Finite Group Databasebeta

← #27182731

Subgroups — #27182731

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
3#20265812C211yesmaximal, characteristic, complemented
4#20265420C212yesmaximal, complemented
5#20265379C211yesmaximal, characteristic, complemented
6#20265369C211yesmaximal, characteristic, complemented
7#20265326C212yesmaximal, complemented
8#20265321C212yesmaximal, complemented
9C4×D808C211yesmaximal, characteristic, complemented
10#20265256C211yesmaximal, characteristic, complemented
11#492×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2212yescomplemented
13C2×D4×D101—21no—
14C2×D4×D101C2212yes—
15#15692282—28no—
16C2×#13108061C2212yes—
17C2×D808C2211yescharacteristic
18C2×#13108059C2212yescomplemented
19C2×#13108057C2212yescomplemented
20#15692240—22no—
21#15692231C2212yescomplemented
22#15692230—21no—
23#15692226—21no—
24#15692226C2212yes—
25#15692223—22no—
26C4×D404C2211yescharacteristic
27C4×D404C2211yescharacteristic
28C4×D404C2211yescharacteristic, complemented
29C4⋊C4×D101—21no—
30#15692189C2211yescharacteristic, complemented
31C4⋊#11717579C2211yescharacteristic, complemented
32#15692179C2212yescomplemented
33#15692177C2211yescharacteristic
34#15692177C2211yescharacteristic, complemented
35#15692176C2212yescomplemented
36#15692167C2212yescomplemented
37C2×SD16×C101C2212yescomplemented
38C2×SD16×C101C2212yescomplemented
39C4⋊Q8×C101C2211yescharacteristic, complemented
40#127×C101C2211yescharacteristic
41C4×C8×C101C2211yescharacteristic, complemented
42C23×D101—22no—
43#13108075—24no—
44#13108075—42no—
45C2×D404—22no—
46C2×D404D411yescharacteristic
47C2×D404C2311yescharacteristic
48C2×D404C2311yescharacteristic
49C2×C22⋊D101—22no—
50C2×C22⋊D101—22no—
51D4×D101—24no—
52D4×D101—42no—
53D4×D101—44no—
54C2×C4×D101—21no—
55C2×C4×D101—21no—
56C2×C4×D101—22no—
57C2×C4×D101C2312yes—
58#13108061—24no—
59D808D414yescomplemented
60#13108059—24no—
61#13108057—24no—
62C2×#11717579C2312yes—
63C4⋊Dic101D411yescharacteristic
64C4⋊Dic101C2311yescharacteristic
65C4⋊Dic101C2311yescharacteristic
66#13108053—22no—
67C22⋊Dic101—22no—
68#13108051—22no—
69#13108051—21no—
70#13108051—21no—
71#13108051D412yes—
72C2×Q8×C101C2312yes—
73C2×D4×C101—21no—
74C2×D4×C101—21no—
75C2×D4×C101C2312yes—
76SD16×C101—24no—
77SD16×C101—42no—
78C2×C8×C101C2311yescharacteristic
79C2×C8×C101C2311yescharacteristic
80C4⋊C4×C101—21no—
81C4⋊C4×C101—21no—
82C42×C101C2311yescharacteristic
83C23×C101—22no—
84C22×D101—21no—
85C22×D101—21no—
86C22×D101—44no—
87C22×D101—44no—
88C22×D101—82no—
89C22×D101C2×D412yes—
90Q8×C101—22no—
91Q8×C101—22no—
92D4×C101—22no—
93D4×C101—22no—
94D4×C101—42no—
95D4×C101—81no—
96C2×C4×C101—22no—
97C2×C4×C101C2×D411yescharacteristic
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2411yescharacteristic
100C22⋊D101—44no—
101C22⋊D101—44no—
102C2×Dic101—21no—
103C2×Dic101—21no—
104C2×Dic101C2×D412yes—
105D404—22no—
106D404—21no—
107D404—21no—
108D404—44no—
109D404C2×D412yes—
110D404C2×D412yescomplemented
111C4×D101—22no—
112C4×D101—22no—
113C4×D101—44no—
114C4×D101—42no—
115C4×D101—42no—
116C8×C101—21no—
117C8×C101C2×D412yescomplemented
118#11717579—22no—
119C22×C101—42no—
120C22×C101—42no—
121C22×C101—42no—
122C22×C101C22×D411yescharacteristic
123C2×D101—22no—
124C2×D101—22no—
125C2×D101—22no—
126C2×D101—22no—
127C2×D101—22no—
128C2×D101—41no—
129C2×D101—41no—
130C2×D101—41no—
131C2×D101—41no—
132C2×D101—88no—
133C4×C101—21no—
134C4×C101—42no—
135C4×C101AGL(1,ℤ/8)12yescomplemented
136C4×C101C22×D411yescharacteristic
137C4×C101H(5,𝔽2)11yescharacteristic
138Dic101—22no—
139Dic101—41no—
140Dic101—41no—
141C2×C101—82no—
142C2×C101D4211yescharacteristic
143C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C101#57711yescharacteristic
145D101—42no—
146D101—42no—
147D101—42no—
148#3272—1011nomaximal, Sylow
