Finite Group Databasebeta

← #29649648

Subgroups — #29649648

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454935C211yesmaximal, characteristic
2#21454898C211yesmaximal, characteristic, complemented
3C4×#14519809C211yesmaximal, characteristic
4#21444697C211yesmaximal, characteristic, complemented
5#21444696C211yesmaximal, characteristic, complemented
6#21444691C211yesmaximal, characteristic, complemented
7#21444688C211yesmaximal, characteristic, complemented
8#21444672C211yesmaximal, characteristic, complemented
9#21444664C211yesmaximal, characteristic, complemented
10#21444659C211yesmaximal, characteristic, complemented
11#21444570C211yesmaximal, characteristic, complemented
12#21444200C211yesmaximal, characteristic
13Q8⋊C8×D61C211yesmaximal, characteristic, complemented
14#21444107C211yesmaximal, characteristic, complemented
15C61×#1584C211yesmaximal, characteristic, complemented
16C2×#14519809C411yescharacteristic
17C4×#11758226C2211yescharacteristic
18C4×Q8×D61C2211yescharacteristic
19#16839893C2211yescharacteristic
20Q8×#92418C2211yescharacteristic
21C8×Dic122—21no—
22C8×Dic122C2211yescharacteristic
23#16839886—21no—
24#16839886C2211yescharacteristic
25#16839869C411yescharacteristic
26#16839866C411yescharacteristic
27#16839865C411yescharacteristic
28#16839861C411yescharacteristic
29C4×#11758202C2211yescharacteristic
30C4×#11758190C2211yescharacteristic
31C4×#11758201C2211yescharacteristic
32#16839841C2211yescharacteristic
33#16839839C2211yescharacteristic
34#16839835—21no—
35#16839832—21no—
36#16839817—21no—
37#16839814—21no—
38#16839814C2211yescharacteristic
39#16839813—21no—
40#16839813C2211yescharacteristic
41#16839803C411yescharacteristic
42#16839784—21no—
43C4⋊C8×D61C2211yescharacteristic
44#16839769C411yescharacteristic
45#16839762C411yescharacteristic
46C4×#11758189C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic
49#16839513C2211yescharacteristic
50#16839512C2211yescharacteristic
51#16839494C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839492C2211yescharacteristic
55#16839491C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic
58#16839484C2211yescharacteristic
59#16839484C2211yescharacteristic
60C61×#446C2211yescharacteristic
61C8×Q8×C61C2211yescharacteristic
62C4×Q16×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×Q8⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic
67C2×#11758226C2×C411yescharacteristic
68C2×Q8×D61C2×C411yescharacteristic
69#14519809—24no—
70C2×#11758202C2×C411yescharacteristic
71C2×#11758190C2×C411yescharacteristic
72C2×#11758201C2×C411yescharacteristic
73C2×#11758189—21no—
74C2×#11758189—21no—
75C2×#11758189C2×C411yescharacteristic
76C2×#11758189C2×C411yescharacteristic
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79Q8×Dic61C2×C411yescharacteristic
80Q8×Dic61C2×C411yescharacteristic
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519742C2×C411yescharacteristic
85C4×D244—21no—
86C4×D244C2311yescharacteristic
87#14519739—21no—
88#14519739C2×C411yescharacteristic
89#14519735—21no—
90#14519735C2×C411yescharacteristic
91C4⋊C4×D61—21no—
92C4⋊C4×D61C2×C411yescharacteristic
93C42×D61—21no—
94C42×D61—21no—
95C42×D61C2311yescharacteristic
96#14519710C2×C411yescharacteristic
97#14519709C2×C411yescharacteristic
98#14519708C2×C411yescharacteristic
99C4⋊#92418—21no—
100C4⋊#92418—21no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519706—21no—
106#14519706D411yescharacteristic
107#14519706D411yescharacteristic
108#14519702C2×C411yescharacteristic
109#14519701C2×C411yescharacteristic
110#14519701C2×C411yescharacteristic
111#14519700C2×C411yescharacteristic
112#14519699C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114#14519689—21no—
115#14519689—21no—
116#14519689—21no—
117#14519689D411yescharacteristic
118#14519689D411yescharacteristic
119#14519688—21no—
120#14519687C2×C411yescharacteristic
121#14519687C2×C411yescharacteristic
122C4×#92418—21no—
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C2×Q16×C61C2×C411yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133#107×C61C2×C411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61C2×C411yescharacteristic
139#103×C61C2×C411yescharacteristic
