Finite Group Databasebeta

← #27181580

Subgroups — #27181580

Order 12928.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#15692283C211yesmaximal, characteristic, complemented
2C2×#15692281C211yesmaximal, characteristic, complemented
3C2×#15692280C211yesmaximal, characteristic, complemented
4#20265723C211yesmaximal, characteristic, complemented
5C2×#15692251C211yesmaximal, characteristic, complemented
6#20265565C212yesmaximal, complemented
7#20265563C212yesmaximal, complemented
8#20265552C212yesmaximal, complemented
9#20265548C212yesmaximal, complemented
10M16×Dic101C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×D404C2211yescharacteristic
14C22×C22⋊D101—22no—
15#15692283—24no—
16#15692281—24no—
17#15692280—24no—
18C2×#13108063C2212yescomplemented
19C2×#13108062C2212yescomplemented
20C2×#13108061C2212yescomplemented
21C2×D808C2212yescomplemented
22C2×#13108059C2212yescomplemented
23C2×#13108059C2212yescomplemented
24C2×#13108058C2211yescharacteristic
25#15692253C2211yescharacteristic, complemented
26#15692251—22no—
27#15692251C2212yes—
28#15692251C2212yes—
29#15692251C2211yescharacteristic, complemented
30#15692244C2211yescharacteristic, complemented
31#15692242C2211yescharacteristic, complemented
32#15692227—22no—
33C4×C22⋊D101—22no—
34C2×C4×Dic101C2211yescharacteristic
35#15692165C2212yescomplemented
36C8×Dic101C2212yescomplemented
37C2×#41×C101C2211yescharacteristic
38C22×D4×C101C2211yescharacteristic
39AGL(1,ℤ/8)×C101—24no—
40C2×SD16×C101C2212yescomplemented
41C2×D8×C101C2212yescomplemented
42C2×M16×C101C2211yescharacteristic
43C23×D101—21no—
44#13108074—22no—
45#13108074—22no—
46C2×Dic202C2311yescharacteristic
47C2×D404—22no—
48C2×D404—22no—
49C2×D404—21no—
50C2×D404C2312yes—
51C2×D404C2311yescharacteristic
52C2×C22⋊D101—21no—
53C2×C22⋊D101—24no—
54C2×C22⋊D101—24no—
55C2×C22⋊D101—22no—
56C2×C22⋊D101—48no—
57C22×Dic101D412yes—
58C2×C4×D101—21no—
59#13108063—24no—
60#13108062—24no—
61#13108061—24no—
62D808—24no—
63#13108059—24no—
64#13108059—24no—
65#13108058D414yescomplemented
66C2×#11717579C2312yes—
67#13108053—22no—
68C22⋊Dic101—22no—
69#13108051—22no—
70C4×Dic101D412yes—
71C4×Dic101C2311yescharacteristic
72C4×Dic101C2311yescharacteristic
73C24×C101—21no—
74#41×C101—22no—
75#41×C101—22no—
76C2×Q8×C101C2311yescharacteristic
77C2×D4×C101—21no—
78C2×D4×C101—22no—
79C2×D4×C101—22no—
80C2×D4×C101—21no—
81C2×D4×C101C2311yescharacteristic
82C2×D4×C101C2312yes—
83C22×C4×C101—21no—
84C22×C4×C101C2311yescharacteristic
85SD16×C101—24no—
86D8×C101—24no—
87M16×C101D414yescomplemented
88C2×C8×C101C2312yes—
89C23×C101—21no—
90C23×C101—22no—
91C23×C101—44no—
92C23×C101—42no—
93C23×C101C2×D411yescharacteristic
94C22×D101—21no—
95C22×D101—22no—
96C22×D101—44no—
97C22×D101—42no—
98Q8×C101—21no—
99Q8×C101—21no—
100D4×C101—21no—
101D4×C101—22no—
102D4×C101—22no—
103D4×C101—21no—
104D4×C101—42no—
105D4×C101—42no—
106C2×C4×C101—21no—
107C2×C4×C101—42no—
108C2×C4×C101C2×D411yescharacteristic
109C2×C4×C101C2×D412yes—
110C2×C4×C101C2×D412yes—
111C2×C4×C101C2411yescharacteristic
112C22⋊D101—48no—
113C22⋊D101—42no—
114C22⋊D101—44no—
115C22⋊D101—48no—
116C2×Dic101—24no—
117C2×Dic101—21no—
118C2×Dic101C2×D412yes—
119C2×Dic101C2×D412yes—
120D404—21no—
121D404—22no—
122D404—21no—
123D404—22no—
124D404—42no—
125C4×D101—42no—
126Dic202—21no—
127Dic202—21no—
128C8×C101C2×D414yescomplemented
129#11717579C2×D414yescomplemented
130C22×C101—22no—
131C22×C101—41no—
132C22×C101—42no—
133C22×C101—44no—
134C22×C101—42no—
135C22×C101—84no—
136C22×C101#12712yes—
137C22×C101C22×D411yescharacteristic
138C2×D101—41no—
139C2×D101—42no—
140C2×D101—44no—
141C2×D101—42no—
142C2×D101—84no—
143C4×C101—41no—
144C4×C101#12712yes—
145C4×C101C22×D411yescharacteristic
146C4×C101C22×D411yescharacteristic
147Dic101—22no—
148Dic101—41no—
149Dic101AGL(1,ℤ/8)14yescomplemented
150C2×C101—22no—
151C2×C101—81no—
152C2×C101—82no—
153C2×C101C2×#12711yescharacteristic
154C2×C101C2×AGL(1,ℤ/8)12yes—
155D101—81no—
156D101—82no—
157#3151—1011nomaximal, Sylow
158C101#315111yescharacteristic, complemented, Sylow
159C2×AGL(1,ℤ/8)—1011no—
160C2×AGL(1,ℤ/8)—1011no—
