Finite Group Databasebeta

← #27184621

Subgroups — #27184621

Order 12928.

1648 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108075C211yesmaximal, characteristic, complemented
2C2×#41×D101C211yesmaximal, characteristic, complemented
3#20265846C211yesmaximal, characteristic, complemented
4Q8×C22⋊D101C211yesmaximal, characteristic, complemented
5#20265839C211yesmaximal, characteristic, complemented
6#20265829C213yesmaximal, complemented
7#20265823C213yesmaximal, complemented
8#20265820C213yesmaximal, complemented
9#20265811C213yesmaximal, complemented
10#20265726C213yesmaximal, complemented
11#20265690C213yesmaximal, complemented
12#20265687C213yesmaximal, complemented
13C4×#13108075C211yesmaximal, characteristic, complemented
14C4×D4×D101C213yesmaximal, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16C2×#13108075—28no—
17C2×#13108075C2211yescharacteristic
18C2×#13108075C2213yescomplemented
19C2×#13108075C2211yescharacteristic, complemented
20C2×#13108075C2213yescomplemented
21C2×#13108074C2213yescomplemented
22C2×#13108073C2213yescomplemented
23#41×D101—28no—
24C2×Q8×D101C2211yescharacteristic, complemented
25C22×D404C2213yescomplemented
26C2×D4×D101C2213yes—
27C2×D4×D101C2213yescomplemented
28C22×C4×D101C2213yescomplemented
29C22×C4×D101C2213yescomplemented
30#15692253C2213yescomplemented
31#15692251C2213yescomplemented
32#15692243C2213yescomplemented
33#15692240C2216yescomplemented
34#15692237C2216yescomplemented
35#15692235C2213yescomplemented
36#15692234C2213yescomplemented
37#15692231C2213yescomplemented
38#15692230C2213yescomplemented
39#15692229C2213yescomplemented
40#15692228C2216yescomplemented
41#15692228C2216yescomplemented
42#15692226C2213yescomplemented
43#15692225C2216yescomplemented
44#15692223C2216yescomplemented
45Q8×Dic101C2211yescharacteristic, complemented
46C4×D404C2213yescomplemented
47C4×D404C2213yescomplemented
48D4×Dic101C2213yescomplemented
49#15692217C2213yescomplemented
50#15692217C2213yescomplemented
51C4×C22⋊D101C2213yes—
52C4×C22⋊D101C2213yescomplemented
53#15692215C2211yescharacteristic
54#15692215C2213yescomplemented
55#15692215C2216yescomplemented
56#15692213C2213yescomplemented
57C4⋊C4×D101C2213yescomplemented
58C22⋊C4×D101C2213yescomplemented
59C22⋊C4×D101C2213yescomplemented
60C42×D101C2213yescomplemented
61C2×#41×C101C2211yescharacteristic, complemented
62C22×Q8×C101C2211yescharacteristic, complemented
63#124×C101C2213yescomplemented
64C22⋊Q8×C101C2213yescomplemented
65#121×C101C2213yescomplemented
66C4×Q8×C101C2211yescharacteristic, complemented
67C4×D4×C101C2213yescomplemented
68C23×D101C2313yes—
69C23×D101C2313yes—
70#13108075—24no—
71#13108075—212no—
72#13108075—212no—
73#13108075D418yescomplemented
74#13108074—212no—
75#13108073—212no—
76C2×Dic202C2313yes—
77Q8×D101—24no—
78C2×D404—212no—
79C2×D404C2313yes—
80C2×D404C2316yes—
81C2×D404C2313yes—
82C2×D404C2313yes—
83C2×C22⋊D101C2313yes—
84C2×C22⋊D101C2316yes—
85C2×C22⋊D101C2316yes—
86C2×C22⋊D101C2311yescharacteristic
87D4×D101—212no—
88D4×D101—212no—
89C22×Dic101C2311yescharacteristic
90C22×Dic101C2313yes—
91C2×C4×D101—212no—
92C2×C4×D101—212no—
