Finite Group Databasebeta

← #27182754

Subgroups — #27182754

Order 12928.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15692283C212yesmaximal, complemented
2#20265848C211yesmaximal, characteristic, complemented
3D4×D404C211yesmaximal, characteristic, complemented
4#20265554C211yesmaximal, characteristic, complemented
5#20265430C212yesmaximal, complemented
6#20265378C211yesmaximal, characteristic, complemented
7#20265373C211yesmaximal, characteristic, complemented
8#20265324C212yesmaximal, complemented
9#20265313C212yesmaximal, complemented
10C4×#13108063C211yesmaximal, characteristic, complemented
11#436×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2212yescomplemented
13C22×D404C2212yes—
14C2×D4×D101—21no—
15#15692283—28no—
16C2×#13108063C2211yescharacteristic
17C2×#13108063C2212yescomplemented
18C2×D808C2212yes—
19C2×D808C2212yescomplemented
20C2×D808C2212yescomplemented
21#15692252C2211yescharacteristic
22#15692252C2211yescharacteristic, complemented
23#15692251—21no—
24#15692230—21no—
25#15692229C2212yes—
26#15692229C2212yescomplemented
27#15692228—22no—
28#15692225—22no—
29C4×Dic202C2211yescharacteristic, complemented
30C4×D404C2211yescharacteristic
31#15692217—21no—
32#15692187C2211yescharacteristic, complemented
33#15692180C2211yescharacteristic, complemented
34#15692177C2211yescharacteristic
35#15692177C2212yescomplemented
36#15692177C2212yescomplemented
37C2×M16×C101C2212yescomplemented
38C4×D4×C101C2211yescharacteristic
39C4⋊C8×C101C2211yescharacteristic, complemented
40C22⋊C8×C101C2212yescomplemented
41C4×C8×C101C2211yescharacteristic, complemented
42C23×D101—22no—
43#13108074—24no—
44#13108074—42no—
45C2×Dic202C2311yescharacteristic
46C2×D404—24no—
47C2×D404—21no—
48C2×D404—22no—
49C2×D404—21no—
50C2×D404—21no—
51C2×D404—21no—
52C2×D404—42no—
53C2×D404C2312yes—
54C2×D404C2312yes—
55C2×D404C2311yescharacteristic
56C2×C22⋊D101—22no—
57C2×C22⋊D101—22no—
58C2×C22⋊D101—22no—
59D4×D101—44no—
60C2×C4×D101—21no—
61C2×C4×D101—22no—
62#13108063—24no—
63#13108063D414yescomplemented
64D808—24no—
65D808—24no—
66D808—42no—
67C4⋊Dic101C2311yescharacteristic
68C4⋊Dic101C2311yescharacteristic
69#13108053—21no—
70#13108051—22no—
71#13108051—21no—
72#13108051—22no—
73C4×Dic101—21no—
74C2×D4×C101D411yescharacteristic
75C22×C4×C101C2312yes—
76M16×C101—24no—
77C2×C8×C101C2311yescharacteristic
78C2×C8×C101C2312yes—
79C2×C8×C101C2311yescharacteristic
80C4⋊C4×C101D411yescharacteristic
81C22⋊C4×C101D412yes—
82C42×C101C2311yescharacteristic
83C23×C101C2×D412yes—
84C22×D101—21no—
85C22×D101—22no—
86C22×D101—22no—
87C22×D101—44no—
88C22×D101—44no—
89C22×D101—82no—
90D4×C101—22no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D412yes—
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2411yescharacteristic
97C22⋊D101—44no—
98C22⋊D101—44no—
99C22⋊D101—82no—
100C2×Dic101—21no—
101C2×Dic101—21no—
102C2×Dic101—21no—
103D404—22no—
104D404—22no—
105D404—21no—
106D404—22no—
107D404—22no—
108D404—44no—
109D404—42no—
110D404—42no—
111D404—81no—
112D404—81no—
113D404C2×D412yes—
114C4×D101—44no—
115C4×D101—42no—
116Dic202—21no—
117Dic202C2×D412yescomplemented
118C8×C101—22no—
119C8×C101—21no—
120C8×C101C2×D412yescomplemented
121C22×C101—22no—
122C22×C101—22no—
123C22×C101—22no—
124C22×C101C22×D411yescharacteristic
125C2×D101—22no—
126C2×D101—41no—
127C2×D101—41no—
128C2×D101—42no—
129C2×D101—42no—
130C2×D101—42no—
131C2×D101—42no—
132C2×D101—42no—
133C2×D101—42no—
134C2×D101—88no—
135C4×C101—22no—
136C4×C101—21no—
137C4×C101AGL(1,ℤ/8)12yescomplemented
138C4×C101C22×D411yescharacteristic
139C4×C101H(5,𝔽2)11yescharacteristic
140Dic101—22no—
141Dic101—41no—
142Dic101—41no—
143C2×C101—42no—
144C2×C101D4211yescharacteristic
145C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C101#57611yescharacteristic
147D101—42no—
148D101—82no—
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#460—1011no—
158#459—1011no—
159#449—1012no—
160#447—1012no—
161C4×SD16—1011no—
162#436D10111yescharacteristic, complemented
