Finite Group Databasebeta

← #27184459

Subgroups — #27184459

Order 12928.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D101C212yesmaximal, complemented
2#20265754C214yesmaximal, complemented
3#20265752C212yesmaximal, complemented
4#20265747C212yesmaximal, complemented
5#20265745C212yesmaximal, complemented
6#20265740C212yesmaximal, complemented
7#20265735C214yesmaximal, complemented
8#20265727C211yesmaximal, characteristic, complemented
9#20265723C212yesmaximal, complemented
10#20265686C218yesmaximal, complemented
11C2×#15692212C211yesmaximal, characteristic, complemented
12#532×C101C211yesmaximal, characteristic, complemented
13C2×#13108075C2212yes—
14C2×#13108074C2214yes—
15C2×#13108074C2212yescomplemented
16C2×#13108073C2212yes—
17C2×#13108073C2214yescomplemented
18#41×D101—216no—
19C2×Q8×D101C2212yescomplemented
20C2×D4×D101C2214yes—
21C2×D4×D101C2212yescomplemented
22C22×C4×D101C2211yescharacteristic
23C22×C4×D101C2214yescomplemented
24#15692255C2211yescharacteristic, complemented
25#15692253C2212yescomplemented
26#15692252C2211yescharacteristic, complemented
27#15692251C2212yescomplemented
28#15692244C2212yescomplemented
29#15692243C2212yescomplemented
30#15692242C2212yescomplemented
31#15692240C2218yescomplemented
32#15692237C2218yescomplemented
33#15692233C2218yescomplemented
34#15692232C2218yescomplemented
35#15692229C2214yescomplemented
36#15692228C2218yescomplemented
37#15692227C2218yescomplemented
38#15692227C2218yescomplemented
39#15692225C2218yescomplemented
40#15692223C2218yescomplemented
41C4×D404C2214yescomplemented
42C4×C22⋊D101C2218yescomplemented
43#15692212C2212yescomplemented
44#15692212C2212yescomplemented
45#15692212C2214yescomplemented
46C2×#13108053C2212yescomplemented
47C2×#13108051C2212yes—
48C2×C4×Dic101C2211yescharacteristic, complemented
49C2×#41×C101C2212yescomplemented
50C4⋊Q8×C101C2211yescharacteristic, complemented
51#127×C101C2211yescharacteristic, complemented
52#124×C101C2212yescomplemented
53#121×C101C2214yescomplemented
54C4×D4×C101C2214yescomplemented
55C2×C42×C101C2211yescharacteristic, complemented
56C23×D101C2314yes—
57C23×D101C2311yescharacteristic
58#13108075—28no—
59#13108074—28no—
60#13108074—216no—
61#13108073—216no—
62#13108073—28no—
63C2×Dic202C2312yes—
64C2×Dic202C2314yes—
65Q8×D101—28no—
66C2×D404C2314yes—
67C2×D404C2312yes—
68C2×D404C2314yes—
69C2×C22⋊D101C2314yes—
70C2×C22⋊D101C2314yes—
71C2×C22⋊D101C2314yes—
72C2×C22⋊D101C2318yes—
73D4×D101—216no—
74D4×D101—28no—
75C22×Dic101C2314yes—
76C22×Dic101C2312yes—
77C22×Dic101C2311yescharacteristic
78C2×C4×D101—216no—
79C2×C4×D101D418yescomplemented
80C2×C4×D101C2314yes—
81C2×C4×D101C2314yes—
82C2×C4×D101C2314yes—
83C2×C4×D101C2312yes—
84C2×C4×D101C2312yes—
85C4⋊Dic101C2314yes—
86#13108053C2314yes—
87#13108053C2314yes—
88#13108053C2318yes—
89C22⋊Dic101C2318yes—
90#13108051C2314yes—
91#13108051C2314yes—
92#13108051C2318yes—
93#13108051C2318yes—
94C4×Dic101C2312yes—
95C4×Dic101C2312yes—
96#41×C101—28no—
97C2×Q8×C101C2312yes—
98C2×D4×C101C2314yes—
99C2×D4×C101C2312yes—
100C2×D4×C101C2314yes—
101C22×C4×C101C2314yes—
102C22×C4×C101C2312yes—
103C22×C4×C101C2311yescharacteristic
104C4⋊C4×C101C2314yes—
105C22⋊C4×C101C2318yes—
106C42×C101C2312yes—
107C42×C101C2311yescharacteristic
108C42×C101C2311yescharacteristic
109C23×C101C2414yes—
