Finite Group Databasebeta

← #27185916

Subgroups — #27185916

Order 12928.

1412 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20266013C211yesmaximal, characteristic, complemented
2C2×#15692302C211yesmaximal, characteristic, complemented
3#20265993C214yesmaximal, complemented
4#20265991C214yesmaximal, complemented
5#20265987C214yesmaximal, complemented
6#20265984C214yesmaximal, complemented
7#20265975C211yesmaximal, characteristic, complemented
8#20265973C214yesmaximal, complemented
9C2×#15692288C211yesmaximal, characteristic, complemented
10C2×#15692285C211yesmaximal, characteristic, complemented
11#137×D101C214yesmaximal, complemented
12#20265943C211yesmaximal, characteristic, complemented
13#575×C101C211yesmaximal, characteristic, complemented
14#15692303C2212yescomplemented
15#15692302—22no—
16#15692302—22no—
17#15692302—21no—
18#15692302C2214yescomplemented
19#15692302C2214yescomplemented
20#15692302C2212yescomplemented
21#15692301—21no—
22#15692301C2212yescomplemented
23#15692300—22no—
24#15692300C2212yescomplemented
25C2×#13108074—22no—
26C2×#13108074—21no—
27C2×#13108074C2211yescharacteristic
28C2×#13108074C2211yescharacteristic
29C2×#13108073—22no—
30C2×#13108073C2212yescomplemented
31#41×D101—22no—
32#41×D101C2214yes—
33C22×Dic202—21no—
34C22×Dic202C2211yescharacteristic, complemented
35C2×Q8×D101C2212yescomplemented
36#15692288C2214yescomplemented
37#15692288C2214yescomplemented
38#15692287C2218yescomplemented
39#15692286C2212yescomplemented
40#15692285C2214yescomplemented
41#15692285C2212yescomplemented
42#15692285C2214yescomplemented
43#15692284C2218yescomplemented
44#15692284C2218yescomplemented
45#15692281C2212yescomplemented
46#15692280C2212yescomplemented
47#15692279C2218yescomplemented
48#15692276C2218yescomplemented
49#15692275C2214yescomplemented
50#15692274C2218yescomplemented
51C2×Dic404C2212yescomplemented
52C2×#13108064C2212yescomplemented
53Q16×D101C2218yescomplemented
54C2×#13108063C2212yescomplemented
55C2×#13108062C2212yescomplemented
56SD16×D101C2218yescomplemented
57#15692263C2214yescomplemented
58#15692261C2214yescomplemented
59C2×#13108058C2211yescharacteristic, complemented
60M16×D101C2214yescomplemented
61C2×#41×C101C2211yescharacteristic
62C2×#41×C101C2211yescharacteristic, complemented
63#137×C101C2212yescomplemented
64#137×C101C2212yescomplemented
65AGL(1,ℤ/8)×C101C2212yescomplemented
66AGL(1,ℤ/8)×C101C2212yescomplemented
67#135×C101C2214yescomplemented
68C2×M16×C101C2211yescharacteristic, complemented
69#13108075—22no—
70#13108075C2314yes—
71#13108074—21no—
72#13108074—21no—
73#13108074—24no—
74#13108074—22no—
75#13108074—24no—
76#13108074—24no—
77#13108074—24no—
78#13108074—22no—
79#13108074D412yes—
80#13108074D412yes—
