Finite Group Databasebeta

← #29667853

Subgroups — #29667853

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456943C211yesmaximal, characteristic, complemented
2#21456761C211yesmaximal, characteristic, complemented
3#21456376C211yesmaximal, characteristic, complemented
4#21456362C211yesmaximal, characteristic, complemented
5#21456270C211yesmaximal, characteristic
6#492×D61C211yesmaximal, characteristic, complemented
7#21445449C211yesmaximal, characteristic, complemented
8#21445445C211yesmaximal, characteristic, complemented
9#21445442C211yesmaximal, characteristic, complemented
10#21445440C211yesmaximal, characteristic, complemented
11#21445409C211yesmaximal, characteristic, complemented
12#21445406C211yesmaximal, characteristic, complemented
13#21445405C211yesmaximal, characteristic, complemented
14C8×#11758189C211yesmaximal, characteristic, complemented
15C61×#1706C211yesmaximal, characteristic, complemented
16C2×#14519801—22no—
17C2×SD16×D61—22no—
18#16840317C2211yescharacteristic
19#16840310C2211yescharacteristic
20C4⋊Q8×D61C2211yescharacteristic
21#127×D61C2211yescharacteristic
22#16840170—22no—
23#16840165C2211yescharacteristic, complemented
24#16840160C2211yescharacteristic, complemented
25#16840160C2211yescharacteristic, complemented
26#16840160C2211yescharacteristic, complemented
27#16840158C2211yescharacteristic, complemented
28#16840158C2211yescharacteristic, complemented
29#16840150—22no—
30#16840138C2211yescharacteristic
31#16840136C2211yescharacteristic
32#16840135C2211yescharacteristic
33#16840135C2211yescharacteristic
34#16840135C2211yescharacteristic
35#16840134C2211yescharacteristic
36#16840134C2211yescharacteristic
37#16840134C2211yescharacteristic
38#16840133—22no—
39#16840112C2211yescharacteristic, complemented
40#16840110C2211yescharacteristic
41#16840108C2211yescharacteristic, complemented
42#16840108C2211yescharacteristic, complemented
43#16840079—22no—
44#16840050—22no—
45#16839988—22no—
46#16839961—22no—
47#16839914—22no—
48SD16×Dic61—22no—
49#16839813C2211yescharacteristic
50#16839803—22no—
51C4×#11758189C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic, complemented
54#16839483C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#500C2211yescharacteristic, complemented
57C61×#493C2211yescharacteristic
58C61×#492C2211yescharacteristic
59C61×#492C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic
61C61×#486C2211yescharacteristic, complemented
62C82×C61C2211yescharacteristic, complemented
63C2×#11758224—21no—
64C2×Q8×D61—21no—
65C2×D4×D61—21no—
66C2×D4×D61—21no—
67C2×D4×D61—21no—
68#14519801—48no—
69C2×#11758190—22no—
70C2×#11758190—22no—
71C2×#11758190—22no—
72C2×#11758200—22no—
73C2×#11758200—22no—
74C2×#11758200—22no—
75C2×#11758201—22no—
76SD16×D61—48no—
77C2×#11758199—22no—
78C2×#11758189D412yes—
79C2×C8×D61D412yes—
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519777C2311yescharacteristic
83#14519776—21no—
84#14519776—21no—
85#14519775—21no—
86#14519774C2311yescharacteristic
87#14519773—21no—
88#14519773—21no—
89#14519773—21no—
90#14519768—21no—
91#14519764—21no—
92#14519757—21no—
93#14519753—21no—
94#14519748—21no—
95#14519748—21no—
96Q8×Dic61—21no—
97#14519742—21no—
98D4×Dic61—21no—
99D4×Dic61—21no—
100#14519735—21no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61—21no—
103C42×D61C2311yescharacteristic
104#14519710—22no—
105#14519709—22no—
106#14519709—22no—
107#14519709—22no—
108#14519708—22no—
109#14519708—22no—
110#14519706D412yes—
111#14519701—22no—
112#14519698—22no—
113#14519698—22no—
114#14519697—22no—
115#14519689D412yes—
116#14519687D412yes—
117C4×#92418C2311yescharacteristic
118C4×#92418C2311yescharacteristic
119C4×#92418C2311yescharacteristic
120C8×Dic61D412yes—
121C2×SD16×C61—22no—
122C2×SD16×C61—22no—
123C2×D8×C61—22no—
124C4⋊Q8×C61C2311yescharacteristic
