Finite Group Databasebeta

← #29662829

Subgroups — #29662829

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456196C211yesmaximal, characteristic, complemented
2#21455826C211yesmaximal, characteristic
3#474×D61C211yesmaximal, characteristic
4D4×#11758189C211yesmaximal, characteristic
5#21445395C211yesmaximal, characteristic, complemented
6#21445393C211yesmaximal, characteristic, complemented
7#21445369C211yesmaximal, characteristic, complemented
8#21445363C211yesmaximal, characteristic, complemented
9#21445270C211yesmaximal, characteristic, complemented
10#21445268C211yesmaximal, characteristic, complemented
11#21445244C211yesmaximal, characteristic, complemented
12#21444568C211yesmaximal, characteristic, complemented
13#21444508C211yesmaximal, characteristic, complemented
14#21444489C211yesmaximal, characteristic, complemented
15C61×#1679C211yesmaximal, characteristic, complemented
16C22×#11758189—21no—
17#16840317C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840129—21no—
21#16840129—21no—
22#16840114C2211yescharacteristic, complemented
23#16840113C2211yescharacteristic, complemented
24#16840112C2211yescharacteristic, complemented
25#16840091—21no—
26#16840053C2211yescharacteristic
27#16840052C2211yescharacteristic
28#16840043C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840041—21no—
31#16840040—21no—
32#16840039C2211yescharacteristic
33#16839964C2211yescharacteristic
34#16839911—21no—
35SD16×Dic61—21no—
36#16839847—21no—
37#16839841C2211yescharacteristic
38#16839839C2211yescharacteristic
39#16839833—21no—
40#16839830—21no—
41D4×#92418C2211yescharacteristic
42#16839814C2211yescharacteristic
43#107×D61—21no—
44#16839784C2211yescharacteristic
45C4⋊C8×D61C2211yescharacteristic
46#16839769—21no—
47#16839769—21no—
48#102×D61—21no—
49#16839724—21no—
50C4×#11758189C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839510C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839509C2211yescharacteristic, complemented
57#16839508C2211yescharacteristic, complemented
58#16839508C2211yescharacteristic, complemented
59#16839507C2211yescharacteristic, complemented
60C61×#487C2211yescharacteristic, complemented
61C61×#474C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#332C2211yescharacteristic, complemented
67C2×Q8×D61—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758201—21no—
71C2×#11758189—21no—
72C2×#11758189—21no—
73C2×#11758189—21no—
74C2×#11758189—44no—
75C2×#11758189D411yescharacteristic
76C2×#11758189D411yescharacteristic
77C22×#92418—21no—
78C2×C8×D61—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519776—21no—
82#14519776—21no—
83#14519775—21no—
84#14519768—21no—
85#14519757—21no—
86#14519753—21no—
87Q8×Dic61—21no—
88#14519742—21no—
89C4×D244C2311yescharacteristic
90D4×Dic61D411yescharacteristic
91#14519739D411yescharacteristic
92C4×C22⋊D61—21no—
93#14519737—21no—
94#14519735—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61D411yescharacteristic
98C22⋊C4×D61—21no—
99C42×D61C2311yescharacteristic
100#14519711—21no—
101#14519710—21no—
102#14519710—21no—
103#14519709—21no—
104#14519708—21no—
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108#14519706—21no—
109#14519706—21no—
110#14519702—21no—
111#14519702—21no—
112#14519701—21no—
113#14519701—21no—
114#14519701—21no—
115#14519700—21no—
116#14519700—21no—
117#14519699—21no—
118#14519699—21no—
119#14519698—21no—
120#14519698—21no—
121#14519697—21no—
122C22⋊#92418—21no—
123#14519689—21no—
124#14519689—21no—
125#14519689—21no—
126#14519687D411yescharacteristic
127#14519687D411yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C2×SD16×C61—21no—
131C22×C8×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133C4⋊Q8×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#107×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C22×Dic61—21no—
152Q8×D61—42no—
153Q8×D61—42no—
154D4×D61—22no—
155D4×D61—41no—
156C2×D244C2×D411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—41no—
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61#4111yescharacteristic
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—41no—