81#13108074C2312yes—
82#13108074C2312yes—
83#13108074C2312yes—
84#13108074C2312yes—
85#13108073—24no—
86#13108073—28no—
87#13108073—28no—
88#13108073—24no—
89#13108073—24no—
90#13108073—22no—
91#13108073C2314yes—
92#13108073C2318yes—
93C2×Dic202—21no—
94C2×Dic202—21no—
95C2×Dic202—21no—
96C2×Dic202—22no—
97C2×Dic202—24no—
98C2×Dic202—24no—
99C2×Dic202—22no—
100C2×Dic202D411yescharacteristic
101C2×Dic202C2312yes—
102C2×Dic202C2312yes—
103C2×Dic202C2312yes—
104C2×Dic202C2311yescharacteristic
105Q8×D101—24no—
106Q8×D101—22no—
107Q8×D101C2318yes—
108Q8×D101C2314yes—
109C2×D404—22no—
110C2×D404D411yescharacteristic
111C2×D404C2311yescharacteristic
112C2×C22⋊D101—21no—
113C2×C22⋊D101—22no—
114C2×C22⋊D101—24no—
115C2×C22⋊D101D412yes—
116D4×D101—24no—
117D4×D101—24no—
118D4×D101—22no—
119D4×D101C2314yes—
120C22×Dic101—21no—
121C22×Dic101—22no—
122C2×C4×D101—22no—
123C2×C4×D101—21no—
124C2×C4×D101—24no—
125C2×C4×D101C2312yes—
126Dic404C2318yes—
127#13108064C2314yes—
128#13108064C2318yes—
129#13108063C2318yes—
130#13108062C2318yes—
131#13108062C2314yes—
132#13108061C2314yes—
133#13108059C2314yes—
134#13108058C2312yes—
135#13108058C2312yes—
136#13108057C2318yes—
137C2×#11717579C2312yes—
138C8×D101C2318yes—
139#41×C101—21no—
140#41×C101—21no—
141#41×C101—21no—
142#41×C101—21no—
143#41×C101C2312yes—
144#41×C101C2312yes—
145#41×C101C2312yes—
146#41×C101C2312yes—
147C2×Q8×C101C2311yescharacteristic
148C2×Q8×C101C2311yescharacteristic
149C2×D4×C101—21no—
150C2×D4×C101—21no—
151C2×D4×C101C2311yescharacteristic
152C2×D4×C101C2311yescharacteristic
153C22×C4×C101—21no—
154C22×C4×C101—21no—
155C22×C4×C101C2311yescharacteristic
156Q16×C101C2314yes—
157SD16×C101C2314yes—
158SD16×C101C2314yes—
159D8×C101C2314yes—
160M16×C101C2312yes—
161M16×C101C2312yes—
162C2×C8×C101C2312yes—
163C23×C101—21no—
164C23×C101—21no—
165C23×C101C2×D411yescharacteristic
166C22×D101—21no—
167C22×D101—24no—
168C22×D101C2×D412yes—
169Q8×C101—21no—
170Q8×C101—21no—
171Q8×C101C2412yes—
172Q8×C101C2412yes—
173D4×C101—22no—
174D4×C101—22no—
175D4×C101—22no—
176D4×C101—22no—
177D4×C101—21no—
178D4×C101—21no—
179D4×C101C2412yes—
180D4×C101C2412yes—
181C2×C4×C101—21no—
182C2×C4×C101—21no—
183C2×C4×C101—22no—
184C2×C4×C101—22no—
185C2×C4×C101—22no—
186C2×C4×C101—22no—
187C2×C4×C101C2×D411yescharacteristic
188C2×C4×C101C2×D411yescharacteristic
189C2×C4×C101C2×D411yescharacteristic
190C2×C4×C101C2411yescharacteristic
191C2×C4×C101C2411yescharacteristic
192C2×C4×C101C2411yescharacteristic
193C22⋊D101—22no—
194C22⋊D101—22no—
195C22⋊D101—24no—
196C22⋊D101—28no—
197C22⋊D101—24no—
198C22⋊D101C2×D414yes—
199C22⋊D101C2×D414yes—
200C2×Dic101—22no—
201C2×Dic101—21no—
202C2×Dic101—22no—
203C2×Dic101—22no—
204C2×Dic101—24no—