125C4⋊Q8×C61C2311yescharacteristic
126#127×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#107×C61—22no—
129#102×C61—22no—
130#102×C61—22no—
131C4×C8×C61C2311yescharacteristic
132C4×C8×C61C2311yescharacteristic
133C4×C8×C61C2311yescharacteristic
134C23×D61—41no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C22×Dic61—41no—
148Q8×D61—44no—
149#11758224—44no—
150D4×D61—44no—
151D4×D61—44no—
152D4×D61—82no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×C4×D61—41no—
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—41no—
164SD16×C61—44no—
165SD16×C61—44no—
166D8×C61—44no—
167C2×C8×C61C2×D412yes—
168C2×C8×C61C2×D412yes—
169C2×C8×C61C2×D412yes—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C42×C61C2411yescharacteristic
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177#11758201—44no—
178#11758200—44no—
179#11758200—44no—
180#11758200—82no—
181#11758199—44no—
182#11758198—41no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190#11758196—41no—
191#11758196—41no—
192#11758196—41no—
193C4×Dic61—41no—
194C4×Dic61C2×D411yescharacteristic
195C4×Dic61C2×D411yescharacteristic
196C4×Dic61C2×D411yescharacteristic
197C2×#92418C2×D412yes—
198C2×#92418C2×D412yes—
199C2×#92418C2×D412yes—
200#11758190—44no—
201#11758190—44no—
202#11758190—82no—
203#11758189SD1618yescomplemented
204C8×D61—24no—
205C23×C61—41no—
206C23×C61—41no—
207C22×D61—41no—
208C22×D61—82no—
209C22×D61—82no—
210C22×D61—82no—
211C22×D61#12711yescharacteristic
212Q8×C61—42no—
213Q8×C61—42no—
214D4×C61—42no—
215D4×C61—42no—
216D4×C61—42no—
217D4×C61—42no—
218D4×C61—42no—
219D4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C22⋊D61—82no—
226C22⋊D61—82no—
227C22⋊D61—82no—
228C22⋊D61—82no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—82no—
233C2×Dic61#12711yescharacteristic
234C2×Dic61#12711yescharacteristic
235C2×Dic61#12711yescharacteristic
236D244—42no—
237D244—42no—
238D244—42no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—82no—
242C4×D61C2×SD1614yes—
243Dic122—42no—
244Dic122—42no—
245Dic122—42no—
246Dic122—82no—
247C8×C61—22no—
248C8×C61C2×SD1614yescomplemented
249C8×C61AGL(1,ℤ/8)14yescomplemented
250#92418—22no—
251#92418—22no—
252#92418C2×SD1614yescomplemented
253#19621—611nomaximal, Sylow
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61C2×#12711yescharacteristic
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—164no—
267C2×D61#49212yes—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61C22×SD1612yes—
271C4×C61C2×AGL(1,ℤ/8)12yes—
272C4×C61#57712yes—
273Dic61—21no—
274Dic61—21no—
275Dic61—81no—
276Dic61—81no—
277Dic61—81no—
278Dic61#49212yes—
279#3227—611no—
280#3208—611no—
281#3167—611no—
282#3165—611no—
283#3151—611no—
284C2×#492—611no—
285#1715—611no—
286#1715—611no—
287#1714—611no—
288#1714—611no—
289#1706—611no—
290#1706D6111yescharacteristic, complemented
291#1705—611no—
292#1705—611no—
293C8×M16—611no—
294C2×C61—161no—
295C2×C61—161no—
296C2×C61C2×#49211yescharacteristic
297C2×C61#315111yescharacteristic
298C2×C61#315411yescharacteristic
299D61—22no—
300D61—161no—
301C2×AGL(1,ℤ/8)—1222no—
302C22×SD16—1222no—
303#533—611no—
304#532—611no—
305C2×C4⋊Q8—611no—
306C2×#127—611no—
307#501—1222no—
308#500—611no—
309#500C2×D6111yescharacteristic, complemented
310#498—611no—
311#498—611no—
312#498—611no—
313#498—611no—
314#498—611no—
315#496—1222no—
316#493—611no—
317#493C2×D6111yescharacteristic
318#492—611no—
319#492—611no—
320#492—611no—
321#492—611no—
322#492—611no—
323#492—611no—
324#492—611no—
325#492—1222no—
326#492C2×D6111yescharacteristic
327#492C2×D6111yescharacteristic, complemented
328#487—611no—
329#486—611no—
330#486—611no—
331#486—611no—
332#486C2×D6111yescharacteristic
333#486C2×D6111yescharacteristic, complemented
334#481—1222no—
335#476—1222no—
336#465—1222no—
337#460—1222no—