169SD16×C61—42no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—42no—
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C42×C61C2411yescharacteristic
183C22⋊Dic61—21no—
184#11758201—42no—
185#11758198—21no—
186#11758198—21no—
187#11758198—41no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—41no—
193C4⋊Dic61C2×D411yescharacteristic
194#11758196—21no—
195#11758196—41no—
196#11758196—41no—
197#11758196—41no—
198C4×Dic61—41no—
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61C2×D411yescharacteristic
201C4×Dic61#4111yescharacteristic
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—42no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208#11758189—22no—
209#11758189—22no—
210#11758189—42no—
211#11758189—42no—
212#11758189—42no—
213C8×D61—42no—
214C23×C61—21no—
215C22×D61—21no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—81no—
219C22×D61#12111yescharacteristic
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—21no—
225D4×C61#13712yes—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C22⋊D61—42no—
235C2×Dic61—21no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61#12111yescharacteristic
240C2×Dic61#12111yescharacteristic
241C2×Dic61#12111yescharacteristic
242D244—21no—
243D244—21no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—41no—
247C4×D61—42no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—42no—
251C4×D61—82no—
252C4×D61#13512yes—
253C4×D61AGL(1,ℤ/8)12yes—
254Dic122—41no—
255Dic122—41no—
256Dic122—82no—
257Dic122—81no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61—41no—
262C8×C61—41no—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#19167—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61C2×#12111yescharacteristic
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—84no—
279C2×D61#46312yes—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61C2×#13511yescharacteristic
285C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C4×C61C2×#13711yescharacteristic
287C4×C61#57511yescharacteristic
288Dic61—21no—
289Dic61—21no—
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293Dic61#46312yes—
294#3139—611no—
295#3101—611no—
296C2×#474—611no—
297D4×M16—611no—
298#1703—611no—
299#1703—611no—
300#1699—611no—
301#1699—611no—
302#1683—611no—
303#1683—611no—
304#1679—611no—
305#1679D6111yescharacteristic, complemented
306#1565—611no—
307#1550—611no—
308#1546—611no—
309C2×C61—42no—
310C2×C61C2×#46311yescharacteristic
311C2×C61#305611yescharacteristic
312C2×C61#306911yescharacteristic
313D61—22no—
314D61—81no—
315C22×M16—1221no—
316#533—611no—
317C2×C4⋊Q8—611no—
318C2×C4×D4—611no—
319#491—1221no—
320#491—1221no—
321#487—611no—
322#487—611no—
323#487—611no—
324#487C2×D6111yescharacteristic, complemented
325#483—1221no—
326#476—611no—
327#476—611no—
328#474—611no—
329#474—611no—
330#474—611no—
331#474—1221no—
332#474—1221no—
333#474C2×D6111yescharacteristic
334#461—611no—
335#461C2×D6111yescharacteristic
336#450—1221no—
337C4×SD16—1221no—
338C4×SD16—1221no—
339#436—611no—
340#436—611no—
341#435—1221no—
342#435—1221no—
343C8×D4—611no—
344C8×D4C2×D6111yescharacteristic
345#431—611no—
346C2×#107—1221no—
347#423—611no—
348C2×C4⋊C8—611no—
349#419—1221no—
350#419—1221no—
351C2×#102—1221no—
352#407—1221no—
353C4×M16—611no—
354#335—611no—
355#335—611no—
356#334—611no—
357#334C2×D6111yescharacteristic, complemented
358#333—611no—
359#333—611no—
360#332—611no—
361#332—611no—
362#332—611no—
363#332—611no—
364#332C2×D6111yescharacteristic, complemented
365#332C2×D6111yescharacteristic, complemented
366C61#1916711yescharacteristic, complemented, Sylow
367C22×Q8—1221no—
368C22×D4—611no—
369C23×C4—1221no—
370C2×SD16—21no—
371C2×SD16—1221no—
372C2×M16—611no—
373C2×M16—611no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16—1221no—
377C2×M16—2444no—
378C22×C8—21no—
379C22×C8—1221no—
380C22×C8—1221no—
381C4⋊Q8—611no—
382C4⋊Q8—611no—
383C4⋊Q8—1221no—
384C4⋊Q8—1221no—
385C4⋊Q8—1221no—
386C4⋊Q8C22×D6111yescharacteristic
387C4⋊Q8C22×D6111yescharacteristic
388#125—1221no—
389C22⋊Q8—1221no—
390C22⋊Q8—1221no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×D4—611no—
394C4×D4—611no—