93C2×C4×D101C2311yescharacteristic
94C2×C4×D101C2313yes—
95C2×C4×D101C2313yes—
96C2×C4×D101C2313yes—
97C2×C4×D101C2313yes—
98C2×C4×D101C2313yes—
99C2×C4×D101C2313yes—
100C2×C4×D101C2313yes—
101C2×C4×D101C2313yes—
102C4⋊Dic101C2313yes—
103C4⋊Dic101C2313yes—
104#13108053C2311yescharacteristic
105#13108053C2313yes—
106#13108053C2316yes—
107C22⋊Dic101C2313yes—
108C22⋊Dic101C2313yes—
109#13108051C2311yescharacteristic
110#13108051C2313yes—
111#13108051C2313yes—
112#13108051C2313yes—
113#13108051C2316yes—
114#13108051C2316yes—
115C4×Dic101C2311yescharacteristic
116C4×Dic101C2313yes—
117C4×Dic101C2313yes—
118#41×C101—24no—
119C2×Q8×C101—24no—
120C2×Q8×C101C2313yes—
121C2×Q8×C101C2311yescharacteristic
122C2×D4×C101C2313yes—
123C2×D4×C101C2313yes—
124C22×C4×C101C2313yes—
125C22×C4×C101C2313yes—
126C4⋊C4×C101C2313yes—
127C4⋊C4×C101C2313yes—
128C22⋊C4×C101C2311yescharacteristic
129C22⋊C4×C101C2313yes—
130C22⋊C4×C101C2316yes—
131C42×C101C2313yes—
132C23×C101C2413yes—
133C23×C101C2411yescharacteristic
134C22×D101—26no—
135C22×D101—26no—
136C22×D101—26no—
137C22×D101—26no—
138C22×D101—43no—
139C22×D101—43no—
140C22×D101C2413yes—
141C22×D101C2413yes—
142C22×D101C2411yescharacteristic
143Q8×C101—26no—
144Q8×C101C2×D414yescomplemented
145D4×C101—26no—
146D4×C101—26no—
147C2×C4×C101—26no—
148C2×C4×C101—26no—
149C2×C4×C101C2411yescharacteristic
150C2×C4×C101C2413yes—
151C2×C4×C101C2413yes—
152C2×C4×C101C2411yescharacteristic
153C2×C4×C101C2413yes—
154C22⋊D101—212no—
155C22⋊D101—26no—
156C22⋊D101—46no—
157C22⋊D101#4114yescomplemented
158C2×Dic101—22no—
159C2×Dic101—26no—
160C2×Dic101C2413yes—
161C2×Dic101C2413yes—
162C2×Dic101C2411yescharacteristic
163C2×Dic101C2411yescharacteristic
164C2×Dic101C2411yescharacteristic
165D404—212no—
166D404—26no—
167D404—26no—
168D404C2×D4112yescomplemented
169C4×D101—26no—
170C4×D101—26no—
171C4×D101—26no—
172C4×D101—26no—
173C4×D101—22no—
174C4×D101—26no—
175C4×D101—26no—
176C4×D101—43no—
177C4×D101C2×D4112yescomplemented
178Dic202—26no—
179C22×C101—21no—
180C22×C101—21no—
181C22×C101—23no—
182C22×C101—23no—
183C22×C101—23no—
184C22×C101C2×#4112yescomplemented
185C22×C101C2511yescharacteristic
186C2×D101—21no—
187C2×D101—23no—
188C2×D101—23no—
189C2×D101—21no—
190C2×D101—23no—
191C2×D101—23no—
192C2×D101—23no—
193C2×D101—412no—
194C2×D101—412no—
195C2×D101C22×D416yes—
196C2×D101C2×#4112yescomplemented
197C4×C101—21no—
198C4×C101—23no—
199C4×C101—23no—
200C4×C101—21no—
201C4×C101C22×D416yes—
202Dic101—23no—
203Dic101—21no—
204Dic101—23no—
205Dic101C22×D412yes—
206Dic101C2×#4112yescomplemented
207C2×C101—22no—
208C2×C101—43no—
209C2×C101C23×D411yescharacteristic
210C2×C101C22×#4111yescharacteristic
211C2×C101#58511yescharacteristic
212D101—22no—
213D101—26no—
214D101—43no—
215D4×#41—1011nomaximal, Sylow
216C101D4×#4111yescharacteristic, complemented, Sylow
217C22×#41—1011no—
218C22×#41—1011no—
219#550—1011no—
220D4×Q8—1011no—
221#548—1011no—
222#548D10111yescharacteristic, complemented
223#547—1013no—
224#546—1013no—
225#546—1013no—
226D42—1013no—
227#532—1013no—
228#525—1013no—
229#525—1013no—
230C4×#41—1011no—
231C2×C4×D4—1013no—
232C2×#41—1011no—