338#450—1222no—
339C4×SD16—1222no—
340#431—611no—
341#428—1222no—
342C4×M16—611no—
343C2×C4×C8—611no—
344#322—611no—
345#322—611no—
346C82—611no—
347C82C2×D6111yescharacteristic, complemented
348C61#1962111yescharacteristic, complemented, Sylow
349C2×#41—1221no—
350C22×Q8—1221no—
351C22×D4—1221no—
352C22×D4—1221no—
353C22×D4—1221no—
354AGL(1,ℤ/8)—2448no—
355C2×SD16—22no—
356C2×SD16—22no—
357C2×SD16—1222no—
358C2×SD16—1222no—
359C2×SD16—1222no—
360C2×SD16—1222no—
361C2×SD16—1222no—
362C2×SD16—1222no—
363C2×SD16—1222no—
364C2×SD16—2448no—
365C2×D8—22no—
366C2×D8—1222no—
367C2×M16—612no—
368C22×C8—612no—
369C4⋊Q8—611no—
370C4⋊Q8—611no—
371C4⋊Q8—611no—
372C4⋊Q8—1221no—
373C4⋊Q8—1221no—
374C4⋊Q8—1221no—
375C4⋊Q8C22×D6111yescharacteristic
376C4⋊Q8C22×D6111yescharacteristic
377#127—611no—
378#127—1221no—
379#127—1221no—
380#127—1221no—
381#127C22×D6111yescharacteristic
382#127C22×D6111yescharacteristic
383#125—1221no—
384#124—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387#121—1221no—
388#121—1221no—
389C4×Q8—1221no—
390C4×Q8—1221no—
391C4×D4—1221no—
392C4×D4—1221no—
393#117—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C42—611no—
397#107—22no—
398#107—1222no—
399#106—1222no—
400#106—1222no—
401#106—1222no—
402#106—1222no—
403#106—1222no—
404C4⋊C8—612no—
405#103—1222no—
406#102—22no—
407#102—22no—
408#102—1222no—
409#102—1222no—
410#102—1222no—
411C22⋊C8—612no—
412#97—612no—
413C4×C8—612no—
414C4×C8—611no—
415C4×C8—611no—
416C4×C8—611no—
417C4×C8C22×D6111yescharacteristic
418C4×C8C22×D6111yescharacteristic
419C4×C8C22×D6111yescharacteristic
420C24—2441no—
421#41—2444no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—2441no—
428C2×Q8—2444no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—4882no—
439C2×D4—2444no—
440C2×D4—2441no—
441C2×D4—2441no—
442C2×D4—2441no—
443C2×D4—2444no—
444C2×D4—2441no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—2441no—
449C22×C4—2441no—
450SD16—44no—
451SD16—44no—
452SD16—4882no—
453SD16—4882no—
454SD16—2444no—
455SD16—2444no—
456SD16—2444no—
457SD16—2444no—
458SD16—2444no—
459D8—44no—
460D8—2444no—
461M16—618no—
462C2×C8—612no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—1224no—
466C2×C8D4×D6112yes—
467C2×C8D4×D6112yes—
468C2×C8D4×D6112yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C42—611no—
487C42—611no—
488C42—611no—
489C42—2441no—
490C42C23×D6111yescharacteristic
491C23—41no—
492C23—41no—
493C23—611no—
494C23—4882no—
495C23—4882no—
496C23—4882no—
497C23—2441no—
498Q8—42no—
499Q8—42no—
500Q8—4882no—
501Q8—2442no—
502Q8—2442no—
503Q8—2442no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—4882no—
511D4—4882no—
512D4—4882no—
513D4—4882no—
514D4—2442no—
515D4—2442no—
516D4—2442no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—611no—
520C2×C4—614no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—4882no—
526C2×C4—4882no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C2×D4×D6111yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C8—22no—
534C8—614no—
535C8—1222no—
536C8—1222no—
537C8SD16×D6114yescomplemented
538C8#1451980114yescomplemented
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—612no—
546C22—1221no—
547C22—1221no—
548C22—9764no—
549C22—4881no—
550C22—4881no—
551C22—4881no—
552C22#127×D6111yescharacteristic
553C4—81no—
554C4—81no—
555C4—612no—
556C4—1221no—
557C4—1221no—
558C4—4881no—
559C4—4881no—
560C4—4881no—
561C4C2×SD16×D6112yes—
562C4C2×#1451980112yes—
563C4#1684056912yes—
564C2—161no—
565C2—161no—
566C2—1222no—
567C2—9761no—
568C2#492×D6111yescharacteristic
569C2#2145627011yescharacteristic
570C2#2145629011yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1706C2
socleC22×C61C2×#127
prime radical#21456270C2
hypercenter#1706D61
hypocenterC61#19621
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1706

