Finite Group Databasebeta

← #27180384

Subgroups — #27180384

Order 12928.

806 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 169 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×#15692280C212yesmaximal, complemented
2C2×#15692229C211yesmaximal, characteristic, complemented
3#20265322C214yesmaximal, complemented
4#20265316C212yesmaximal, complemented
5#20265311C212yesmaximal, complemented
6#20265178C212yesmaximal, complemented
7#20265141C211yesmaximal, characteristic, complemented
8C22×#41×C101C211yesmaximal, characteristic, complemented
9C22×D404C2211yescharacteristic, complemented
10C22×C22⋊D101—21no—
11#15692280—28no—
12C2×#13108061C2214yescomplemented
13C2×#13108059C2214yescomplemented
14C2×#13108058C2212yescomplemented
15#15692229—21no—
16#15692229—21no—
17#15692229C2212yes—
18#15692229C2212yescomplemented
19C2×C4⋊Dic101C2211yescharacteristic, complemented
20C2×#13108051—21no—
21#15692179C2214yescomplemented
22#15692176C2214yescomplemented
23C22⋊#11717579C2212yescomplemented
24C22⋊#11717579C2212yescomplemented
25C2×#41×C101—28no—
26C2×#41×C101—21no—
27C2×#41×C101—21no—
28C2×#41×C101C2212yes—
29C2×#41×C101C2212yescomplemented
30C22×Q8×C101C2211yescharacteristic, complemented
31C22×D4×C101—21no—
32C22×D4×C101C2211yescharacteristic, complemented
33C23×C4×C101—21no—
34C23×C4×C101C2211yescharacteristic
35C23×D101—21no—
36C2×D404—24no—
37C2×D404—21no—
38C2×D404C2312yes—
39C2×C22⋊D101—22no—
40C2×C22⋊D101—22no—
41C2×C22⋊D101—44no—
42C22×Dic101—21no—
43#13108061—28no—
44#13108059—28no—
45#13108058—24no—
46C2×#11717579C2314yes—
47C4⋊Dic101—21no—
48C4⋊Dic101C2312yes—
49#13108051—22no—
50#13108051—22no—
51C24×C101—21no—
52C24×C101D411yescharacteristic
53#41×C101—28no—
54#41×C101—28no—
55#41×C101—24no—
56#41×C101—24no—
57#41×C101—24no—
58#41×C101D418yescomplemented
59C2×Q8×C101—24no—
60C2×Q8×C101—21no—
61C2×Q8×C101C2312yes—
62C2×D4×C101—28no—
63C2×D4×C101—22no—
64C2×D4×C101—22no—
65C2×D4×C101—24no—
66C2×D4×C101—21no—
67C2×D4×C101C2312yes—
68C22×C4×C101—28no—
69C22×C4×C101—21no—
70C22×C4×C101—22no—
71C22×C4×C101—22no—
72C22×C4×C101—24no—
73C22×C4×C101D411yescharacteristic
74C22×C4×C101D411yescharacteristic
75C22×C4×C101D411yescharacteristic
76C22×C4×C101C2311yescharacteristic
77C22×C4×C101C2311yescharacteristic
78C22×C4×C101C2311yescharacteristic
79C23×C101—24no—
80C23×C101—22no—
81C23×C101—22no—
82C23×C101—22no—
83C23×C101—41no—
84C23×C101—42no—
85C23×C101—42no—
86C23×C101C2×D411yescharacteristic
87C23×C101C2×D411yescharacteristic
88C23×C101C2×D411yescharacteristic
89C22×D101—22no—
90C22×D101—42no—
91C22×D101—42no—
92C22×D101—42no—
93Q8×C101—24no—
94Q8×C101—22no—
95Q8×C101C2×D414yescomplemented
96D4×C101—28no—
97D4×C101—24no—
98D4×C101—28no—
99D4×C101—22no—
100D4×C101C2×D414yescomplemented
101C2×C4×C101—24no—
102C2×C4×C101—28no—
103C2×C4×C101—22no—
104C2×C4×C101—22no—
105C2×C4×C101—22no—
106C2×C4×C101—28no—
107C2×C4×C101—22no—
108C2×C4×C101—22no—
109C2×C4×C101—22no—
110C2×C4×C101C2×D411yescharacteristic
111C2×C4×C101C2×D411yescharacteristic
112C2×C4×C101C2×D411yescharacteristic
113C2×C4×C101C2×D414yes—
114C2×C4×C101C2411yescharacteristic
115C22⋊D101—44no—
116C22⋊D101—44no—
117C2×Dic101—22no—
118C2×Dic101—42no—
119D404—24no—
120D404—42no—
121#11717579—24no—
122C22×C101—21no—
123C22×C101—21no—
124C22×C101—21no—
125C22×C101—21no—
126C22×C101—21no—
127C22×C101—24no—
128C22×C101—21no—
129C22×C101—21no—
130C22×C101—44no—
131C22×C101—48no—
132C22×C101—42no—
133C22×C101—42no—
134C22×C101#12012yes—
135C22×C101AGL(1,ℤ/8)12yescomplemented
136C22×C101C22×D411yescharacteristic
137C2×D101—42no—
138C2×D101—42no—
139C2×D101—42no—
140C2×D101—84no—
141C4×C101—21no—
142C4×C101—24no—
143C4×C101—21no—
144C4×C101#12012yes—
145C4×C101C22×D412yes—
146Dic101—42no—
147C2×C101—22no—
148C2×C101—22no—
149C2×C101—41no—
150C2×C101—44no—
151C2×C101C2×#12011yescharacteristic
152C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C101#57511yescharacteristic
154D101—82no—
155#3006—1011nomaximal, Sylow
156C101#300611yescharacteristic, complemented, Sylow