205C2×Dic101—28no—
206C2×Dic101C2×D412yes—
207D404—24no—
208D404—21no—
209D404C2×D412yes—
210D404C2×D412yes—
211D404C2412yes—
212C4×D101—24no—
213C4×D101—28no—
214C4×D101—22no—
215C4×D101—22no—
216C4×D101—24no—
217C4×D101C2×D414yes—
218C4×D101C2414yes—
219Dic202—22no—
220Dic202—22no—
221Dic202—22no—
222Dic202—21no—
223Dic202—28no—
224Dic202—24no—
225Dic202C2×D412yes—
226Dic202C2×D412yes—
227Dic202C2414yes—
228Dic202C2412yes—
229C8×C101C2414yes—
230#11717579C2414yes—
231C22×C101—22no—
232C22×C101—22no—
233C22×C101—41no—
234C22×C101—42no—
235C22×C101—81no—
236C22×C101C22×D411yescharacteristic
237C22×C101C22×D411yescharacteristic
238C22×C101C22×D411yescharacteristic
239C2×D101—22no—
240C2×D101—42no—
241C2×D101—84no—
242C2×D101—81no—
243C2×D101C22×D414yes—
244C4×C101—22no—
245C4×C101—22no—
246C4×C101C22×D411yescharacteristic
247C4×C101C22×D411yescharacteristic
248C4×C101C22×D411yescharacteristic
249C4×C101C2511yescharacteristic
250Dic101—24no—
251Dic101—22no—
252Dic101C22×D414yes—
253C2×C101—21no—
254C2×C101—21no—
255C2×C101—21no—
256C2×C101—42no—
257C2×C101—42no—
258C2×C101C23×D411yescharacteristic
259D101—24no—
260D101—42no—
261#3586—1011nomaximal, Sylow
262C101#358611yescharacteristic, complemented, Sylow
263#585—1011no—
264C2×2−1+4—1011no—
265#578—1014no—
266#578—1014no—
267#577—1014no—
268#577—1014no—
269#575—1011no—
270#575—1014no—
271#575D10111yescharacteristic, complemented
272C2×#137—1011no—
273C2×#137—1014no—
274C2×#137—1011no—
275#568—1011no—
2762−1+4—1014no—
2772−1+4—1014no—
2782−1+4—1012no—
2792−1+4—1012no—
2802−1+4—2021no—
2812−1+4—2022no—
2822−1+4—2022no—
283H(5,𝔽2)—1012no—
284H(5,𝔽2)—1012no—
285H(5,𝔽2)—2022no—
286H(5,𝔽2)—2021no—
287C2×#41—1011no—
288C2×#41—1014no—
289C2×#41—1012no—
290C2×#41—1011no—
291C2×#41—2021no—
292C2×#41—2022no—
293C2×#41—2022no—
294C2×#41—2022no—
295C2×#41C2×D10111yescharacteristic
296C2×#41C2×D10111yescharacteristic, complemented
297C22×Q8—1012no—
298C22×Q8—1011no—
299C22×Q8—2021no—
300#137—1014no—
301#137—1014no—
302#137—1012no—
303#137—1012no—
304#137—1018no—
305#137—1018no—
306#137—1018no—
307#137—1014no—
308#137—1014no—
309#137C2×D10112yescomplemented
310#137C2×D10112yescomplemented
311AGL(1,ℤ/8)—1012no—
312AGL(1,ℤ/8)—1012no—
313AGL(1,ℤ/8)—1018no—
314AGL(1,ℤ/8)C2×D10112yescomplemented
315AGL(1,ℤ/8)C2×D10112yescomplemented
316#135—1014no—
317#135—1018no—
318#135—1018no—
319#135C2×D10114yescomplemented
320C2×Q16—1012no—
321C2×Q16—1018no—
322C2×Q16—1012no—
323C2×SD16—1012no—
324C2×SD16—1018no—
325C2×SD16—1012no—
326#131—1014no—
327#131—1014no—
328C2×M16—1011no—
329C2×M16—1014no—
330C2×M16C2×D10111yescharacteristic, complemented
331#41—21no—
332#41—21no—
333#41—21no—
334#41—21no—
335#41—1012no—
336#41—1012no—
337#41—1012no—