233C2×#41—1013no—
234C2×#41—1013no—
235C2×#41—1011no—
236C2×#41—1013no—
237C2×#41—1013no—
238C2×#41—2028no—
239C2×#41—2028no—
240C2×#41C2×D10111yescharacteristic, complemented
241C22×Q8—1011no—
242C22×Q8C2×D10111yescharacteristic, complemented
243C22×D4—1013no—
244C22×D4—1013no—
245C22×D4—1013no—
246C23×C4—1013no—
247C23×C4—1013no—
248C4⋊Q8—1013no—
249#127—1013no—
250#124—1013no—
251#124C2×D10113yescomplemented
252#123—1016no—
253#123—1016no—
254C22⋊Q8—1013no—
255C22⋊Q8—1013no—
256C22⋊Q8—1013no—
257C22⋊Q8C2×D10113yescomplemented
258#121—1013no—
259#121—1013no—
260#121—1013no—
261#121—1016no—
262#121—1016no—
263#121C2×D10113yescomplemented
264#120—1016no—
265#120—1016no—
266C4×Q8—1011no—
267C4×Q8C2×D10111yescharacteristic, complemented
268C4×D4—1013no—
269C4×D4—1013no—
270C4×D4—1013no—
271C4×D4—1013no—
272C4×D4—1013no—
273C4×D4—1011no—
274C4×D4—1013no—
275C4×D4—1016no—
276C4×D4—1013no—
277C4×D4—1013no—
278C4×D4C2×D10113yescomplemented
279#117—1013no—
280C2×C4⋊C4—1013no—
281C2×C22⋊C4—1013no—
282C2×C22⋊C4—1013no—
283C2×C42—1013no—
284C24—1013no—
285C24—1013no—
286#41—24no—
287#41—1018no—
288#41—20212no—
289#41—20212no—
290#41—20212no—
291#41—20212no—
292#41—2024no—
293C2×Q8—24no—
294C2×Q8—1013no—
295C2×Q8—2024no—
296C2×Q8C22×D10113yes—
297C2×Q8C22×D10111yescharacteristic
298C2×D4—1011no—
299C2×D4—1016no—
300C2×D4—1013no—
301C2×D4—1016no—
302C2×D4—1013no—
303C2×D4—1013no—
304C2×D4—1016no—
305C2×D4—1013no—
306C2×D4—20212no—
307C2×D4—20212no—
308C2×D4—20212no—
309C2×D4C22×D10113yes—
310C2×D4C22×D10113yes—
311C22×C4—1013no—
312C22×C4—1011no—
313C22×C4—1013no—
314C22×C4—1013no—
315C22×C4—1013no—
316C22×C4—1013no—
317C22×C4—1013no—
318C22×C4—1011no—
319C22×C4—1013no—
320C22×C4—1013no—
321C22×C4—1013no—
322C22×C4—20212no—
323C22×C4—20212no—
324C22×C4C22×D10113yes—
325C22×C4C22×D10113yes—
326C4⋊C4—1011no—
327C4⋊C4—1013no—
328C4⋊C4—1016no—
329C4⋊C4—1013no—
330C4⋊C4—1013no—
331C4⋊C4C22×D10113yes—
332C4⋊C4C22×D10113yes—
333C22⋊C4—1013no—
334C22⋊C4—1013no—
335C22⋊C4—1011no—
336C22⋊C4—1013no—
337C22⋊C4—1013no—
338C22⋊C4—1013no—
339C22⋊C4—1016no—
340C22⋊C4—1016no—
341C22⋊C4C22×D10111yescharacteristic
342C22⋊C4C22×D10113yes—
343C22⋊C4C22×D10116yes—
344C42—1011no—
345C42—1013no—
346C42—1013no—
347C42C22×D10113yes—
348C23—1011no—
349C23—1013no—
350C23—1013no—
351C23—4043no—
352C23—4043no—
353C23—2026no—
354C23—2026no—
355C23—2026no—
356C23—2026no—
357C23C23×D10113yes—
358C23C23×D10111yescharacteristic
359Q8—26no—
360Q8—2026no—
361Q8D4×D10114yescomplemented
362D4—26no—
363D4—26no—
364D4—1014no—
365D4—10112no—
366D4—4046no—
367D4—20212no—
368D4—2026no—
369D4—2026no—
370D4—20212no—
371D4—2026no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—1011no—
375C2×C4—1011no—
376C2×C4—10112no—
377C2×C4—1013no—
378C2×C4—1011no—
379C2×C4—1013no—
380C2×C4—4043no—
381C2×C4—2022no—
382C2×C4—2026no—
383C2×C4—2026no—
384C2×C4—2026no—
385C2×C4—2026no—
386C2×C4—2022no—
387C2×C4—2026no—
388C2×C4—2026no—
389C2×C4—2026no—
390C2×C4C23×D10111yescharacteristic
391C2×C4C23×D10113yes—
392C2×C4C23×D10111yescharacteristic
393C2×C4C23×D10113yes—
394C2×C4C23×D10113yes—
395C22—21no—
