Finite Group Databasebeta

← #541×D101 #27184494

Subgroups — #541×D101 #27184494

Order 12928.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265790C211yesmaximal, characteristic, complemented
2#20265789C211yesmaximal, characteristic, complemented
3#20265788C211yesmaximal, characteristic, complemented
4#20265787C212yesmaximal, complemented
5#20265786C214yesmaximal, complemented
6#20265785C212yesmaximal, complemented
7#20265784C214yesmaximal, complemented
8C4⋊Q8×D101C211yesmaximal, characteristic, complemented
9#126×D101C214yesmaximal
10#125×D101C214yesmaximal, complemented
11#125×D101C211yesmaximal, characteristic, complemented
12C22⋊Q8×D101C212yesmaximal
13C4×Q8×D101C212yesmaximal, complemented
14#117×D101C211yesmaximal, characteristic
15#541×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2212yes—
17C22×C4×D101C2211yescharacteristic
18#15692255C2211yescharacteristic, complemented
19#15692254C2214yescomplemented
20#15692253C2212yescomplemented
21#15692250C2214yes—
22#15692250C2214yes—
23#15692250C2214yes—
24#15692249C2214yes—
25#15692248C2214yes—
26#15692248C2214yes—
27#15692248C2214yes—
28#15692247C2214yescomplemented
29#15692247C2211yescharacteristic, complemented
30#15692246C2214yescomplemented
31#15692246C2212yescomplemented
32#15692246C2214yescomplemented
33#15692245C2214yescomplemented
34#15692245C2214yescomplemented
35#15692245C2214yescomplemented
36#15692245C2214yescomplemented
37#15692245C2214yescomplemented
38#15692235C2212yes—
39#15692234C2212yes—
40#15692233C2214yes—
41#15692232C2214yes—
42#15692231C2212yes—
43Q8×Dic101C2212yescomplemented
44C4×Dic202C2214yescomplemented
45C4×Dic202C2212yescomplemented
46#15692220C2214yescomplemented
47#15692220C2212yescomplemented
48#15692214C2211yescharacteristic
49#15692213C2212yes—
50#15692212C2211yescharacteristic
51#15692212C2211yescharacteristic
52#15692211C2212yes—
53C4⋊C4×D101C2212yes—
54C4⋊C4×D101C2212yes—
55C4⋊C4×D101C2214yes—
56C4⋊C4×D101C2214yes—
57C4⋊C4×D101C2214yes—
58C4⋊C4×D101C2214yes—
59C22⋊C4×D101C2212yes—
60C22⋊C4×D101C2214yes—
61C42×D101C2211yescharacteristic
62C42×D101C2211yescharacteristic
63C42×D101C2214yes—
64C4⋊Q8×C101C2211yescharacteristic, complemented
65#126×C101C2214yes—
66#125×C101C2214yescomplemented
67#125×C101C2211yescharacteristic, complemented
68C22⋊Q8×C101C2212yes—
69C4×Q8×C101C2212yescomplemented
70#117×C101C2211yescharacteristic
71C23×D101C2311yescharacteristic
72C2×Dic202C2314yes—
73C2×Dic202C2312yes—
74Q8×D101—28no—
75C22×Dic101C2311yescharacteristic
76C2×C4×D101—24no—
77C2×C4×D101C2311yescharacteristic
78C2×C4×D101C2311yescharacteristic
79C2×C4×D101C2311yescharacteristic
80C2×C4×D101C2314yes—
81C2×C4×D101C2312yes—
82C2×C4×D101C2312yes—
83C2×C4×D101C2314yes—
84C2×C4×D101C2311yescharacteristic
85C4⋊Dic101C2312yes—
86C4⋊Dic101C2312yes—
87C4⋊Dic101C2314yes—
88C4⋊Dic101C2314yes—
89C4⋊Dic101C2314yes—
90C4⋊Dic101C2314yes—
91#13108053C2312yes—
92#13108053C2312yes—
93#13108053C2314yes—
94#13108053C2314yes—
95#13108053C2314yes—
96#13108053C2314yes—
97#13108053C2314yes—
98#13108053C2314yes—
99#13108053C2314yes—
100#13108053C2314yes—
101#13108053C2314yes—
102C22⋊Dic101C2312yes—
103C22⋊Dic101C2314yes—
104#13108051C2312yes—
105#13108051C2312yes—
106#13108051C2314yes—
107#13108051C2314yes—
108C4×Dic101C2312yes—
109C4×Dic101C2312yes—
110C4×Dic101C2311yescharacteristic
