Finite Group Databasebeta

← #27181617

Subgroups — #27181617

Order 12928.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15692275C211yesmaximal, characteristic, complemented
3C22×#13108059C211yesmaximal, characteristic, complemented
4#20265744C211yesmaximal, characteristic, complemented
5#20265741C211yesmaximal, characteristic, complemented
6#20265565C212yesmaximal, complemented
7#20265563C212yesmaximal, complemented
8#20265558C212yesmaximal, complemented
9#20265552C212yesmaximal, complemented
10#20265135C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×D404C2211yescharacteristic
14#15692283—24no—
15#15692275—24no—
16C2×#13108064C2211yescharacteristic, complemented
17C2×#13108063C2212yescomplemented
18C2×#13108062C2211yescharacteristic, complemented
19C2×#13108061C2211yescharacteristic, complemented
20C2×D808C2212yescomplemented
21C2×#13108059—24no—
22C2×#13108059C2212yes—
23C2×#13108059C2212yes—
24C2×#13108059C2211yescharacteristic, complemented
25C22×#11717579C2211yescharacteristic
26#15692251C2212yescomplemented
27#15692251C2212yescomplemented
28#15692244C2212yescomplemented
29#15692242C2212yescomplemented
30#15692229—22no—
31#15692226—22no—
32#15692224—22no—
33#15692223—22no—
34#15692214C2211yescharacteristic
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×C4×D101—21no—
56#13108064—22no—
57#13108063—24no—
58#13108062—22no—
59#13108061—22no—
60D808—24no—
61#13108059—24no—
62#13108059—24no—
63#13108059—22no—
64C2×#11717579D414yescomplemented
65C2×#11717579C2311yescharacteristic
66C2×#11717579C2311yescharacteristic
67C4⋊Dic101D412yes—
68C22⋊Dic101—24no—
69C22⋊Dic101D412yes—
70#13108051—24no—
71C4×Dic101C2312yes—
72C24×C101—21no—
73#41×C101—22no—
74#41×C101—22no—
75C2×Q8×C101C2311yescharacteristic
76C2×D4×C101—21no—
77C2×D4×C101—22no—
78C2×D4×C101—22no—
79C2×D4×C101—21no—
80C2×D4×C101C2311yescharacteristic
81C2×D4×C101C2312yes—
82C22×C4×C101—21no—
83C22×C4×C101C2311yescharacteristic
84SD16×C101—24no—
85D8×C101—24no—
86M16×C101D414yescomplemented
87C2×C8×C101C2312yes—
88C23×C101—21no—
89C23×C101—22no—
90C23×C101—42no—
91C23×C101—82no—
92C23×C101C2×D411yescharacteristic
93C22×D101—22no—
94C22×D101—21no—
95C22×D101—42no—
96C22×D101—82no—
97Q8×C101—21no—
98Q8×C101—21no—
99D4×C101—21no—
100D4×C101—22no—
101D4×C101—22no—
102D4×C101—21no—
103D4×C101—42no—
104D4×C101—42no—
105C2×C4×C101—21no—
106C2×C4×C101—42no—
107C2×C4×C101C2×D411yescharacteristic
108C2×C4×C101C2×D412yes—
109C2×C4×C101C2×D412yes—
110C2×C4×C101C2411yescharacteristic
111C22⋊D101—42no—
112C22⋊D101—84no—
113C22⋊D101—84no—
114C2×Dic101—21no—
115C2×Dic101C2×D414yes—
116D404—22no—
117D404—21no—
118D404—21no—
119D404—22no—
120D404—42no—
121C4×D101—42no—
122Dic202—21no—
123Dic202—21no—
124C8×C101C2×D414yes—
125#11717579C2×D412yes—
126#11717579C2×D412yes—
127C22×C101—22no—
128C22×C101—41no—
129C22×C101—42no—
130C22×C101—42no—
131C22×C101—44no—
132C22×C101—84no—
133C22×C101#12712yes—
134C22×C101C22×D411yescharacteristic
135C2×D101—41no—
136C2×D101—42no—
137C2×D101—42no—
138C2×D101—44no—
139C2×D101—84no—
140C4×C101—41no—
141C4×C101#12712yes—
142C4×C101C22×D411yescharacteristic
143C4×C101C22×D411yescharacteristic
144Dic101—24no—
145Dic101—41no—
146C2×C101—22no—
147C2×C101—81no—
148C2×C101—82no—
149C2×C101C2×#12711yescharacteristic
150C2×C101#57612yes—
151D101—81no—
152D101—82no—
153#3155—1011nomaximal, Sylow
154C101#315511yescharacteristic, complemented, Sylow
155C2×AGL(1,ℤ/8)—1011no—
156C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
157C2×#135—1011no—
158C22×D8—1011no—
159#535—1011no—
160#535—1011no—
161#496—1012no—
162#496—1012no—
163#495—1012no—
164#493—1012no—
