Finite Group Databasebeta

← #27182747

Subgroups — #27182747

Order 12928.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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
2#20265849C211yesmaximal, characteristic, complemented
3#20265812C211yesmaximal, characteristic, complemented
4#20265553C211yesmaximal, characteristic, complemented
5#20265429C212yesmaximal, complemented
6#20265373C211yesmaximal, characteristic, complemented
7#20265321C212yesmaximal, complemented
8#20265312C212yesmaximal, complemented
9C4×#13108061C211yesmaximal, characteristic, complemented
10#20265254C211yesmaximal, characteristic, complemented
11#460×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2212yescomplemented
13C2×D4×D101—21no—
14C2×D4×D101C2212yes—
15#15692282—28no—
16C2×#13108061C2211yescharacteristic
17C2×D808C2212yes—
18C2×#13108059C2212yescomplemented
19C2×#13108059C2212yescomplemented
20C2×#13108057C2212yescomplemented
21#15692252—21no—
22#15692252C2211yescharacteristic, complemented
23#15692230—22no—
24#15692230C2212yescomplemented
25#15692226—21no—
26#15692226C2212yes—
27#15692223—22no—
28C4×D404—21no—
29C4×D404C2211yescharacteristic
30C4×D404C2211yescharacteristic
31#15692186C2211yescharacteristic, complemented
32#15692179C2211yescharacteristic, complemented
33#15692177C2212yescomplemented
34#15692175C2211yescharacteristic
35#15692167C2212yescomplemented
36C4×#11717579C2211yescharacteristic, complemented
37C2×SD16×C101C2212yescomplemented
38#127×C101C2211yescharacteristic
39C4×Q8×C101C2211yescharacteristic, complemented
40C4⋊C8×C101C2211yescharacteristic, complemented
41#102×C101C2212yescomplemented
42C23×D101—22no—
43#13108075—24no—
44#13108075—42no—
45C2×D404—22no—
46C2×D404—21no—
47C2×D404—22no—
48C2×D404—21no—
49C2×D404D411yescharacteristic
50C2×D404C2312yes—
51C2×D404C2311yescharacteristic
52C2×C22⋊D101—22no—
53C2×C22⋊D101—22no—
54D4×D101—24no—
55D4×D101—42no—
56D4×D101—44no—
57C2×C4×D101—21no—
58C2×C4×D101—22no—
59C2×C4×D101C2312yes—
60#13108061D414yescomplemented
61D808—24no—
62#13108059—24no—
63#13108059—42no—
64#13108057—24no—
65C2×#11717579C2311yescharacteristic
66C2×#11717579C2311yescharacteristic
67C4⋊Dic101D411yescharacteristic
68C4⋊Dic101C2311yescharacteristic
69C22⋊Dic101—22no—
70#13108051—21no—
71#13108051—22no—
72#13108051D412yes—
73C2×Q8×C101C2311yescharacteristic
74C2×D4×C101—21no—
75C2×D4×C101—21no—
76C2×D4×C101C2312yes—
77SD16×C101—24no—
78C2×C8×C101C2312yes—
79C4⋊C4×C101—21no—
80C4⋊C4×C101C2311yescharacteristic
81C42×C101—21no—
82C42×C101C2311yescharacteristic
83C23×C101—22no—
84C22×D101—21no—
85C22×D101—22no—
86C22×D101—44no—
87C22×D101—44no—
88C22×D101—82no—
89C22×D101C2×D412yes—
90Q8×C101—21no—
91Q8×C101C2×D412yescomplemented
92D4×C101—22no—
93D4×C101—22no—
94D4×C101—42no—
95D4×C101—81no—
96C2×C4×C101—21no—
97C2×C4×C101—21no—
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101C2411yescharacteristic
101C22⋊D101—44no—
102C22⋊D101—44no—
103C2×Dic101—21no—
104C2×Dic101C2×D412yes—
105D404—22no—
106D404—22no—
107D404—21no—
108D404—22no—
109D404—44no—
110D404—42no—
111D404—81no—
112D404—82no—
113D404C2×D412yes—
114C4×D101—22no—
115C4×D101—22no—
116C4×D101—44no—
117C4×D101—42no—
118C8×C101—22no—
119#11717579—21no—
120#11717579C2×D412yescomplemented
121C22×C101—42no—
122C22×C101—42no—
123C22×C101—42no—
124C22×C101C22×D411yescharacteristic
125C2×D101—22no—
126C2×D101—22no—
127C2×D101—22no—
128C2×D101—22no—
129C2×D101—41no—
130C2×D101—41no—
131C2×D101—42no—
132C2×D101—42no—
133C2×D101—42no—
134C2×D101—88no—
135C4×C101—22no—
136C4×C101—21no—
137C4×C101—41no—
138C4×C101AGL(1,ℤ/8)12yescomplemented
139C4×C101C22×D411yescharacteristic
140C4×C101H(5,𝔽2)11yescharacteristic
141Dic101—22no—
142Dic101—41no—
143C2×C101—82no—
144C2×C101D4211yescharacteristic
145C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C101#57611yescharacteristic
147D101—42no—
148D101—42no—
149D101—82no—
150#3274—1011nomaximal, Sylow
151C101#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1012no—
153#550—1011no—
154D42—1011no—
155#493—1011no—