111C4×Dic101C2311yescharacteristic
112C4×Dic101C2314yes—
113C4×Dic101C2314yes—
114C4×Dic101C2314yes—
115C2×Q8×C101C2312yes—
116C22×C4×C101C2311yescharacteristic
117C4⋊C4×C101C2312yes—
118C4⋊C4×C101C2312yes—
119C4⋊C4×C101C2314yes—
120C4⋊C4×C101C2314yes—
121C4⋊C4×C101C2314yes—
122C4⋊C4×C101C2314yes—
123C22⋊C4×C101C2312yes—
124C22⋊C4×C101C2314yes—
125C42×C101C2311yescharacteristic
126C42×C101C2311yescharacteristic
127C42×C101C2314yes—
128C23×C101C2411yescharacteristic
129C22×D101—22no—
130C22×D101—24no—
131C22×D101C2411yescharacteristic
132C22×D101C2411yescharacteristic
133Q8×C101—24no—
134C2×C4×C101—22no—
135C2×C4×C101C2412yes—
136C2×C4×C101C2412yes—
137C2×C4×C101C2414yes—
138C2×C4×C101C2414yes—
139C2×C4×C101C2411yescharacteristic
140C2×C4×C101C2411yescharacteristic
141C2×Dic101—22no—
142C2×Dic101C2412yes—
143C2×Dic101C2412yes—
144C2×Dic101C2414yes—
145C2×Dic101C2414yes—
146C2×Dic101C2411yescharacteristic
147C2×Dic101C2411yescharacteristic
148C4×D101—22no—
149C4×D101—28no—
150C4×D101—24no—
151C4×D101—24no—
152C4×D101—28no—
153C4×D101—22no—
154C4×D101—22no—
155C4×D101#4114yes—
156Dic202—24no—
157Dic202—28no—
158C22×C101—21no—
159C22×C101—22no—
160C22×C101C2511yescharacteristic
161C2×D101—21no—
162C2×D101—22no—
163C2×D101—44no—
164C2×D101C2×#4112yes—
165C2×D1012−1+414yes—
166C4×C101—24no—
167C4×C101—21no—
168C4×C101—22no—
169C4×C101—22no—
170C4×C101—24no—
171C4×C101C2×#4112yes—
172Dic101—24no—
173Dic101—21no—
174Dic101—22no—
175Dic101—22no—
176Dic101—24no—
177Dic101C2×#4112yes—
178C2×C101—41no—
179C2×C101C22×#4111yescharacteristic
180C2×C101C2×2−1+412yes—
181D101—41no—
182D101#54114yesdirect product factor
183C2×#541—1011nomaximal, Sylow
184C101C2×#54111yescharacteristic, complemented, Sylow
185#541—1011no—
186#541—1014no—
187#541—1014no—
188#541—1011no—
189#541—1012no—
190#541—1012no—
191#541—1011no—
192#541D10111yescharacteristic, direct product factor
193C2×C4⋊Q8—1011no—
194C2×#126—1014no—
195C2×#125—1011no—
196C2×#125—1014no—
197C2×C22⋊Q8—1012no—
198C2×C4×Q8—1012no—
199C2×#117—1011no—
200C22×Q8—1012no—
201C23×C4—1011no—
202C4⋊Q8—1012no—
203C4⋊Q8—1014no—
204C4⋊Q8—1011no—
205C4⋊Q8C2×D10111yescharacteristic, complemented
206#126—1014no—
207#126—1014no—
208#126—1014no—
209#126—1014no—
210#126—1014no—
211#126—1014no—
212#126—1014no—
213#126C2×D10114yes—
214#125—1014no—
215#125—1014no—
216#125—1014no—
217#125—1014no—
218#125—1014no—
219#125—1012no—
220#125—1014no—
221#125—1014no—
222#125—1011no—
223#125—1014no—
224#125C2×D10111yescharacteristic, complemented
225#125C2×D10114yescomplemented
226C22⋊Q8—1012no—
227C22⋊Q8—1012no—
228C22⋊Q8—1014no—
229C22⋊Q8—1012no—
230C22⋊Q8—1014no—
231C22⋊Q8C2×D10112yes—
232C4×Q8—1012no—
233C4×Q8—1014no—
234C4×Q8—1012no—
235C4×Q8—1014no—
236C4×Q8—1012no—
237C4×Q8C2×D10112yescomplemented
238#117—1011no—
239#117—1012no—
240#117—1012no—
241#117—1011no—
242#117—1011no—
243#117C2×D10111yescharacteristic
244C2×C4⋊C4—1012no—
245C2×C4⋊C4—1012no—
246C2×C4⋊C4—1014no—
247C2×C4⋊C4—1014no—
248C2×C4⋊C4—1014no—
249C2×C4⋊C4—1014no—
250C2×C22⋊C4—1012no—
251C2×C22⋊C4—1014no—
252C2×C42—1011no—
253C2×C42—1011no—
254C2×C42—1014no—
255C24—1011no—
256C2×Q8—1014no—
257C2×Q8—1012no—
258C2×Q8—2028no—
259C2×Q8C22×D10112yes—
260C22×C4—1011no—
261C22×C4—1011no—
262C22×C4—1012no—
263C22×C4—1012no—
264C22×C4—1014no—
265C22×C4—1014no—
266C22×C4—1011no—