Complements

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

NH ≅ G/Ntype
1. #21456943C2semidirect
2. #21456761C2semidirect
3. #21456376C2semidirect
4. #21456362C2semidirect
6. #492×D61C2semidirect
7. #21445449C2semidirect
8. #21445445C2semidirect
9. #21445442C2semidirect
10. #21445440C2semidirect
11. #21445409C2semidirect
12. #21445406C2semidirect
13. #21445405C2semidirect
14. C8×#11758189C2semidirect
15. C61×#1706C2semidirect
23. #16840165C22semidirect
24. #16840160C22semidirect
25. #16840160C22semidirect
26. #16840160C22semidirect
27. #16840158C22semidirect
28. #16840158C22semidirect
39. #16840112C22semidirect
41. #16840108C22semidirect
42. #16840108C22semidirect
53. #16839484C22semidirect
54. #16839483C22semidirect
55. C8×#92418C22semidirect
56. C61×#500C22semidirect
59. C61×#492C22semidirect
61. C61×#486C22semidirect
62. C82×C61C22semidirect
203. #11758189SD16semidirect
248. C8×C61C2×SD16semidirect
249. C8×C61AGL(1,ℤ/8)semidirect
252. #92418C2×SD16semidirect
290. #1706D61semidirect
309. #500C2×D61semidirect
327. #492C2×D61semidirect
333. #486C2×D61semidirect
347. C82C2×D61semidirect
348. C61#19621semidirect
537. C8SD16×D61semidirect
538. C8#14519801semidirect

Maximal subgroups

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

subgroupindex
#214569432
#214567612
#214563762
#214563622
#214562702
#492×D612
#214454492
#214454452
#214454422
#214454402
#214454092
#214454062
#214454052
C8×#117581892
C61×#17062
#1962161