110C23×C101C2411yescharacteristic
111C22×D101—28no—
112C22×D101—42no—
113C22×D101—48no—
114C22×D101C2×D414yes—
115C22×D101C2414yes—
116C22×D101C2412yes—
117Q8×C101—24no—
118D4×C101—28no—
119D4×C101—24no—
120D4×C101—44no—
121C2×C4×C101—28no—
122C2×C4×C101—42no—
123C2×C4×C101C2×D414yes—
124C2×C4×C101C2412yes—
125C2×C4×C101C2412yes—
126C2×C4×C101C2414yes—
127C2×C4×C101C2411yescharacteristic
128C2×C4×C101C2411yescharacteristic
129C22⋊D101—28no—
130C22⋊D101—28no—
131C22⋊D101—28no—
132C22⋊D101—48no—
133C2×Dic101—28no—
134C2×Dic101—42no—
135C2×Dic101C2×D414yes—
136C2×Dic101C2414yes—
137C2×Dic101C2414yes—
138C2×Dic101C2412yes—
139D404—28no—
140D404—24no—
141D404—44no—
142C4×D101—28no—
143C4×D101—28no—
144C4×D101—28no—
145C4×D101C2×D418yes—
146C4×D101C2×D418yes—
147Dic202—24no—
148Dic202—28no—
149C22×C101—22no—
150C22×C101—24no—
151C22×C101—28no—
152C22×C101C22×D412yes—
153C22×C101C2511yescharacteristic
154C2×D101—24no—
155C2×D101—24no—
156C2×D101—28no—
157C2×D101—44no—
158C2×D101—416no—
159C2×D101C22×D414yes—
160C4×C101—24no—
161C4×C101—22no—
162C4×C101—22no—
163C4×C101C22×D412yes—
164C4×C101C22×D412yes—
165Dic101—24no—
166Dic101—24no—
167Dic101C22×D414yes—
168C2×C101—22no—
169C2×C101—44no—
170C2×C101C23×D411yescharacteristic
171C2×C101#58512yes—
172D101—24no—
173D101—44no—
174#3449—1011nomaximal, Sylow
175C101#344911yescharacteristic, complemented, Sylow
176C22×#41—1012no—
177#537—1014no—
178#536—1012no—
179#536—1012no—
180#535—1012no—
181#535—1012no—
182#534—1014no—
183#532—1012no—
184#532—1011no—
185#532D10111yescharacteristic, complemented
186#525—1018no—
187C2×#117—1011no—
188C2×#41—1012no—
189C2×#41—1012no—
190C2×#41—1014no—
191C2×#41—1014no—
192C2×#41—1012no—
193C2×#41—20216no—
194C2×#41C2×D10112yescomplemented
195C22×Q8—1012no—
196C22×D4—1014no—
197C22×D4—1012no—
198C23×C4—1011no—
199C23×C4—1014no—
200C4⋊Q8—1012no—
201C4⋊Q8—1011no—
202C4⋊Q8C2×D10111yescharacteristic, complemented
203#127—1012no—
204#127—1011no—
205#127C2×D10111yescharacteristic, complemented
206#124—1012no—
207#124—1012no—
208#124—1012no—
209#124C2×D10112yescomplemented
210#123—1018no—
211#123—1018no—
212C22⋊Q8—1018no—
213C22⋊Q8—1018no—
214#121—1018no—
215#121—1018no—
216#121—1014no—
217#121—1018no—
218#121C2×D10114yescomplemented
219#120—1018no—
220#120—1018no—
221C4×D4—1018no—
222C4×D4—1014no—
223C4×D4C2×D10114yescomplemented
224#117—1014no—
225#117—1012no—
226#117—1012no—
227C2×C4⋊C4—1012no—
228C2×C22⋊C4—1012no—
229C2×C42—1011no—
230C2×C42C2×D10111yescharacteristic, complemented
231C24—1011no—
232C24—1014no—
233#41—28no—
234#41—20216no—
235#41—20216no—
236#41—2028no—
237#41—2028no—
238#41—2028no—
239C2×Q8—1014no—
240C2×Q8—1012no—
241C2×Q8—2028no—
242C2×Q8C22×D10112yes—
243C2×D4—1018no—
244C2×D4—1014no—
245C2×D4—1014no—
246C2×D4—1014no—
247C2×D4—1014no—
248C2×D4—1014no—
249C2×D4—1012no—
250C2×D4—2028no—
251C2×D4—20216no—
252C2×D4C22×D10114yes—
253C2×D4C22×D10112yes—
254C2×D4C22×D10114yes—
255C22×C4—1011no—
256C22×C4—1018no—
257C22×C4—1012no—
258C22×C4—1012no—
259C22×C4—1012no—
260C22×C4—1014no—
261C22×C4—1014no—
262C22×C4—1014no—
263C22×C4—1014no—
264C22×C4—20216no—
265C22×C4C22×D10114yes—
266C22×C4C22×D10112yes—
267C22×C4C22×D10111yescharacteristic
268C4⋊C4—1014no—
269C4⋊C4—1014no—
270C4⋊C4—1018no—
271C4⋊C4—1014no—
272C4⋊C4C22×D10114yes—
