Finite Group Databasebeta

← #27182735

Subgroups — #27182735

Order 12928.

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

normal subgroups only

This group has 546 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#20265830C211yesmaximal, characteristic, complemented
3D4×D404C211yesmaximal, characteristic, complemented
4#20265551C211yesmaximal, characteristic, complemented
5#20265426C212yesmaximal, complemented
6#20265373C211yesmaximal, characteristic, complemented
7#20265326C212yesmaximal, complemented
8#20265325C212yesmaximal, complemented
9C4×#13108059C211yesmaximal, characteristic, complemented
10#20265254C211yesmaximal, characteristic, complemented
11#461×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2212yescomplemented
13C22×D404—21no—
14C2×D4×D101C2212yes—
15#15692282—28no—
16C2×#13108061C2212yescomplemented
17C2×#13108061C2212yescomplemented
18C2×D808C2212yescomplemented
19C2×#13108059C2211yescharacteristic
20C2×#13108057C2212yescomplemented
21#15692252C2211yescharacteristic
22#15692240—22no—
23#15692231C2212yescomplemented
24#15692230—21no—
25#15692230C2212yes—
26#15692229—21no—
27#15692225—22no—
28C4×D404C2211yescharacteristic
29C4×D404C2211yescharacteristic, complemented
30C4⋊C4×D101—21no—
31#15692188C2211yescharacteristic, complemented
32#15692177C2212yescomplemented
33#15692176C2211yescharacteristic
34#15692175C2211yescharacteristic, complemented
35#15692167C2212yescomplemented
36C4×#11717579C2211yescharacteristic, complemented
37C2×SD16×C101C2212yes—
38C4⋊Q8×C101C2211yescharacteristic, complemented
39C4×D4×C101C2211yescharacteristic
40C4⋊C8×C101C2211yescharacteristic, complemented
41#103×C101C2212yescomplemented
42C23×D101—22no—
43#13108075—24no—
44#13108075—42no—
45C2×D404—22no—
46C2×D404—22no—
47C2×D404—21no—
48C2×D404—21no—
49C2×D404—44no—
50C2×D404D411yescharacteristic
51C2×D404C2312yes—
52C2×D404C2311yescharacteristic
53C2×C22⋊D101—22no—
54D4×D101—24no—
55D4×D101—42no—
56C2×C4×D101—21no—
57C2×C4×D101—22no—
58C2×C4×D101C2312yes—
59#13108061—24no—
60#13108061—42no—
61D808—24no—
62#13108059D414yescomplemented
63#13108057—24no—
64C2×#11717579C2311yescharacteristic
65C2×#11717579C2311yescharacteristic
66C4⋊Dic101D411yescharacteristic
67C4⋊Dic101C2311yescharacteristic
68#13108053—22no—
69#13108051—22no—
70#13108051—21no—
71#13108051—22no—
72#13108051D412yes—
73C2×Q8×C101C2312yes—
74C2×D4×C101C2311yescharacteristic
75C22×C4×C101—21no—
76SD16×C101—24no—
77C2×C8×C101C2312yes—
78C4⋊C4×C101—21no—
79C4⋊C4×C101—21no—
80C4⋊C4×C101C2311yescharacteristic
81C22⋊C4×C101—21no—
82C42×C101C2311yescharacteristic
83C23×C101—21no—
84C22×D101—21no—
85C22×D101—22no—
86C22×D101—44no—
87C22×D101—44no—
88C22×D101—82no—
89C22×D101C2×D412yes—
90Q8×C101—22no—
91Q8×C101—22no—
92D4×C101—21no—
93D4×C101C2×D412yes—
94C2×C4×C101—21no—
95C2×C4×C101—22no—
96C2×C4×C101—42no—
97C2×C4×C101C2×D411yescharacteristic
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2411yescharacteristic
100C22⋊D101—44no—
101C2×Dic101—21no—
102C2×Dic101C2×D412yes—
103D404—22no—
104D404—22no—
105D404—21no—
106D404—22no—
107D404—44no—
108D404—44no—
109D404—42no—
110D404—81no—
111D404C2×D412yescomplemented
112C4×D101—22no—
113C4×D101—22no—
114C4×D101—44no—
115C4×D101—42no—
116C8×C101—22no—
117#11717579—21no—
118#11717579C2×D412yescomplemented
119C22×C101—22no—
120C22×C101—41no—
121C22×C101—41no—
122C22×C101C22×D411yescharacteristic
123C2×D101—22no—
124C2×D101—22no—
125C2×D101—22no—
126C2×D101—22no—
127C2×D101—41no—
128C2×D101—41no—
129C2×D101—42no—
130C2×D101—42no—
131C2×D101—42no—
132C2×D101—88no—
133C4×C101—21no—
134C4×C101—42no—
135C4×C101—41no—
136C4×C101AGL(1,ℤ/8)12yescomplemented
137C4×C101C22×D411yescharacteristic
138C4×C101H(5,𝔽2)11yescharacteristic
139Dic101—22no—