165#405—1011no—
166C2×#41—1011no—
167C2×#41C2×D10111yescharacteristic
168C22×D4—1011no—
169C22×D4C2×D10111yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—2024no—
172#135—2024no—
173C2×Q16—1011no—
174C2×SD16—1011no—
175C2×SD16—1011no—
176C2×SD16—1012no—
177C2×SD16C2×D10112yescomplemented
178C2×D8—1012no—
179C2×D8—1012no—
180C2×D8—1011no—
181C2×D8—1012no—
182C2×D8—2024no—
183C2×D8C2×D10112yescomplemented
184C2×M16C2×D10111yescharacteristic
185C22×C8—1011no—
186#127—1012no—
187#127—1012no—
188#124—1012no—
189#124—1012no—
190#121—2022no—
191#121—2022no—
192#120—2022no—
193#120—2022no—
194#117—1011no—
195#97—1012no—
196C4×C8—1012no—
197C24—21no—
198C24—2021no—
199#41—22no—
200#41—22no—
201#41—2022no—
202#41—2022no—
203C2×Q8—1011no—
204C2×Q8C22×D10111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1012no—
210C2×D4—1011no—
211C2×D4—2024no—
212C2×D4—2024no—
213C2×D4—2021no—
214C2×D4—2021no—
215C2×D4—2022no—
216C2×D4—2022no—
217C2×D4C22×D10112yes—
218C2×D4C22×D10111yescharacteristic
219C22×C4—21no—
220C22×C4—2021no—
221C22×C4C22×D10111yescharacteristic
222Q16—2022no—
223SD16—24no—
224SD16—2022no—
225SD16—2022no—
226SD16—2024no—
227D8—24no—
228D8—2024no—
229D8—2024no—
230D8—2022no—
231D8—2024no—
232M16C22⋊D10114yescomplemented
233C2×C8—1014no—
234C2×C8—1011no—
235C2×C8—1011no—
236C2×C8C22×D10112yes—
237C4⋊C4—1012no—
238C22⋊C4—1012no—
239C22⋊C4—2024no—
240C22⋊C4—2024no—
241C42—1012no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—8082no—
247C23—4042no—
248C23—2022no—
249C23—2021no—
250C23D4×D10111yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2021no—
254Q8—2021no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—8084no—
262D4—8084no—
263D4—4042no—
264D4—4042no—
265D4—2022no—
266D4—2021no—
267D4—2021no—
268D4—2022no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—1014no—
272C2×C4—4042no—
273C2×C4—2021no—
274C2×C4D4×D10112yes—
275C2×C4D4×D10111yescharacteristic
276C2×C4C2×C22⋊D10112yes—
277C2×C4C23×D10111yescharacteristic
278C8—1012no—
279C8—1012no—
280C8C2×C22⋊D10114yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—8084no—
288C22—4041no—
289C22—4042no—
290C22—4042no—
291C22—4044no—
292C22#1569225112yes—
293C22C2×D4×D10111yescharacteristic
294C4—41no—
295C4—4041no—
296C4—2024no—
297C4#1569225112yes—
298C4C2×D4×D10111yescharacteristic
299C4C22×C22⋊D10111yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—8082no—
304C2—8081no—
305C2C2×#1569225111yescharacteristic
306C2#2026598112yes—
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#3155
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×#15692275C2semidirect
3. C22×#13108059C2semidirect
4. #20265744C2semidirect
5. #20265741C2semidirect
6. #20265565C2semidirect
7. #20265563C2semidirect
8. #20265558C2semidirect
9. #20265552C2semidirect
10. #20265135C2semidirect
11. C2×AGL(1,ℤ/8)×C101C2semidirect
16. C2×#13108064C22semidirect
17. C2×#13108063C22semidirect
18. C2×#13108062C22semidirect
19. C2×#13108061C22semidirect
20. C2×D808C22semidirect
24. C2×#13108059C22semidirect
26. #15692251C22semidirect
27. #15692251C22semidirect
28. #15692244C22semidirect
29. #15692242C22semidirect
35. #15692165C22semidirect
36. C8×Dic101C22semidirect
40. C2×SD16×C101C22semidirect
41. C2×D8×C101C22semidirect
64. C2×#11717579D4semidirect
86. M16×C101D4semidirect
154. C101#3155semidirect
156. C2×AGL(1,ℤ/8)D101semidirect
177. C2×SD16C2×D101semidirect
183. C2×D8C2×D101semidirect
232. M16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
C2×#156922752
C22×#131080592
#202657442
#202657412
#202655652
#202655632
#202655582
#202655522
#202651352
C2×AGL(1,ℤ/8)×C1012
#3155101