273C22⋊C4—1018no—
274C22⋊C4—1014no—
275C22⋊C4—1014no—
276C22⋊C4—1018no—
277C22⋊C4—1018no—
278C22⋊C4C22×D10118yes—
279C42—1012no—
280C42—1012no—
281C42C22×D10111yescharacteristic
282C42C22×D10111yescharacteristic
283C42C22×D10112yes—
284C23—1014no—
285C23—1012no—
286C23—1014no—
287C23—4042no—
288C23—4048no—
289C23—2028no—
290C23C23×D10114yes—
291C23C23×D10111yescharacteristic
292Q8—24no—
293Q8—2028no—
294Q8—2024no—
295D4—28no—
296D4—24no—
297D4—44no—
298D4—4044no—
299D4—4048no—
300D4—2028no—
301D4—2028no—
302D4—2028no—
303D4—2028no—
304D4—2024no—
305C2×C4—28no—
306C2×C4—42no—
307C2×C4—1014no—
308C2×C4—1012no—
309C2×C4—1018no—
310C2×C4—1018no—
311C2×C4—1014no—
312C2×C4—1014no—
313C2×C4—4042no—
314C2×C4—2028no—
315C2×C4—2028no—
316C2×C4—2028no—
317C2×C4—2028no—
318C2×C4D4×D10114yes—
319C2×C4C23×D10112yes—
320C2×C4C23×D10112yes—
321C2×C4C23×D10114yes—
322C2×C4C23×D10111yescharacteristic
323C2×C4C23×D10111yescharacteristic
324C22—22no—
325C22—24no—
326C22—28no—
327C22—1014no—
328C22—4044no—
329C22—40416no—
330C22—2024no—
331C22—2028no—
332C22—2024no—
333C22C2×D4×D10112yes—
334C22C24×D10111yescharacteristic
335C4—24no—
336C4—22no—
337C4—22no—
338C4—1014no—
339C4—2024no—
340C4—2024no—
341C4C2×D4×D10112yes—
342C4C2×D4×D10112yes—
343C2—22no—
344C2—44no—
345C2—4044no—
346C2—2024no—
347C2C22×D4×D10111yescharacteristic
348C2#2026601312yes—
subgroupG/H
centerC2×C4C23×D101
commutator, socle, cosocleC22×C101C25
FrattiniC22C24×D101
Fitting#532×C101C2
hypercenter#532D101
hypocenterC101#3449
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#41×D101C2semidirect
2. #20265754C2semidirect
3. #20265752C2semidirect
4. #20265747C2semidirect
5. #20265745C2semidirect
6. #20265740C2semidirect
7. #20265735C2semidirect
8. #20265727C2semidirect
9. #20265723C2semidirect
10. #20265686C2semidirect
11. C2×#15692212C2semidirect
12. #532×C101C2semidirect
15. C2×#13108074C22semidirect
17. C2×#13108073C22semidirect
19. C2×Q8×D101C22semidirect
21. C2×D4×D101C22semidirect
23. C22×C4×D101C22semidirect
24. #15692255C22semidirect
25. #15692253C22semidirect
26. #15692252C22semidirect
27. #15692251C22semidirect
28. #15692244C22semidirect
29. #15692243C22semidirect
30. #15692242C22semidirect
31. #15692240C22semidirect
32. #15692237C22semidirect
33. #15692233C22semidirect
34. #15692232C22semidirect
35. #15692229C22semidirect
36. #15692228C22semidirect
37. #15692227C22semidirect
38. #15692227C22semidirect
39. #15692225C22semidirect
40. #15692223C22semidirect
41. C4×D404C22semidirect
42. C4×C22⋊D101C22semidirect
43. #15692212C22semidirect
44. #15692212C22semidirect
45. #15692212C22semidirect
46. C2×#13108053C22semidirect
48. C2×C4×Dic101C22semidirect
49. C2×#41×C101C22semidirect
50. C4⋊Q8×C101C22semidirect
51. #127×C101C22semidirect
52. #124×C101C22semidirect
53. #121×C101C22semidirect
54. C4×D4×C101C22semidirect
55. C2×C42×C101C22semidirect
79. C2×C4×D101D4semidirect
175. C101#3449semidirect
185. #532D101semidirect
194. C2×#41C2×D101semidirect
202. C4⋊Q8C2×D101semidirect
205. #127C2×D101semidirect
209. #124C2×D101semidirect
218. #121C2×D101semidirect
223. C4×D4C2×D101semidirect
230. C2×C42C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1012
#202657542
#202657522
#202657472
#202657452
#202657402
#202657352
#202657272
#202657232
#202656862
C2×#156922122
#532×C1012
#3449101