157C22×#41D10111yescharacteristic, complemented
158C2×AGL(1,ℤ/8)—1012no—
159C2×#121—1011no—
160#449—1014no—
161#448—1012no—
162#447—1012no—
163#417—1012no—
164#407—1011no—
165C2×#41—28no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41C2×D10112yes—
169C2×#41C2×D10112yescomplemented
170C22×Q8C2×D10111yescharacteristic, complemented
171C22×D4—21no—
172C22×D4—1011no—
173C22×D4—2021no—
174C22×D4C2×D10111yescharacteristic, complemented
175C23×C4—21no—
176C23×C4C2×D10111yescharacteristic
177AGL(1,ℤ/8)—2028no—
178C2×SD16—1014no—
179C2×D8—1014no—
180C2×M16—1012no—
181#121—1012no—
182#121—1012no—
183#121—2021no—
184#121—2021no—
185C2×C4⋊C4—1011no—
186C2×C22⋊C4—2021no—
187#103—1014no—
188#102—1014no—
189C22⋊C8—1012no—
190C22⋊C8—1012no—
191C24—21no—
192C24—2021no—
193C24C22⋊D10111yescharacteristic
194#41—28no—
195#41—28no—
196#41—24no—
197#41—24no—
198#41—24no—
199#41C22⋊D10118yescomplemented
200C2×Q8—24no—
201C2×Q8—21no—
202C2×Q8C22×D10112yes—
203C2×D4—28no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—1012no—
209C2×D4—4044no—
210C2×D4—2022no—
211C2×D4—2022no—
212C2×D4—2021no—
213C2×D4—2024no—
214C2×D4C22×D10112yes—
215C22×C4—28no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—2021no—
221C22×C4C22⋊D10111yescharacteristic
222C22×C4C22⋊D10111yescharacteristic
223C22×C4C22⋊D10111yescharacteristic
224C22×C4C22×D10111yescharacteristic
225C22×C4C22×D10111yescharacteristic
226C22×C4C22×D10111yescharacteristic
227SD16—2028no—
228D8—2028no—
229M16—2024no—
230C2×C8—1014no—
231C4⋊C4—1012no—
232C4⋊C4—2021no—
233C22⋊C4—2022no—
234C22⋊C4—2022no—
235C23—24no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—4042no—
243C23—4042no—
244C23—4042no—
245C23—2022no—
246C23C2×C22⋊D10111yescharacteristic
247C23C2×C22⋊D10111yescharacteristic
248C23C2×C22⋊D10111yescharacteristic
249Q8—24no—
250Q8—22no—
251Q8C2×C22⋊D10114yescomplemented
252D4—28no—
253D4—24no—
254D4—28no—
255D4—22no—
256D4—4044no—
257D4—4044no—
258D4—4042no—
259D4—2024no—
260D4C2×C22⋊D10114yescomplemented
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—28no—
269C2×C4—22no—
270C2×C4—4042no—
271C2×C4—2022no—
272C2×C4C2×C22⋊D10111yescharacteristic
273C2×C4C2×C22⋊D10111yescharacteristic
274C2×C4C2×C22⋊D10114yes—
275C2×C4C2×C22⋊D10111yescharacteristic
276C2×C4C23×D10111yescharacteristic
277C8—2024no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—48no—
288C22—42no—
289C22—42no—
290C22—8084no—
291C22—4042no—
292C22—4042no—
293C22—4042no—
294C22#1569222412yes—
295C22#1569228012yescomplemented
296C22C22×C22⋊D10111yescharacteristic
297C4—21no—
298C4—24no—
299C4—21no—
300C4—4042no—
301C4#1569222412yes—
302C4C22×C22⋊D10112yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—44no—
307C2—8082no—
308C2C2×#1569222411yescharacteristic
309C2C2×#1569228011yescharacteristic
310C2#2026597511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×#41×C101C2
socleC22×C101C22×D4
hypercenterC22×#41D101
hypocenterC101#3006
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. C2×#15692229C2semidirect
3. #20265322C2semidirect
4. #20265316C2semidirect
5. #20265311C2semidirect
6. #20265178C2semidirect
7. #20265141C2semidirect
8. C22×#41×C101C2semidirect
9. C22×D404C22semidirect
12. C2×#13108061C22semidirect
13. C2×#13108059C22semidirect
14. C2×#13108058C22semidirect
18. #15692229C22semidirect
19. C2×C4⋊Dic101C22semidirect
21. #15692179C22semidirect
22. #15692176C22semidirect
23. C22⋊#11717579C22semidirect
24. C22⋊#11717579C22semidirect
29. C2×#41×C101C22semidirect
30. C22×Q8×C101C22semidirect
32. C22×D4×C101C22semidirect
58. #41×C101D4semidirect
95. Q8×C101C2×D4semidirect
100. D4×C101C2×D4semidirect
135. C22×C101AGL(1,ℤ/8)semidirect
156. C101#3006semidirect
157. C22×#41D101semidirect
169. C2×#41C2×D101semidirect
170. C22×Q8C2×D101semidirect
174. C22×D4C2×D101semidirect
199. #41C22⋊D101semidirect
251. Q8C2×C22⋊D101semidirect
260. D4C2×C22⋊D101semidirect
295. C22#15692280semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
C2×#156922292
#202653222
#202653162
#202653112
#202651782
#202651412
C22×#41×C1012
#3006101