Finite Group Databasebeta

← #27182734

Subgroups — #27182734

Order 12928.

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

normal subgroups only

This group has 546 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#20265832C211yesmaximal, characteristic, complemented
3D4×D404C211yesmaximal, characteristic, complemented
4#20265550C211yesmaximal, characteristic, complemented
5#20265423C212yesmaximal, complemented
6#20265383C211yesmaximal, characteristic, complemented
7#20265373C211yesmaximal, characteristic, complemented
8#20265330C212yesmaximal, complemented
9#20265324C212yesmaximal, complemented
10C4×D808C211yesmaximal, characteristic, complemented
11#436×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2212yescomplemented
13C22×D404C2212yes—
14C2×D4×D101—21no—
15#15692283—28no—
16C2×#13108063C2212yes—
17C2×#13108063C2212yescomplemented
18C2×#13108063C2212yescomplemented
19C2×D808C2211yescharacteristic
20C2×D808C2212yescomplemented
21#15692255C2211yescharacteristic, complemented
22#15692252C2211yescharacteristic
23#15692237—22no—
24#15692234C2212yescomplemented
25#15692230—21no—
26#15692229C2212yes—
27#15692226—21no—
28#15692225—22no—
29C4×D404C2211yescharacteristic
30C4×D404C2211yescharacteristic, complemented
31C4⋊C4×D101—21no—
32#15692189C2211yescharacteristic, complemented
33#15692180C2212yescomplemented
34#15692177C2211yescharacteristic
35#15692177C2211yescharacteristic, complemented
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×Dic202C2312yes—
46C2×D404—24no—
47C2×D404—21no—
48C2×D404—22no—
49C2×D404—21no—
50C2×D404—42no—
51C2×D404C2312yes—
52C2×D404C2311yescharacteristic
53C2×D404C2311yescharacteristic
54C2×C22⋊D101—22no—
55C2×C22⋊D101—22no—
56D4×D101—44no—
57C2×C4×D101—21no—
58C2×C4×D101—22no—
59C2×C4×D101—21no—
60#13108063—24no—
61#13108063—24no—
62#13108063—42no—
63D808—24no—
64D808D414yescomplemented
65C4⋊Dic101—21no—
66C4⋊Dic101—21no—
67C4⋊Dic101C2311yescharacteristic
68C4⋊Dic101C2311yescharacteristic
69#13108053—22no—
70C22⋊Dic101—22no—
71#13108051—21no—
72#13108051—21no—
73#13108051—22no—
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—21no—
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—
99C2×Dic101—21no—
100C2×Dic101—21no—
101C2×Dic101—22no—
102D404—22no—
103D404—21no—
104D404—21no—
105D404—22no—
106D404—44no—
107D404—42no—
108D404—81no—
109D404C2×D412yes—
110D404C2×D412yescomplemented
111C4×D101—44no—
112C4×D101—42no—
113C4×D101—42no—
114Dic202—22no—
115Dic202—22no—
116C8×C101—22no—
117C8×C101—21no—
118C8×C101C2×D412yescomplemented
119C22×C101—22no—
120C22×C101—22no—
121C22×C101—22no—
122C22×C101C22×D411yescharacteristic
123C2×D101—22no—
124C2×D101—22no—
125C2×D101—41no—
126C2×D101—41no—
127C2×D101—41no—
128C2×D101—41no—
129C2×D101—42no—
130C2×D101—42no—
131C2×D101—42no—
132C2×D101—88no—
133C4×C101—22no—
134C4×C101—21no—
135C4×C101AGL(1,ℤ/8)12yescomplemented
136C4×C101C22×D411yescharacteristic
137C4×C101H(5,𝔽2)11yescharacteristic
138Dic101—42no—
139Dic101—41no—
140Dic101—41no—
141C2×C101—42no—
142C2×C101D4211yescharacteristic
143C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C101#57711yescharacteristic
145D101—42no—
146D101—42no—
147D101—82no—
148#3272—1011nomaximal, Sylow
149C101#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1012no—
151#547—1011no—
152D42—1011no—
153#492—1011no—
154#468—1012no—
155#461—1011no—
156#459—1011no—
157#450—1012no—
158#449—1012no—
159C4×D8—1011no—
160#436D10111yescharacteristic, complemented
161C2×#41—1012no—
162C22×D4—1012no—
163C22×D4—2021no—
164AGL(1,ℤ/8)—2028no—
165C2×SD16—1012no—
166C2×SD16—1012no—
167C2×SD16—1012no—
168C2×D8—1011no—
169C2×D8—1012no—
170C2×M16C2×D10112yescomplemented
171C4⋊Q8—1011no—