161C2×AGL(1,ℤ/8)—1011no—
162C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
163#532—1011no—
164C2×#127—1011no—
165#496—1012no—
166#496—1012no—
167#493—1012no—
168#492—1012no—
169C4×M16—1011no—
170C2×#41—1011no—
171C2×#41C2×D10111yescharacteristic
172C22×D4—1011no—
173C22×D4—2022no—
174C22×D4C2×D10111yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2024no—
177AGL(1,ℤ/8)—2024no—
178AGL(1,ℤ/8)—2024no—
179C2×SD16—1012no—
180C2×SD16—1012no—
181C2×SD16—1012no—
182C2×SD16C2×D10112yescomplemented
183C2×D8—1012no—
184C2×D8—1012no—
185C2×D8—1012no—
186C2×D8C2×D10112yescomplemented
187C2×M16—1011no—
188C2×M16C2×D10111yescharacteristic
189C4⋊Q8—1011no—
190#127—1012no—
191#127—1012no—
192#127—1011no—
193#127—2022no—
194#124—1011no—
195#124—1011no—
196#121—2022no—
197C4×D4—2022no—
198C2×C42—1011no—
199#97—1012no—
200C4×C8—1012no—
201C24—21no—
202C24—2021no—
203#41—22no—
204#41—22no—
205#41—2022no—
206#41—2022no—
207C2×Q8—1011no—
208C2×Q8C22×D10111yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—1011no—
214C2×D4—1012no—
215C2×D4—4048no—
216C2×D4—2022no—
217C2×D4—2024no—
218C2×D4—2024no—
219C2×D4—2021no—
220C2×D4—2021no—
221C2×D4—2022no—
222C2×D4—2022no—
223C2×D4C22×D10111yescharacteristic
224C2×D4C22×D10112yes—
225C22×C4—21no—
226C22×C4—1012no—
227C22×C4—2021no—
228C22×C4C22×D10111yescharacteristic
229SD16—24no—
230SD16—2024no—
231SD16—2024no—
232SD16—2024no—
233D8—24no—
234D8—2024no—
235D8—2024no—
236D8—2024no—
237M16—1014no—
238M16C22⋊D10114yescomplemented
239C2×C8—1012no—
240C2×C8C22×D10112yes—
241C4⋊C4—2022no—
242C22⋊C4—2022no—
243C22⋊C4—2022no—
244C42—1012no—
245C42—1011no—
246C42—1011no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—4044no—
252C23—4042no—
253C23—2021no—
254C23—2022no—
255C23D4×D10111yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2021no—
259Q8—2021no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—4048no—
267D4—4044no—
268D4—4042no—
269D4—4048no—
270D4—4042no—
271D4—2021no—
272D4—2022no—
273D4—2021no—
274D4—2022no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—1012no—
278C2×C4—1012no—
279C2×C4—4042no—
280C2×C4—2024no—
281C2×C4—2021no—
282C2×C4D4×D10112yes—
283C2×C4D4×D10111yescharacteristic
284C2×C4C2×C22⋊D10112yes—
285C2×C4C23×D10111yescharacteristic
286C8—1014no—
287C8C2×C22⋊D10114yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—8084no—
295C22—4044no—
296C22—4042no—
297C22—4041no—
298C22—4042no—
299C22#1569225112yes—
300C22C2×D4×D10111yescharacteristic
301C4—41no—
302C4—1014no—
303C4—4041no—
304C4—2022no—
305C4#1569225112yes—
306C4C2×D4×D10111yescharacteristic
307C4C22×C22⋊D10111yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—8082no—
312C2—8081no—
313C2C2×#1569225111yescharacteristic
314C2AGL(1,ℤ/8)×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×AGL(1,ℤ/8)×C101C2
socleC22×C101C22×D4
hypercenterC2×AGL(1,ℤ/8)D101
hypocenterC101#3151
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15692283C2semidirect
2. C2×#15692281C2semidirect
3. C2×#15692280C2semidirect
4. #20265723C2semidirect
5. C2×#15692251C2semidirect
6. #20265565C2semidirect
7. #20265563C2semidirect
8. #20265552C2semidirect
9. #20265548C2semidirect
10. M16×Dic101C2semidirect
11. C2×AGL(1,ℤ/8)×C101C2semidirect
18. C2×#13108063C22semidirect
19. C2×#13108062C22semidirect
20. C2×#13108061C22semidirect
21. C2×D808C22semidirect
22. C2×#13108059C22semidirect
23. C2×#13108059C22semidirect
25. #15692253C22semidirect
29. #15692251C22semidirect
30. #15692244C22semidirect
31. #15692242C22semidirect
35. #15692165C22semidirect
36. C8×Dic101C22semidirect
40. C2×SD16×C101C22semidirect
41. C2×D8×C101C22semidirect
65. #13108058D4semidirect
87. M16×C101D4semidirect
128. C8×C101C2×D4semidirect
129. #11717579C2×D4semidirect
149. Dic101AGL(1,ℤ/8)semidirect
158. C101#3151semidirect
162. C2×AGL(1,ℤ/8)D101semidirect
182. C2×SD16C2×D101semidirect
186. C2×D8C2×D101semidirect
238. M16C22⋊D101semidirect
287. C8C2×C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
C2×#156922812
C2×#156922802
#202657232
C2×#156922512
#202655652
#202655632
#202655522
#202655482
M16×Dic1012
C2×AGL(1,ℤ/8)×C1012
#3151101