338#41—1012no—
339#41—1014no—
340#41—1018no—
341#41—1014no—
342#41—1012no—
343#41—1012no—
344#41—2024no—
345#41—2024no—
346#41—2022no—
347#41—2024no—
348#41—2024no—
349#41—2022no—
350#41—2024no—
351#41—2024no—
352#41—2024no—
353#41—2028no—
354#41—2028no—
355#41—2022no—
356#41—2022no—
357#41—2021no—
358#41—2021no—
359#41C22×D10112yes—
360#41C22×D10112yes—
361#41C22×D10112yes—
362#41C22×D10112yes—
363C2×Q8—1011no—
364C2×Q8—1018no—
365C2×Q8—1014no—
366C2×Q8—1012no—
367C2×Q8—1012no—
368C2×Q8—1011no—
369C2×Q8—1012no—
370C2×Q8—2022no—
371C2×Q8—2024no—
372C2×Q8—2024no—
373C2×Q8—2022no—
374C2×Q8—2024no—
375C2×Q8—2022no—
376C2×Q8—2021no—
377C2×Q8—2021no—
378C2×Q8—2021no—
379C2×Q8C22×D10111yescharacteristic
380C2×Q8C22×D10111yescharacteristic
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—1012no—
384C2×D4—1014no—
385C2×D4—1011no—
386C2×D4—1011no—
387C2×D4—2022no—
388C2×D4—2024no—
389C2×D4—2022no—
390C2×D4—2024no—
391C2×D4—2024no—
392C2×D4—2022no—
393C2×D4—2021no—
394C2×D4C22×D10111yescharacteristic
395C2×D4C22×D10111yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—1012no—
399C22×C4—2024no—
400C22×C4—2022no—
401C22×C4—2022no—
402C22×C4—2021no—
403C22×C4—2021no—
404C22×C4C22×D10111yescharacteristic
405Q16—1018no—
406Q16—1014no—
407Q16—1018no—
408Q16C22×D10114yes—
409SD16—1014no—
410SD16—1018no—
411SD16—1014no—
412SD16—1018no—
413SD16C22×D10114yes—
414SD16C22×D10114yes—
415D8—1014no—
416D8C22×D10114yes—
417M16—1012no—
418M16—1012no—
419M16—1018no—
420M16C22×D10112yes—
421M16C22×D10112yes—
422C2×C8—1012no—
423C2×C8—1018no—
424C2×C8C22×D10112yes—
425C23—21no—
426C23—21no—
427C23—1012no—
428C23—2024no—
429C23—2021no—
430C23D4×D10111yescharacteristic
431Q8—21no—
432Q8—21no—
433Q8—1012no—
434Q8—1012no—
435Q8—1014no—
436Q8—1012no—
437Q8—2028no—
438Q8—2024no—
439Q8—2022no—
440Q8—2022no—
441Q8—2022no—
442Q8—2021no—
443Q8C23×D10112yes—
444Q8C23×D10112yes—
445D4—22no—
446D4—22no—
447D4—22no—
448D4—22no—
449D4—21no—
450D4—21no—
451D4—1014no—
452D4—1014no—
453D4—1012no—
454D4—1012no—
455D4—1012no—
456D4—2024no—
457D4—2024no—
458D4—2024no—
459D4—2028no—
460D4—2022no—
461D4—2022no—
462D4—2021no—
463D4C23×D10112yes—
464D4C23×D10112yes—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—1012no—
472C2×C4—1014no—
473C2×C4—1014no—
474C2×C4—2024no—
475C2×C4—2024no—
476C2×C4—2028no—
477C2×C4—2024no—
478C2×C4—2028no—
479C2×C4—2022no—
480C2×C4—2021no—
481C2×C4—2022no—
482C2×C4—2022no—
483C2×C4—2022no—
484C2×C4—2022no—
485C2×C4D4×D10111yescharacteristic
486C2×C4D4×D10111yescharacteristic
487C2×C4D4×D10111yescharacteristic
488C2×C4C23×D10111yescharacteristic
489C2×C4C23×D10111yescharacteristic
490C2×C4C23×D10111yescharacteristic
491C8—1014no—