156#469—1012no—
157#460D10111yescharacteristic, complemented
158#459—1011no—
159#449—1012no—
160#447—1012no—
161C4×SD16—1011no—
162#436—1011no—
163C2×#41—1012no—
164C22×D4—1012no—
165C22×D4—2021no—
166AGL(1,ℤ/8)—2028no—
167C2×SD16—1011no—
168C2×SD16C2×D10112yescomplemented
169C2×D8—1012no—
170C2×D8—1012no—
171C2×D8—1012no—
172C2×M16—1012no—
173#127—1011no—
174#127—2021no—
175#127C2×D10111yescharacteristic
176#121—1012no—
177#121—1012no—
178#121—2022no—
179#121—2021no—
180#120—2022no—
181C4×Q8C2×D10111yescharacteristic, complemented
182C4×D4—1011no—
183C4×D4—1011no—
184C4×D4—2021no—
185#106—1011no—
186C4⋊C8C2×D10111yescharacteristic, complemented
187#103—1011no—
188#102—1012no—
189#102—1011no—
190#102C2×D10112yescomplemented
191C22⋊C8—1012no—
192C4×C8—1011no—
193C24—2022no—
194#41—4042no—
195#41—2024no—
196C2×Q8C22×D10111yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—1011no—
200C2×D4—1011no—
201C2×D4—1012no—
202C2×D4—4044no—
203C2×D4—4042no—
204C2×D4—2022no—
205C2×D4—2022no—
206C2×D4—2022no—
207C2×D4—2024no—
208C2×D4—2022no—
209C2×D4—2021no—
210C2×D4—2021no—
211C2×D4C22×D10112yes—
212C22×C4—1012no—
213C22×C4—2022no—
214C22×C4—2021no—
215SD16—24no—
216SD16—1014no—
217D8—4042no—
218D8—2024no—
219D8—2024no—
220M16—2024no—
221C2×C8—1011no—
222C2×C8—1011no—
223C2×C8C22×D10112yes—
224C4⋊C4—21no—
225C4⋊C4—1011no—
226C4⋊C4—1011no—
227C4⋊C4C22×D10111yescharacteristic
228C22⋊C4—1012no—
229C22⋊C4—2022no—
230C22⋊C4—2022no—
231C22⋊C4—2021no—
232C42—21no—
233C42C22×D10111yescharacteristic
234C23—22no—
235C23—8082no—
236C23—1012no—
237C23—4044no—
238C23—4044no—
239C23—2022no—
240C23—2021no—
241Q8—21no—
242Q8D4×D10112yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—8081no—
248D4—8082no—
249D4—1012no—
250D4—4044no—
251D4—4044no—
252D4—4044no—
253D4—4042no—
254D4—2022no—
255D4—2022no—
256D4—2021no—
257D4—2022no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—1012no—
261C2×C4—4044no—
262C2×C4—4042no—
263C2×C4—2021no—
264C2×C4—2022no—
265C2×C4—2022no—
266C2×C4D4×D10111yescharacteristic
267C2×C4D4×D10111yescharacteristic
268C2×C4C23×D10111yescharacteristic
269C8—22no—
270C8—1012no—
271C8—2021no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—8088no—
276C22—4042no—
277C22—4042no—
278C22—4042no—
279C22—4041no—
280C22—4041no—
281C22—2022no—
282C22—2022no—
283C22—2022no—
284C22—2022no—
285C22C2×D4×D10111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—4041no—
290C4—2022no—
291C4#1569228212yescomplemented
292C4C2×D4×D10111yescharacteristic
293C4#1569230111yescharacteristic
294C2—82no—
295C2—8082no—
296C2—4042no—
297C2—4042no—
298C2#2026581111yescharacteristic
299C2C2×#1569228211yescharacteristic
300C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#460×C101C2
socleC22×C101C22×D4
hypercenter#460D101
hypocenterC101#3274
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. #20265849C2semidirect
3. #20265812C2semidirect
4. #20265553C2semidirect
5. #20265429C2semidirect
6. #20265373C2semidirect
7. #20265321C2semidirect
8. #20265312C2semidirect
9. C4×#13108061C2semidirect
10. #20265254C2semidirect
11. #460×C101C2semidirect
12. C2×#13108075C22semidirect
18. C2×#13108059C22semidirect
19. C2×#13108059C22semidirect
20. C2×#13108057C22semidirect
22. #15692252C22semidirect
24. #15692230C22semidirect
31. #15692186C22semidirect
32. #15692179C22semidirect
33. #15692177C22semidirect
35. #15692167C22semidirect
36. C4×#11717579C22semidirect
37. C2×SD16×C101C22semidirect
39. C4×Q8×C101C22semidirect
40. C4⋊C8×C101C22semidirect
41. #102×C101C22semidirect
60. #13108061D4semidirect
91. Q8×C101C2×D4semidirect
120. #11717579C2×D4semidirect
138. C4×C101AGL(1,ℤ/8)semidirect
151. C101#3274semidirect
157. #460D101semidirect
168. C2×SD16C2×D101semidirect
181. C4×Q8C2×D101semidirect
186. C4⋊C8C2×D101semidirect
190. #102C2×D101semidirect
242. Q8D4×D101semidirect
291. C4#15692282semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
#202658492
#202658122
#202655532
#202654292
#202653732
#202653212
#202653122
C4×#131080612
#202652542
#460×C1012
#3274101