267C22×C4—1011no—
268C22×C4—1011no—
269C22×C4—2024no—
270C22×C4C22×D10111yescharacteristic
271C4⋊C4—1012no—
272C4⋊C4—1012no—
273C4⋊C4—1014no—
274C4⋊C4—1014no—
275C4⋊C4—1014no—
276C4⋊C4—1014no—
277C4⋊C4—1014no—
278C4⋊C4—1014no—
279C4⋊C4—1014no—
280C4⋊C4—1014no—
281C4⋊C4—1014no—
282C4⋊C4—1012no—
283C4⋊C4—1012no—
284C4⋊C4—1014no—
285C4⋊C4—1014no—
286C4⋊C4—1014no—
287C4⋊C4—1014no—
288C4⋊C4C22×D10112yes—
289C4⋊C4C22×D10112yes—
290C4⋊C4C22×D10114yes—
291C4⋊C4C22×D10114yes—
292C4⋊C4C22×D10114yes—
293C4⋊C4C22×D10114yes—
294C22⋊C4—1012no—
295C22⋊C4—1014no—
296C22⋊C4—1012no—
297C22⋊C4—1012no—
298C22⋊C4—1014no—
299C22⋊C4—1014no—
300C22⋊C4C22×D10112yes—
301C22⋊C4C22×D10114yes—
302C42—1012no—
303C42—1011no—
304C42—1012no—
305C42—1011no—
306C42—1014no—
307C42—1014no—
308C42—1014no—
309C42C22×D10111yescharacteristic
310C42C22×D10111yescharacteristic
311C42C22×D10114yes—
312C23—1011no—
313C23—1011no—
314C23—2022no—
315C23—2024no—
316C23C23×D10111yescharacteristic
317Q8—24no—
318Q8—2028no—
319Q8—2024no—
320C2×C4—22no—
321C2×C4—1011no—
322C2×C4—1014no—
323C2×C4—1012no—
324C2×C4—1012no—
325C2×C4—1014no—
326C2×C4—1014no—
327C2×C4—1011no—
328C2×C4—2022no—
329C2×C4—2022no—
330C2×C4—2028no—
331C2×C4—2024no—
332C2×C4—2024no—
333C2×C4—2028no—
334C2×C4—2022no—
335C2×C4—2022no—
336C2×C4C23×D10112yes—
337C2×C4C23×D10112yes—
338C2×C4C23×D10114yes—
339C2×C4C23×D10114yes—
340C2×C4C23×D10111yescharacteristic
341C2×C4C23×D10111yescharacteristic
342C22—21no—
343C22—22no—
344C22—1012no—
345C22—1014no—
346C22—4044no—
347C22—2021no—
348C22—2022no—
349C22C24×D10111yescharacteristic
350C4—21no—
351C4—24no—
352C4—22no—
353C4—22no—
354C4—24no—
355C4—1012no—
356C4—2021no—
357C4—2024no—
358C4—2022no—
359C4—2022no—
360C4—2024no—
361C4#41×D10112yes—
362C2—41no—
363C2—1014no—
364C2—4041no—
365C2C2×#41×D10111yescharacteristic
366C22−1+4×D10112yes—
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
Fitting#541×C101C2
prime radicalC23×D101C23
hypercenter#541D101
hypocenterC101C2×#541
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #20265790C2semidirect
2. #20265789C2semidirect
3. #20265788C2semidirect
4. #20265787C2semidirect
5. #20265786C2semidirect
6. #20265785C2semidirect
7. #20265784C2semidirect
8. C4⋊Q8×D101C2semidirect
10. #125×D101C2semidirect
11. #125×D101C2semidirect
13. C4×Q8×D101C2semidirect
15. #541×C101C2semidirect
18. #15692255C22semidirect
19. #15692254C22semidirect
20. #15692253C22semidirect
28. #15692247C22semidirect
29. #15692247C22semidirect
30. #15692246C22semidirect
31. #15692246C22semidirect
32. #15692246C22semidirect
33. #15692245C22semidirect
34. #15692245C22semidirect
35. #15692245C22semidirect
36. #15692245C22semidirect
37. #15692245C22semidirect
43. Q8×Dic101C22semidirect
44. C4×Dic202C22semidirect
45. C4×Dic202C22semidirect
46. #15692220C22semidirect
47. #15692220C22semidirect
64. C4⋊Q8×C101C22semidirect
66. #125×C101C22semidirect
67. #125×C101C22semidirect
69. C4×Q8×C101C22semidirect
182. D101#541direct (N × H)
184. C101C2×#541semidirect
192. #541D101direct (N × H)
205. C4⋊Q8C2×D101semidirect
224. #125C2×D101semidirect
225. #125C2×D101semidirect
237. C4×Q8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657902
#202657892
#202657882
#202657872
#202657862
#202657852
#202657842
C4⋊Q8×D1012
#126×D1012
#125×D1012
C22⋊Q8×D1012
C4×Q8×D1012
#117×D1012
#541×C1012
C2×#541101