Finite Group Databasebeta

← #27182733

Subgroups — #27182733

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×#15692279C212yesmaximal, complemented
2#20265831C211yesmaximal, characteristic, complemented
3D4×D404C211yesmaximal, characteristic, complemented
4#20265550C211yesmaximal, characteristic, complemented
5#20265424C212yesmaximal, complemented
6#20265382C211yesmaximal, characteristic, complemented
7#20265372C211yesmaximal, characteristic, complemented
8#20265329C212yesmaximal, complemented
9#20265323C212yesmaximal, complemented
10#20265256C211yesmaximal, characteristic, complemented
11C4×D8×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2212yescomplemented
13C22×D404—21no—
14C2×D4×D101C2212yes—
15#15692279—28no—
16C2×#13108063C2212yescomplemented
17C2×#13108063C2212yescomplemented
18C2×#13108062C2212yes—
19C2×#13108059C2212yescomplemented
20C2×#13108057C2212yescomplemented
21#15692255C2211yescharacteristic, complemented
22#15692252C2211yescharacteristic
23#15692239—22no—
24#15692235C2212yescomplemented
25#15692230C2212yes—
26#15692229—21no—
27#15692229—21no—
28#15692225—22no—
29C4×D404C2211yescharacteristic
30C2×C4⋊Dic101—21no—
31C4⋊#11717579C2211yescharacteristic, complemented
32#15692178C2212yescomplemented
33#15692175C2212yescomplemented
34#15692167C2212yescomplemented
35C2×D8×C101C2211yescharacteristic
36C4×D4×C101C2211yescharacteristic
37C4×D4×C101C2211yescharacteristic, complemented
38#107×C101C2211yescharacteristic, complemented
39#102×C101C2211yescharacteristic
40#102×C101C2211yescharacteristic, complemented
41C4×C8×C101C2211yescharacteristic, complemented
42C23×D101—22no—
43#13108073—24no—
44#13108073—42no—
45C2×Dic202C2312yes—
46C2×D404—22no—
47C2×D404—21no—
48C2×D404—21no—
49C2×D404—44no—
50C2×D404D411yescharacteristic
51C2×D404C2312yes—
52C2×C22⋊D101—22no—
53C2×C22⋊D101—22no—
54D4×D101—24no—
55D4×D101—42no—
56C22×Dic101—22no—
57C2×C4×D101C2312yes—
58#13108063—24no—
59#13108063—42no—
60#13108062—24no—
61#13108059—24no—
62#13108057—24no—
63C2×#11717579C2312yes—
64C4⋊Dic101—21no—
65C4⋊Dic101—21no—
66C4⋊Dic101—22no—
67C4⋊Dic101D411yescharacteristic
68#13108051—22no—
69#13108051—22no—
70#13108051D412yes—
71C2×D4×C101C2311yescharacteristic
72C2×D4×C101C2311yescharacteristic
73C22×C4×C101—21no—
74C22×C4×C101—21no—
75D8×C101D414yescomplemented
76C2×C8×C101C2311yescharacteristic
77C2×C8×C101C2311yescharacteristic
78C4⋊C4×C101C2311yescharacteristic
79C4⋊C4×C101C2311yescharacteristic
80C22⋊C4×C101—21no—
81C22⋊C4×C101—21no—
82C42×C101C2311yescharacteristic
83C23×C101—21no—
84C23×C101—21no—
85C22×D101—22no—
86C22×D101—44no—
87C22×D101—44no—
88C22×D101—82no—
89C22×D101C2×D412yes—
90D4×C101—21no—
91D4×C101—21no—
92D4×C101C2×D412yes—
93D4×C101C2×D412yescomplemented
94C2×C4×C101—21no—
95C2×C4×C101—21no—
96C2×C4×C101—42no—
97C2×C4×C101—42no—
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101C2411yescharacteristic
101C22⋊D101—44no—
102C22⋊D101—44no—
103C2×Dic101—22no—
104C2×Dic101—44no—
105C2×Dic101C2×D412yes—
106D404—22no—
107D404—22no—
108D404—22no—
109D404—44no—
110D404—42no—
111D404—81no—
112C4×D101—22no—
113C4×D101—22no—
114Dic202—22no—
115Dic202—22no—
116C8×C101—21no—
117C8×C101C2×D412yescomplemented
118#11717579—22no—
119C22×C101—22no—
120C22×C101—22no—
121C22×C101—41no—
122C22×C101—41no—
123C22×C101—41no—
124C22×C101—41no—
125C22×C101C22×D411yescharacteristic
126C2×D101—22no—
127C2×D101—22no—
128C2×D101—22no—
129C2×D101—42no—
130C2×D101—42no—
131C2×D101—42no—
132C2×D101—88no—
133C4×C101—21no—
134C4×C101—41no—
135C4×C101—41no—
136C4×C101AGL(1,ℤ/8)12yescomplemented
137C4×C101C22×D411yescharacteristic
138C4×C101H(5,𝔽2)11yescharacteristic
139Dic101—22no—
140Dic101—42no—
141C2×C101—42no—
142C2×C101—42no—
143C2×C101D4211yescharacteristic
144C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C101#57711yescharacteristic
