#202655662
#202655642
#202655522
#202655482
#202651292
#496×C1012
#3151101