492C8C23×D10114yes—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—42no—
497C22—81no—
498C22—8081no—
499C22—8084no—
500C22—1014no—
501C22—4042no—
502C22—2022no—
503C22C2×D4×D10111yescharacteristic
504C22C2×D4×D10111yescharacteristic
505C22C2×D4×D10111yescharacteristic
506C4—22no—
507C4—22no—
508C4—1014no—
509C4—2022no—
510C4—2024no—
511C4C2×D4×D10111yescharacteristic
512C4C2×D4×D10111yescharacteristic
513C4C2×D4×D10111yescharacteristic
514C4C24×D10111yescharacteristic
515C2—21no—
516C2—21no—
517C2—21no—
518C2—42no—
519C2—42no—
520C2—4042no—
521C2—2024no—
522C2C22×D4×D10111yescharacteristic
subgroupG/H
centerC2C22×D4×D101
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
Fitting#575×C101C2
socleC2×C101C23×D4
hypercenter#575D101
hypocenterC101#3586
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20266013C2semidirect
2. C2×#15692302C2semidirect
3. #20265993C2semidirect
4. #20265991C2semidirect
5. #20265987C2semidirect
6. #20265984C2semidirect
7. #20265975C2semidirect
8. #20265973C2semidirect
9. C2×#15692288C2semidirect
10. C2×#15692285C2semidirect
11. #137×D101C2semidirect
12. #20265943C2semidirect
13. #575×C101C2semidirect
14. #15692303C22semidirect
18. #15692302C22semidirect
19. #15692302C22semidirect
20. #15692302C22semidirect
22. #15692301C22semidirect
24. #15692300C22semidirect
30. C2×#13108073C22semidirect
34. C22×Dic202C22semidirect
35. C2×Q8×D101C22semidirect
36. #15692288C22semidirect
37. #15692288C22semidirect
38. #15692287C22semidirect
39. #15692286C22semidirect
40. #15692285C22semidirect
41. #15692285C22semidirect
42. #15692285C22semidirect
43. #15692284C22semidirect
44. #15692284C22semidirect
45. #15692281C22semidirect
46. #15692280C22semidirect
47. #15692279C22semidirect
48. #15692276C22semidirect
49. #15692275C22semidirect
50. #15692274C22semidirect
51. C2×Dic404C22semidirect
52. C2×#13108064C22semidirect
53. Q16×D101C22semidirect
54. C2×#13108063C22semidirect
55. C2×#13108062C22semidirect
56. SD16×D101C22semidirect
57. #15692263C22semidirect
58. #15692261C22semidirect
59. C2×#13108058C22semidirect
60. M16×D101C22semidirect
62. C2×#41×C101C22semidirect
63. #137×C101C22semidirect
64. #137×C101C22semidirect
65. AGL(1,ℤ/8)×C101C22semidirect
66. AGL(1,ℤ/8)×C101C22semidirect
67. #135×C101C22semidirect
68. C2×M16×C101C22semidirect
262. C101#3586semidirect
271. #575D101semidirect
296. C2×#41C2×D101semidirect
309. #137C2×D101semidirect
310. #137C2×D101semidirect
314. AGL(1,ℤ/8)C2×D101semidirect
315. AGL(1,ℤ/8)C2×D101semidirect
319. #135C2×D101semidirect
330. C2×M16C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202660132
C2×#156923022
#202659932
#202659912
#202659872
#202659842
#202659752
#202659732
C2×#156922882
C2×#156922852
#137×D1012
#202659432
#575×C1012
#3586101