163C2×#41—1012no—
164C22×D4—1012no—
165C22×D4—2021no—
166AGL(1,ℤ/8)—2028no—
167C2×SD16—1011no—
168C2×SD16—1012no—
169C2×D8—1012no—
170C2×D8—1012no—
171C2×D8—1012no—
172C2×M16C2×D10112yescomplemented
173#127—1011no—
174#127—1011no—
175#127—2021no—
176#121—1012no—
177#121—1012no—
178#121—2021no—
179#121—2022no—
180#120—2022no—
181C4×Q8—1011no—
182C4×D4—1011no—
183C4×D4—2021no—
184C4×D4C2×D10111yescharacteristic
185#106—1011no—
186C4⋊C8C2×D10111yescharacteristic, complemented
187#103—1011no—
188#102—1011no—
189#102—1012no—
190#102—1012no—
191C22⋊C8C2×D10112yescomplemented
192C4×C8C2×D10111yescharacteristic, complemented
193C24—2022no—
194#41—4042no—
195#41—2024no—
196C2×Q8—1011no—
197C2×D4—1011no—
198C2×D4—1012no—
199C2×D4—1012no—
200C2×D4—4042no—
201C2×D4—4044no—
202C2×D4—2024no—
203C2×D4—2022no—
204C2×D4—2022no—
205C2×D4—2022no—
206C2×D4—2021no—
207C2×D4—2021no—
208C2×D4—2021no—
209C2×D4—2022no—
210C2×D4—2021no—
211C2×D4D40411yescharacteristic
212C22×C4—2021no—
213C22×C4—2022no—
214C22×C4C22×D10112yes—
215SD16—1014no—
216SD16—2024no—
217D8—4042no—
218D8—2024no—
219D8—2024no—
220M16—24no—
221C2×C8C22×D10111yescharacteristic
222C2×C8C22×D10112yes—
223C2×C8C22×D10111yescharacteristic
224C4⋊C4—1011no—
225C4⋊C4—1011no—
226C4⋊C4—2021no—
227C4⋊C4D40411yescharacteristic
228C22⋊C4—2021no—
229C22⋊C4—2022no—
230C22⋊C4—2022no—
231C22⋊C4D40412yes—
232C42—2021no—
233C42C22×D10111yescharacteristic
234C23—8082no—
235C23—4044no—
236C23—4044no—
237C23—2022no—
238C23—2022no—
239C23—2021no—
240C23C2×D40412yes—
241Q8—1012no—
242Q8—2021no—
243D4—22no—
244D4—8081no—
245D4—8081no—
246D4—8082no—
247D4—1012no—
248D4—4044no—
249D4—4044no—
250D4—4044no—
251D4—4042no—
252D4—4042no—
253D4—2022no—
254D4—2022no—
255D4—2022no—
256D4—2022no—
257D4—2021no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—4044no—
261C2×C4—4042no—
262C2×C4—2021no—
263C2×C4—2021no—
264C2×C4—2021no—
265C2×C4C2×D40411yescharacteristic
266C2×C4C2×D40412yes—
267C2×C4C2×D40411yescharacteristic
268C2×C4C23×D10111yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D10112yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—8088no—
276C22—4042no—
277C22—4042no—
278C22—4042no—
279C22—4042no—
280C22—4042no—
281C22—4042no—
282C22—4041no—
283C22—4041no—
284C22—2022no—
285C22C22×D40411yescharacteristic
286C4—22no—
287C4—21no—
288C4—4041no—
289C4—4041no—
290C4—2022no—
291C4#1569228312yescomplemented
292C4C2×D4×D10111yescharacteristic
293C4#1569230111yescharacteristic
294C2—42no—
295C2—8082no—
296C2—8082no—
297C2—4042no—
298C2D4×D40411yescharacteristic
299C2C2×#1569228311yescharacteristic
300C2#2026598211yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#436×C101C2
socleC22×C101C22×D4
hypercenter#436D101
hypocenterC101#3274
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#15692283C2semidirect
2. #20265848C2semidirect
3. D4×D404C2semidirect
4. #20265554C2semidirect
5. #20265430C2semidirect
6. #20265378C2semidirect
7. #20265373C2semidirect
8. #20265324C2semidirect
9. #20265313C2semidirect
10. C4×#13108063C2semidirect
11. #436×C101C2semidirect
12. C2×#13108074C22semidirect
17. C2×#13108063C22semidirect
19. C2×D808C22semidirect
20. C2×D808C22semidirect
22. #15692252C22semidirect
26. #15692229C22semidirect
29. C4×Dic202C22semidirect
32. #15692187C22semidirect
33. #15692180C22semidirect
35. #15692177C22semidirect
36. #15692177C22semidirect
37. C2×M16×C101C22semidirect
39. C4⋊C8×C101C22semidirect
40. C22⋊C8×C101C22semidirect
41. C4×C8×C101C22semidirect
63. #13108063D4semidirect
117. Dic202C2×D4semidirect
120. C8×C101C2×D4semidirect
137. C4×C101AGL(1,ℤ/8)semidirect
151. C101#3274semidirect
162. #436D101semidirect
172. C2×M16C2×D101semidirect
186. C4⋊C8C2×D101semidirect
191. C22⋊C8C2×D101semidirect
192. C4×C8C2×D101semidirect
271. C8D4×D101semidirect
291. C4#15692283semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
#202658482
D4×D4042
#202655542
#202654302
#202653782
#202653732
#202653242
#202653132
C4×#131080632
#436×C1012
#3274101