140#97×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C2×Q8×C61C22×C411yescharacteristic
147#11758226—22no—
148#11758226—22no—
149Q8×D61—22no—
150Q8×D61M1614yescomplemented
151C2×D244—21no—
152C2×D244C22×C411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61C22×C411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61#4111yescharacteristic
161C2×Dic122—21no—
162C2×Dic122C22×C411yescharacteristic
163Q16×C61—22no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C22×C411yescharacteristic
168C2×C8×C61C22×C411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C411yescharacteristic
174C4⋊C4×C61C22×C411yescharacteristic
175C42×C61—21no—
176C42×C61—21no—
177C42×C61C2411yescharacteristic
178#11758202—22no—
179#11758201—22no—
180#11758198—21no—
181#11758198—21no—
182#11758198—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61C22×C411yescharacteristic
186C4⋊Dic61C22×C411yescharacteristic
187#11758196—21no—
188#11758196—21no—
189#11758196—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61C22×C411yescharacteristic
196C4×Dic61C2×D411yescharacteristic
197C4×Dic61#4111yescharacteristic
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C22×C411yescharacteristic
202C2×#92418C22×C411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205#11758190—22no—
206#11758189—22no—
207#11758189—22no—
208#11758189—81no—
209#11758189—81no—
210C8×D61—42no—
211C8×D61—81no—
212C22×D61—21no—
213C22×D61C4×D411yescharacteristic
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61C2×M1612yes—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61C4×D411yescharacteristic
229C2×Dic61C4×D411yescharacteristic
230C2×Dic61C4×D411yescharacteristic
231D244—21no—
232D244—21no—
233D244—42no—
234C4×D61—24no—
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—41no—
238C4×D61—41no—
239C4×D61—41no—
240C4×D61—42no—
241C4×D61—81no—
242C4×D61—81no—
243C4×D61C2×M1612yes—
244C4×D61#13112yes—
245Dic122—21no—
246Dic122—24no—
247Dic122C2×M1612yes—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—21no—
252C8×C61—41no—
253C8×C61—41no—
254C8×C61—41no—
255#92418—22no—
256#92418—21no—
257#92418—21no—
258#92418—21no—
259#92418—21no—
260#92418—41no—
261#92418—41no—
262#17874—611nomaximal, Sylow
263C22×C61C2×C4×D411yescharacteristic
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—41no—
269C2×D61#43512yes—
270C4×C61—22no—
271C4×C61—21no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61C22×M1611yescharacteristic
276C4×C61C2×#13111yescharacteristic
277C4×C61#57711yescharacteristic
278C4×C61#57811yescharacteristic
279Dic61—22no—
280Dic61—21no—
281Dic61—21no—
282Dic61—41no—
283Dic61—41no—
284Dic61#43512yes—
285#2989—611no—
286#2982—611no—
287C4×#137—611no—
288#1588—611no—
289#1588—611no—
290#1587—611no—
291#1587—611no—
292#1584—611no—
293#1584D6111yescharacteristic, complemented
294#1583—611no—
295#1583—611no—
296#1565—611no—
297#1476—611no—
298C2×Q8⋊C8—611no—
299#1451—611no—
300C2×C61C2×#43511yescharacteristic
301C2×C61#294711yescharacteristic
302C2×C61#295611yescharacteristic
303D61—22no—
304D61—81no—
305C2×#137—611no—
306C4×#41—611no—
307C2×C4×Q8—611no—
308#446—611no—
309#446C2×D6111yescharacteristic
310C8×Q8—611no—
311C8×Q8—611no—
312C8×Q8—611no—
313C8×Q8—1221no—
314C8×Q8—1221no—
315C8×Q8C2×D6111yescharacteristic
316#441—611no—
317#441—611no—
318#440—611no—
319#440—611no—
320C4×Q16—611no—
321C4×Q16C2×D6111yescharacteristic
322C4×SD16—611no—
323C4×SD16—611no—
324#436—611no—
325#436—611no—
326#435—1221no—
327#435—1221no—
328#432—1221no—
329#431—611no—
330#431—611no—
331#431—1221no—
332#431—1221no—
333#428—611no—
334#423—1221no—
335C2×C4⋊C8—611no—
336#419—611no—
337#417—611no—
338C4×M16—611no—
339C2×C4×C8—611no—
340#335—611no—
341#335C2×D6111yescharacteristic
342#334—611no—
343#334—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348Q8⋊C8C2×D6111yescharacteristic
349Q8⋊C8C2×D6111yescharacteristic
350D4⋊C8—611no—
351D4⋊C8—611no—
352#322—611no—
353#322—611no—
354#322—611no—
355#322C2×D6111yescharacteristic
356C61#1787411yescharacteristic, complemented, Sylow
357C2×#41—611no—
358C22×Q8—611no—
359#137—1224no—
360C2×Q16—611no—
361C2×Q16C4×D6111yescharacteristic
362C2×SD16—611no—
363C2×SD16—611no—
364C2×M16—611no—
365C2×M16—611no—
366C2×M16—1221no—
367C2×M16—1221no—
368C22×C8—1221no—
369C22×C8—1221no—