396C22—21no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—1016no—
401C22—1012no—
402C22—40412no—
403C22—40412no—
404C22—2023no—
405C22—2023no—
406C22—2023no—
407C22—2021no—
408C22—2023no—
409C22—2023no—
410C22—2021no—
411C22#41×D10112yescomplemented
412C22C24×D10111yescharacteristic
413C4—21no—
414C4—23no—
415C4—23no—
416C4—21no—
417C4—1012no—
418C4—1012no—
419C4—2021no—
420C4—2023no—
421C4—2023no—
422C4C2×D4×D10116yes—
423C2—22no—
424C2—43no—
425C2—4043no—
426C2—2022no—
427C2—2026no—
428C2C22×D4×D10111yescharacteristic
429C2C2×#41×D10111yescharacteristic
430C2#2026601411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
Fitting#548×C101C2
hypercenter#548D101
hypocenterC101D4×#41
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C22×#13108075C2semidirect
2. C2×#41×D101C2semidirect
3. #20265846C2semidirect
4. Q8×C22⋊D101C2semidirect
5. #20265839C2semidirect
6. #20265829C2semidirect
7. #20265823C2semidirect
8. #20265820C2semidirect
9. #20265811C2semidirect
10. #20265726C2semidirect
11. #20265690C2semidirect
12. #20265687C2semidirect
13. C4×#13108075C2semidirect
14. C4×D4×D101C2semidirect
15. #548×C101C2semidirect
18. C2×#13108075C22semidirect
19. C2×#13108075C22semidirect
20. C2×#13108075C22semidirect
21. C2×#13108074C22semidirect
22. C2×#13108073C22semidirect
24. C2×Q8×D101C22semidirect
25. C22×D404C22semidirect
27. C2×D4×D101C22semidirect
28. C22×C4×D101C22semidirect
29. C22×C4×D101C22semidirect
30. #15692253C22semidirect
31. #15692251C22semidirect
32. #15692243C22semidirect
33. #15692240C22semidirect
34. #15692237C22semidirect
35. #15692235C22semidirect
36. #15692234C22semidirect
37. #15692231C22semidirect
38. #15692230C22semidirect
39. #15692229C22semidirect
40. #15692228C22semidirect
41. #15692228C22semidirect
42. #15692226C22semidirect
43. #15692225C22semidirect
44. #15692223C22semidirect
45. Q8×Dic101C22semidirect
46. C4×D404C22semidirect
47. C4×D404C22semidirect
48. D4×Dic101C22semidirect
49. #15692217C22semidirect
50. #15692217C22semidirect
52. C4×C22⋊D101C22semidirect
54. #15692215C22semidirect
55. #15692215C22semidirect
56. #15692213C22semidirect
57. C4⋊C4×D101C22semidirect
58. C22⋊C4×D101C22semidirect
59. C22⋊C4×D101C22semidirect
60. C42×D101C22semidirect
61. C2×#41×C101C22semidirect
62. C22×Q8×C101C22semidirect
63. #124×C101C22semidirect
64. C22⋊Q8×C101C22semidirect
65. #121×C101C22semidirect
66. C4×Q8×C101C22semidirect
67. C4×D4×C101C22semidirect
73. #13108075D4semidirect
144. Q8×C101C2×D4semidirect
157. C22⋊D101#41semidirect
168. D404C2×D4semidirect
177. C4×D101C2×D4semidirect
184. C22×C101C2×#41semidirect
196. C2×D101C2×#41semidirect
206. Dic101C2×#41semidirect
216. C101D4×#41semidirect
222. #548D101semidirect
240. C2×#41C2×D101semidirect
242. C22×Q8C2×D101semidirect
251. #124C2×D101semidirect
257. C22⋊Q8C2×D101semidirect
263. #121C2×D101semidirect
267. C4×Q8C2×D101semidirect
278. C4×D4C2×D101semidirect
361. Q8D4×D101semidirect
411. C22#41×D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080752
C2×#41×D1012
#202658462
Q8×C22⋊D1012
#202658392
#202658292
#202658232
#202658202
#202658112
#202657262
#202656902
#202656872
C4×#131080752
C4×D4×D1012
#548×C1012
D4×#41101