172#127—1011no—
173#123—2022no—
174C22⋊Q8—1012no—
175#121—1012no—
176#121—2021no—
177#121—2021no—
178#120—2022no—
179C4×D4—1011no—
180C4×D4—1011no—
181C4×D4C2×D10111yescharacteristic
182C2×C4⋊C4—2021no—
183#107—1011no—
184C4⋊C8C2×D10111yescharacteristic, complemented
185#103—1012no—
186#102—1011no—
187#102—1011no—
188#102—1012no—
189C22⋊C8C2×D10112yescomplemented
190C4×C8C2×D10111yescharacteristic, complemented
191C24—2022no—
192#41—4042no—
193#41—2024no—
194C2×Q8—1012no—
195C2×D4—1011no—
196C2×D4—1011no—
197C2×D4—1012no—
198C2×D4—4042no—
199C2×D4—4044no—
200C2×D4—2024no—
201C2×D4—2022no—
202C2×D4—2022no—
203C2×D4—2021no—
204C2×D4—2022no—
205C2×D4—2021no—
206C2×D4D40411yescharacteristic
207C22×C4—2021no—
208C22×C4—2022no—
209C22×C4—2021no—
210C22×C4C22×D10112yes—
211SD16—4042no—
212SD16—2024no—
213SD16—2024no—
214D8—1014no—
215D8—2024no—
216M16—24no—
217C2×C8C22×D10111yescharacteristic
218C2×C8C22×D10112yes—
219C2×C8C22×D10111yescharacteristic
220C4⋊C4—1011no—
221C4⋊C4—1011no—
222C4⋊C4—2022no—
223C4⋊C4—2021no—
224C4⋊C4—2021no—
225C4⋊C4D40411yescharacteristic
226C22⋊C4—2022no—
227C22⋊C4—2021no—
228C22⋊C4—2021no—
229C22⋊C4—2022no—
230C22⋊C4D40412yes—
231C42C22×D10111yescharacteristic
232C23—8082no—
233C23—4044no—
234C23—4044no—
235C23—2022no—
236C23—2021no—
237C23—2021no—
238C23C2×D40412yes—
239Q8—2022no—
240Q8—2022no—
241D4—22no—
242D4—8081no—
243D4—1012no—
244D4—1012no—
245D4—4044no—
246D4—4044no—
247D4—4044no—
248D4—4042no—
249D4—2022no—
250D4—2022no—
251D4—2021no—
252D4—2021no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—4044no—
256C2×C4—4042no—
257C2×C4—4042no—
258C2×C4—2021no—
259C2×C4—2021no—
260C2×C4—2022no—
261C2×C4C2×D40411yescharacteristic
262C2×C4C2×D40412yes—
263C2×C4C2×D40411yescharacteristic
264C2×C4C23×D10111yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D10112yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—8088no—
272C22—4042no—
273C22—4042no—
274C22—4042no—
275C22—4041no—
276C22—4041no—
277C22—4041no—
278C22—4041no—
279C22—2022no—
280C22—2022no—
281C22C22×D40411yescharacteristic
282C4—21no—
283C4—22no—
284C4—4042no—
285C4—4041no—
286C4—4041no—
287C4#1569228312yescomplemented
288C4C2×D4×D10111yescharacteristic
289C4#1569230111yescharacteristic
290C2—42no—
291C2—8082no—
292C2—4042no—
293C2—4042no—
294C2D4×D40411yescharacteristic
295C2C2×#1569228311yescharacteristic
296C2#2026598911yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#436×C101C2
socleC22×C101C22×D4
hypercenter#436D101
hypocenterC101#3272
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. #20265832C2semidirect
3. D4×D404C2semidirect
4. #20265550C2semidirect
5. #20265423C2semidirect
6. #20265383C2semidirect
7. #20265373C2semidirect
8. #20265330C2semidirect
9. #20265324C2semidirect
10. C4×D808C2semidirect
11. #436×C101C2semidirect
12. C2×#13108074C22semidirect
17. C2×#13108063C22semidirect
18. C2×#13108063C22semidirect
20. C2×D808C22semidirect
21. #15692255C22semidirect
24. #15692234C22semidirect
30. C4×D404C22semidirect
32. #15692189C22semidirect
33. #15692180C22semidirect
35. #15692177C22semidirect
36. #15692177C22semidirect
37. C2×M16×C101C22semidirect
39. C4⋊C8×C101C22semidirect
40. C22⋊C8×C101C22semidirect
41. C4×C8×C101C22semidirect
64. D808D4semidirect
110. D404C2×D4semidirect
118. C8×C101C2×D4semidirect
135. C4×C101AGL(1,ℤ/8)semidirect
149. C101#3272semidirect
160. #436D101semidirect
170. C2×M16C2×D101semidirect
184. C4⋊C8C2×D101semidirect
189. C22⋊C8C2×D101semidirect
190. C4×C8C2×D101semidirect
267. C8D4×D101semidirect
287. C4#15692283semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
#202658322
D4×D4042
#202655502
#202654232
#202653832
#202653732
#202653302
#202653242
C4×D8082
#436×C1012
#3272101