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#461D10111yescharacteristic, complemented
156#459—1011no—
157#450—1012no—
158#449—1012no—
159C4×D8—1011no—
160#436—1011no—
161C2×#41—1012no—
162C22×D4—1012no—
163C22×D4—2021no—
164AGL(1,ℤ/8)—2028no—
165C2×SD16—1012no—
166C2×SD16—1012no—
167C2×SD16C2×D10112yes—
168C2×D8—1011no—
169C2×D8—1012no—
170C2×M16—1012no—
171C4⋊Q8C2×D10111yescharacteristic, complemented
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#103C2×D10112yescomplemented
186#102—1011no—
187#102—1012no—
188#102—1011no—
189C22⋊C8—1012no—
190C4×C8—1011no—
191C24—2022no—
192#41—4042no—
193#41—2024no—
194C2×Q8C22×D10112yes—
195C2×D4—1011no—
196C2×D4—1011no—
197C2×D4—1012no—
198C2×D4—4044no—
199C2×D4—4042no—
200C2×D4—2022no—
201C2×D4—2022no—
202C2×D4—2022no—
203C2×D4—2024no—
204C2×D4—2021no—
205C2×D4—2021no—
206C2×D4C22×D10111yescharacteristic
207C22×C4—21no—
208C22×C4—1012no—
209C22×C4—2022no—
210C22×C4—2021no—
211SD16—24no—
212SD16—4042no—
213SD16—2024no—
214D8—1014no—
215D8—2024no—
216M16—2024no—
217C2×C8—1011no—
218C2×C8—1011no—
219C2×C8C22×D10112yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—1011no—
223C4⋊C4—1011no—
224C4⋊C4—2022no—
225C4⋊C4C22×D10111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1012no—
228C22⋊C4—2022no—
229C22⋊C4—2022no—
230C22⋊C4—2021no—
231C42C22×D10111yescharacteristic
232C23—21no—
233C23—8082no—
234C23—1012no—
235C23—4044no—
236C23—4044no—
237C23—2022no—
238C23—2021no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—8081no—
243D4—1012no—
244D4—4044no—
245D4—4044no—
246D4—4044no—
247D4—4042no—
248D4—2022no—
249D4—2022no—
250D4—2021no—
251D4—2022no—
252D4D4×D10112yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—1012no—
257C2×C4—4044no—
258C2×C4—4042no—
259C2×C4—2021no—
260C2×C4—2022no—
261C2×C4—2022no—
262C2×C4D4×D10111yescharacteristic
263C2×C4D4×D10111yescharacteristic
264C2×C4C23×D10111yescharacteristic
265C8—22no—
266C8—1012no—
267C8—2021no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—8088no—
272C22—4042no—
273C22—4042no—
274C22—4042no—
275C22—4041no—
276C22—4041no—
277C22—2022no—
278C22—2022no—
279C22—2022no—
280C22—2022no—
281C22C2×D4×D10111yescharacteristic
282C4—21no—
283C4—42no—
284C4—41no—
285C4—4041no—
286C4—2022no—
287C4#1569228212yescomplemented
288C4C2×D4×D10111yescharacteristic
289C4#1569230111yescharacteristic
290C2—42no—
291C2—8082no—
292C2—4042no—
293C2—4042no—
294C2#2026581111yescharacteristic
295C2C2×#1569228211yescharacteristic
296C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#461×C101C2
socleC22×C101C22×D4
hypercenter#461D101
hypocenterC101#3272
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. #20265830C2semidirect
3. D4×D404C2semidirect
4. #20265551C2semidirect
5. #20265426C2semidirect
6. #20265373C2semidirect
7. #20265326C2semidirect
8. #20265325C2semidirect
9. C4×#13108059C2semidirect
10. #20265254C2semidirect
11. #461×C101C2semidirect
12. C2×#13108075C22semidirect
16. C2×#13108061C22semidirect
17. C2×#13108061C22semidirect
18. C2×D808C22semidirect
20. C2×#13108057C22semidirect
23. #15692231C22semidirect
29. C4×D404C22semidirect
31. #15692188C22semidirect
32. #15692177C22semidirect
34. #15692175C22semidirect
35. #15692167C22semidirect
36. C4×#11717579C22semidirect
38. C4⋊Q8×C101C22semidirect
40. C4⋊C8×C101C22semidirect
41. #103×C101C22semidirect
62. #13108059D4semidirect
111. D404C2×D4semidirect
118. #11717579C2×D4semidirect
136. C4×C101AGL(1,ℤ/8)semidirect
149. C101#3272semidirect
155. #461D101semidirect
171. C4⋊Q8C2×D101semidirect
184. C4⋊C8C2×D101semidirect
185. #103C2×D101semidirect
287. C4#15692282semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
#202658302
D4×D4042
#202655512
#202654262
#202653732
#202653262
#202653252
C4×#131080592
#202652542
#461×C1012
#3272101