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×D8D10111yescharacteristic, complemented
160#436—1011no—
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—1012no—
169C2×D8C2×D10111yescharacteristic
170C2×M16—1012no—
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×D4C2×D10111yescharacteristic
181C4×D4C2×D10111yescharacteristic, complemented
182C2×C4⋊C4—2021no—
183#107C2×D10111yescharacteristic, complemented
184C4⋊C8—1011no—
185#103—1012no—
186#102—1012no—
187#102C2×D10111yescharacteristic
188#102C2×D10111yescharacteristic, complemented
189C22⋊C8—1012no—
190C4×C8C2×D10111yescharacteristic, complemented
191C24—2022no—
192#41—4042no—
193#41—2024no—
194C2×Q8—1012no—
195C2×D4—1011no—
196C2×D4—1012no—
197C2×D4—4044no—
198C2×D4—4042no—
199C2×D4—2022no—
200C2×D4—2022no—
201C2×D4—2022no—
202C2×D4—2024no—
203C2×D4—2021no—
204C2×D4—2021no—
205C2×D4C22×D10111yescharacteristic
206C2×D4C22×D10111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—1012no—
210C22×C4—2022no—
211SD16—4042no—
212SD16—2024no—
213SD16—2024no—
214D8—2024no—
215D8D40414yescomplemented
216M16—2024no—
217C2×C8—1012no—
218C2×C8C22×D10111yescharacteristic
219C2×C8C22×D10111yescharacteristic
220C4⋊C4—1011no—
221C4⋊C4—2022no—
222C4⋊C4—2021no—
223C4⋊C4—2021no—
224C4⋊C4C22×D10111yescharacteristic
225C4⋊C4C22×D10111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—1012no—
229C22⋊C4—2022no—
230C22⋊C4—2022no—
231C42C22×D10111yescharacteristic
232C23—21no—
233C23—21no—
234C23—8082no—
235C23—1012no—
236C23—4044no—
237C23—4044no—
238C23—2022no—
239Q8—2022no—
240Q8—2022no—
241D4—21no—
242D4—21no—
243D4—8081no—
244D4—4044no—
245D4—4044no—
246D4—4044no—
247D4—4042no—
248D4—2022no—
249D4—2022no—
250D4—2022no—
251D4C2×D40412yes—
252D4C2×D40412yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—1012no—
258C2×C4—4044no—
259C2×C4—2022no—
260C2×C4—2022no—
261C2×C4—2022no—
262C2×C4D4×D10111yescharacteristic
263C2×C4D4×D10111yescharacteristic
264C2×C4C23×D10111yescharacteristic
265C8—21no—
266C8—2022no—
267C8C2×D40412yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—8088no—
275C22—4042no—
276C22—4042no—
277C22—4042no—
278C22—2022no—
279C22—2022no—
280C22—2022no—
281C22C2×D4×D10111yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—4042no—
286C4—2022no—
287C4#1569227912yescomplemented
288C4C22×D40411yescharacteristic
289C4#1569230111yescharacteristic
290C2—42no—
291C2—42no—
292C2—8082no—
293C2—4042no—
294C2D4×D40411yescharacteristic
295C2C2×#1569227911yescharacteristic
296C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×D8×C101C2
socleC22×C101C22×D4
hypercenterC4×D8D101
hypocenterC101#3272
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. #20265831C2semidirect
3. D4×D404C2semidirect
4. #20265550C2semidirect
5. #20265424C2semidirect
6. #20265382C2semidirect
7. #20265372C2semidirect
8. #20265329C2semidirect
9. #20265323C2semidirect
10. #20265256C2semidirect
11. C4×D8×C101C2semidirect
12. C2×#13108073C22semidirect
16. C2×#13108063C22semidirect
17. C2×#13108063C22semidirect
19. C2×#13108059C22semidirect
20. C2×#13108057C22semidirect
21. #15692255C22semidirect
24. #15692235C22semidirect
31. C4⋊#11717579C22semidirect
32. #15692178C22semidirect
33. #15692175C22semidirect
34. #15692167C22semidirect
37. C4×D4×C101C22semidirect
38. #107×C101C22semidirect
40. #102×C101C22semidirect
41. C4×C8×C101C22semidirect
75. D8×C101D4semidirect
93. D4×C101C2×D4semidirect
117. C8×C101C2×D4semidirect
136. C4×C101AGL(1,ℤ/8)semidirect
149. C101#3272semidirect
159. C4×D8D101semidirect
181. C4×D4C2×D101semidirect
183. #107C2×D101semidirect
188. #102C2×D101semidirect
190. C4×C8C2×D101semidirect
215. D8D404semidirect
252. D4C2×D404semidirect
267. C8C2×D404semidirect
287. C4#15692279semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
#202658312
D4×D4042
#202655502
#202654242
#202653822
#202653722
#202653292
#202653232
#202652562
C4×D8×C1012
#3272101