149C101#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1012no—
151#547—1011no—
152D42—1011no—
153#492D10111yescharacteristic, complemented
154#468—1012no—
155#461—1011no—
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×SD16C2×D10112yescomplemented
167C2×SD16C2×D10112yescomplemented
168C2×D8—1012no—
169C2×D8—1011no—
170C2×M16—1012no—
171C4⋊Q8C2×D10111yescharacteristic, complemented
172#127C2×D10111yescharacteristic
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×D4—1011no—
182C2×C4⋊C4—2021no—
183#107—1011no—
184C4⋊C8—1011no—
185#103—1012no—
186#102—1012no—
187#102—1011no—
188#102—1011no—
189C22⋊C8—1012no—
190C4×C8C2×D10111yescharacteristic, complemented
191C24—2022no—
192#41—4042no—
193#41—2024no—
194C2×Q8C22×D10112yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—1011no—
198C2×D4—1011no—
199C2×D4—1011no—
200C2×D4—4044no—
201C2×D4—4042no—
202C2×D4—2022no—
203C2×D4—2022no—
204C2×D4—2024no—
205C2×D4—2022no—
206C2×D4C22×D10112yes—
207C22×C4—1012no—
208C22×C4—2022no—
209C22×C4—2021no—
210C22×C4—2021no—
211SD16—24no—
212SD16—42no—
213SD16—2024no—
214D8—1014no—
215D8—2024no—
216M16—2024no—
217C2×C8—1012no—
218C2×C8C22×D10111yescharacteristic
219C2×C8C22×D10111yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—1011no—
223C4⋊C4—1011no—
224C4⋊C4—1011no—
225C4⋊C4—2022no—
226C22⋊C4—1012no—
227C22⋊C4—2022no—
228C22⋊C4—2022no—
229C22⋊C4—2021no—
230C22⋊C4—2021no—
231C42C22×D10111yescharacteristic
232C23—22no—
233C23—8082no—
234C23—1012no—
235C23—4044no—
236C23—4044no—
237C23—2021no—
238C23—2021no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—1012no—
246D4—1012no—
247D4—4044no—
248D4—4044no—
249D4—4044no—
250D4—2021no—
251D4—2021no—
252D4—2022no—
253C2×C4—22no—
254C2×C4—1012no—
255C2×C4—4044no—
256C2×C4—4042no—
257C2×C4—4042no—
258C2×C4—2021no—
259C2×C4—2021no—
260C2×C4—2022no—
261C2×C4—2022no—
262C2×C4D4×D10111yescharacteristic
263C2×C4D4×D10111yescharacteristic
264C2×C4C23×D10111yescharacteristic
265C8—21no—
266C8—2022no—
267C8D4×D10112yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—8088no—
272C22—4041no—
273C22—4041no—
274C22—4041no—
275C22—4041no—
276C22—2022no—
277C22—2022no—
278C22—2022no—
279C22—2022no—
280C22—2022no—
281C22C2×D4×D10111yescharacteristic
282C4—21no—
283C4—42no—
284C4—4041no—
285C4—4041no—
286C4—2022no—
287C4#1569228212yescomplemented
288C4C2×D4×D10111yescharacteristic
289C4#1569230011yescharacteristic
290C2—82no—
291C2—4042no—
292C2—4042no—
293C2—4042no—
294C2#2026581211yescharacteristic
295C2C2×#1569228211yescharacteristic
296C2#2026598311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#492×C101C2
socleC22×C101C22×D4
hypercenter#492D101
hypocenterC101#3272
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. #20265830C2semidirect
3. #20265812C2semidirect
4. #20265420C2semidirect
5. #20265379C2semidirect
6. #20265369C2semidirect
7. #20265326C2semidirect
8. #20265321C2semidirect
9. C4×D808C2semidirect
10. #20265256C2semidirect
11. #492×C101C2semidirect
12. C2×#13108075C22semidirect
18. C2×#13108059C22semidirect
19. C2×#13108057C22semidirect
21. #15692231C22semidirect
28. C4×D404C22semidirect
30. #15692189C22semidirect
31. C4⋊#11717579C22semidirect
32. #15692179C22semidirect
34. #15692177C22semidirect
35. #15692176C22semidirect
36. #15692167C22semidirect
37. C2×SD16×C101C22semidirect
38. C2×SD16×C101C22semidirect
39. C4⋊Q8×C101C22semidirect
41. C4×C8×C101C22semidirect
59. D808D4semidirect
110. D404C2×D4semidirect
117. C8×C101C2×D4semidirect
135. C4×C101AGL(1,ℤ/8)semidirect
149. C101#3272semidirect
153. #492D101semidirect
166. C2×SD16C2×D101semidirect
167. C2×SD16C2×D101semidirect
171. C4⋊Q8C2×D101semidirect
190. C4×C8C2×D101semidirect
267. C8D4×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
#202658122
#202654202
#202653792
#202653692
#202653262
#202653212
C4×D8082
#202652562
#492×C1012
#3272101