395C4×D4—611no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4C22×D6111yescharacteristic
399#117—1221no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C42—611no—
405#107—21no—
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#106—1221no—
410#106—1221no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8C22×D6111yescharacteristic
417C4⋊C8C22×D6111yescharacteristic
418C4⋊C8C22×D6111yescharacteristic
419#103—21no—
420#103—21no—
421#103—21no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#102—21no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435C22⋊C8—21no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440#97—611no—
441#97—611no—
442C4×C8—611no—
443C4×C8—1221no—
444C4×C8C22×D6111yescharacteristic
445C24—1221no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—2441no—
451C2×Q8—2442no—
452C2×Q8—2442no—
453C2×D4—611no—
454C2×D4—1221no—
455C2×D4—1222no—
456C2×D4—2441no—
457C2×D4D4×D6111yescharacteristic
458C22×C4—21no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4—2441no—
469SD16—42no—
470SD16—2442no—
471M16—1222no—
472M16—1222no—
473M16—2442no—
474M16—2442no—
475M16—2442no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—42no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—2442no—
486C2×C8—2442no—
487C2×C8D4×D6111yescharacteristic
488C2×C8D4×D6111yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—611no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4D4×D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—2441no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—2441no—
513C42C23×D6111yescharacteristic
514C23—21no—
515C23—611no—
516C23—1221no—
517C23—4881no—
518C23—2442no—
519C23—2442no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—4881no—
525Q8—4882no—
526Q8—2441no—
527Q8—2441no—
528D4—21no—
529D4—1221no—
530D4—1221no—
531D4—2442no—
532D4#1451980612yes—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—611no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—4882no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C8—21no—
559C8—21no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—1221no—
564C8—1221no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—612no—
572C22—1221no—
573C22—1221no—
574C22—4884no—
575C22—2441no—
576C22—2441no—
577C22—2441no—
578C22#121×D6111yescharacteristic
579C4—21no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—612no—
584C4—1221no—
585C4—1221no—
586C4—4881no—
587C4—4881no—
588C4—2441no—
589C4#135×D6111yescharacteristic
590C4AGL(1,ℤ/8)×D6111yescharacteristic
591C4C2×#1451980611yescharacteristic
592C4#1684055911yescharacteristic
593C2—42no—
594C2—1222no—
595C2—4881no—
596C2#463×D6111yescharacteristic
597C2#2145544211yescharacteristic
598C2#2145555411yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1679C2
socleC22×C61C2×#121
prime radicalC2×D4×D61D4
hypercenter#1679D61
hypocenterC61#19167
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1679

Complements

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

NH ≅ G/Ntype
1. #21456196C2semidirect
5. #21445395C2semidirect
6. #21445393C2semidirect
7. #21445369C2semidirect
8. #21445363C2semidirect
9. #21445270C2semidirect
10. #21445268C2semidirect
11. #21445244C2semidirect
12. #21444568C2semidirect
13. #21444508C2semidirect
14. #21444489C2semidirect
15. C61×#1679C2semidirect
22. #16840114C22semidirect
23. #16840113C22semidirect
24. #16840112C22semidirect
51. #16839515C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839510C22semidirect
55. #16839510C22semidirect
56. #16839509C22semidirect
57. #16839508C22semidirect
58. #16839508C22semidirect
59. #16839507C22semidirect
60. C61×#487C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
66. C61×#332C22semidirect
305. #1679D61semidirect
324. #487C2×D61semidirect
357. #334C2×D61semidirect
364. #332C2×D61semidirect
365. #332C2×D61semidirect
366. C61#19167semidirect

Maximal subgroups

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

subgroupindex
#214561962
#214558262
#474×D612
D4×#117581892
#214453952
#214453932
#214453692
#214453632
#214452702
#214452682
#214452442
#214445682
#214445082
#214444892
C61×#16792
#1916761