370C4×Q8—611no—
371C4×Q8—611no—
372C4×Q8—611no—
373C4×Q8—611no—
374C4×Q8—1221no—
375C4×Q8—1221no—
376C4×Q8C22×D6111yescharacteristic
377C4×Q8C22×D6111yescharacteristic
378C4×D4—611no—
379C4×D4—611no—
380C4×D4—1221no—
381C4×D4—1221no—
382#117—611no—
383#117—1221no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C42—611no—
387C2×C42—1221no—
388C2×C42—1221no—
389#107—611no—
390#107C4×D6111yescharacteristic
391#106—611no—
392#106—611no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—1221no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405C4⋊C8C22×D6111yescharacteristic
406#103—611no—
407#103—611no—
408#103—611no—
409#103—611no—
410#103C4×D6111yescharacteristic
411#103C4×D6111yescharacteristic
412#102—611no—
413#102—611no—
414C22⋊C8—611no—
415C22⋊C8—611no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419#97—21no—
420#97—611no—
421#97—611no—
422#97—1221no—
423C4×C8—21no—
424C4×C8—611no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—1221no—
428C4×C8—1221no—
429C4×C8—1221no—
430C4×C8—1221no—
431C4×C8—1221no—
432C4×C8C22×D6111yescharacteristic
433C4×C8C22×D6111yescharacteristic
434C4×C8C22×D6111yescharacteristic
435#41—1222no—
436#41—1222no—
437C2×Q8—614no—
438C2×Q8—611no—
439C2×Q8—1221no—
440C2×Q8—1222no—
441C2×Q8C2×C4×D6111yescharacteristic
442C2×Q8C2×C4×D6111yescharacteristic
443C2×D4—611no—
444C2×D4—1221no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453Q16—22no—
454Q16—1222no—
455SD16—1222no—
456SD16—1222no—
457M16—1222no—
458M16—1222no—
459M16—4881no—
460M16—4881no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—4881no—
472C2×C8—2442no—
473C2×C8C2×C4×D6111yescharacteristic
474C2×C8C2×C4×D6111yescharacteristic
475C2×C8D4×D6111yescharacteristic
476C2×C8D4×D6111yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—611no—
480C4⋊C4—611no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4C2×C4×D6111yescharacteristic
487C4⋊C4C2×C4×D6111yescharacteristic
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C42—21no—
492C42—21no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—1221no—
497C42—1221no—
498C42—1221no—
499C42—1221no—
500C42—1221no—
501C42C23×D6111yescharacteristic
502C23—611no—
503C23—1221no—
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8—612no—
508Q8—1224no—
509Q8—1221no—
510Q8M16×D6112yes—
511D4—1221no—
512D4—1221no—
513D4—2442no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—611no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—1224no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—4881no—
530C2×C4—4881no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4C22×C4×D6111yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4#41×D6111yescharacteristic
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—1222no—
546C8—1221no—
547C8—1221no—
548C8—1221no—
549C8—1221no—
550C8—2441no—
551C8—2441no—
552C22—612no—
553C22—1221no—
554C22—1221no—
555C22—2441no—
556C22—2441no—
557C22—2441no—
558C22C4×D4×D6111yescharacteristic
559C4—21no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—612no—
565C4—1221no—
566C4—1221no—
567C4—1222no—
568C4—2441no—
569C4—2441no—
570C4C2×M16×D6111yescharacteristic
571C4#131×D6111yescharacteristic
572C4#1684056911yescharacteristic
573C4#1684057511yescharacteristic
574C2—1222no—
575C2—4881no—
576C2#435×D6111yescharacteristic
577C2#2145465711yescharacteristic
578C2#2145472711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1584C2
socleC22×C61C2×C4×D4
prime radicalC2×#11758226C2×C4
cosocleC42×C61C24
hypercenter#1584D61
hypocenterC61#17874
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
2. #21454898C2semidirect
4. #21444697C2semidirect
5. #21444696C2semidirect
6. #21444691C2semidirect
7. #21444688C2semidirect
8. #21444672C2semidirect
9. #21444664C2semidirect
10. #21444659C2semidirect
11. #21444570C2semidirect
13. Q8⋊C8×D61C2semidirect
14. #21444107C2semidirect
15. C61×#1584C2semidirect
150. Q8×D61M16semidirect
293. #1584D61semidirect
356. C61#17874semidirect

Maximal subgroups

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

subgroupindex
#214549352
#214548982
C4×#145198092
#214446972
#214446962
#214446912
#214446882
#214446722
#214446642
#214446592
#214445702
#214442002
Q8⋊C8×D612
#214441072
C61×#15842
#1787461