Finite Group Databasebeta

← #29663397

Subgroups — #29663397

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457145C211yesmaximal, characteristic, complemented
2#21456394C211yesmaximal, characteristic, complemented
3#21454795C211yesmaximal, characteristic, complemented
4#21445361C211yesmaximal, characteristic, complemented
5#21445357C211yesmaximal, characteristic, complemented
6#21445354C211yesmaximal, characteristic, complemented
7#21445350C211yesmaximal, characteristic, complemented
8#21445221C211yesmaximal, characteristic, complemented
9#21445216C211yesmaximal, characteristic, complemented
10#21445177C211yesmaximal, characteristic, complemented
11#21445172C211yesmaximal, characteristic, complemented
12C2×#16839514C211yesmaximal, characteristic, complemented
13#21444514C211yesmaximal, characteristic, complemented
14#21444512C211yesmaximal, characteristic, complemented
15C61×#3247C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840138C2211yescharacteristic
19#16840110C2211yescharacteristic
20#16840108C2211yescharacteristic
21#16840059—21no—
22#16840057C2211yescharacteristic
23#16840052C2211yescharacteristic
24#16839996—21no—
25#16839996—21no—
26#16839996—21no—
27#16839993—21no—
28#16839974—21no—
29#16839962C2211yescharacteristic
30#16839957C2211yescharacteristic
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33C4×#11758199—21no—
34C4×#11758199—21no—
35#16839840C2211yescharacteristic
36#16839834—21no—
37D4×#92418—21no—
38D4×#92418C2211yescharacteristic
39#16839823—21no—
40#16839819C2211yescharacteristic
41C2×C4⋊#92418C2211yescharacteristic
42#16839718—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839514C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839506C2211yescharacteristic
53#16839506C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#478—21no—
59C61×#477C2211yescharacteristic
60C61×#476C2211yescharacteristic
61C61×#466—21no—
62C61×#465—21no—
63C61×#463—21no—
64C61×#460C2211yescharacteristic
65C61×#459C2211yescharacteristic
66C2×C4⋊C8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758201—21no—
69C2×#11758199—21no—
70C2×#11758199—21no—
71C2×#11758199—21no—
72C2×#11758194—21no—
73C22×#92418—21no—
74C22×#92418—21no—
75C22×#92418D411yescharacteristic
76C22×#92418D411yescharacteristic
77#14519777C2311yescharacteristic
78#14519774C2311yescharacteristic
79#14519765—21no—
80#14519751—21no—
81#14519751—21no—
82C4×D244—21no—
83C4×D244—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—21no—
87#14519708—21no—
88C4⋊#92418—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519698—21no—
97#14519698—21no—
98#14519698—21no—
99#14519697—21no—
100#14519697—21no—
101#14519697—21no—
102#14519697—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106#14519688—21no—
107#14519688—21no—
108C4×#92418—21no—
109C4×#92418D411yescharacteristic
110C4×#92418D411yescharacteristic
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×SD16×C61—21no—
115C2×D8×C61—21no—
116C22×C8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118#127×C61C2311yescharacteristic
119#126×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61—21no—
122#124×C61D411yescharacteristic
123#123×C61—21no—
124C22⋊Q8×C61D411yescharacteristic
125#121×C61—21no—
126#121×C61—21no—
127#121×C61D411yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#107×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143#102×C61—21no—
144#41×C61—82no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61#4111yescharacteristic
158C2×C22⋊D61—41no—
159#11758222—82no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×C4×D61—41no—
164C2×Dic122—21no—
165SD16×C61—81no—
166D8×C61—81no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—81no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61#4111yescharacteristic
189C42×C61C2411yescharacteristic
190#11758201—81no—
191#11758199—42no—
192#11758199—42no—
193#11758199—81no—
194#11758198—41no—
195#11758198—41no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198#11758194—42no—
199C2×#92418—22no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—41no—
205C2×#92418—42no—
206C2×#92418—81no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C23×C61—21no—
212C23×C61—41no—
213C23×C61#12111yescharacteristic
214C22×D61—41no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—81no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—42no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—81no—
226D4×C61—82no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—82no—
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61#12111yescharacteristic
241C2×C4×C61#12111yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C22⋊D61—82no—
246C2×Dic61—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—81no—
252C4×D61—82no—
253Dic122—81no—
254C8×C61—41no—
255C8×C61—41no—
256#92418—21no—
257#92418—21no—
258#92418—21no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418#13512yes—
264#19222—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61#47212yes—
274C22×C61C2×#12111yescharacteristic
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—81no—
286C4×C61C2×#13511yescharacteristic
287C4×C61#57511yescharacteristic
288C4×C61#57611yescharacteristic
289C4×C61#57811yescharacteristic
290Dic61—81no—
291#3247—611no—
292#3247D6111yescharacteristic, complemented
293#3169—611no—
294#2966—611no—
295#1698—611no—
296#1698—611no—
297#1697—611no—
298#1697—611no—
299#1676—611no—
300#1675—611no—
301#1668—611no—
302#1667—611no—
303C2×#335—611no—
304#1551—611no—
305#1551—611no—
306C2×C61—22no—
307C2×C61—81no—
308C2×C61—161no—
309C2×C61#303511yescharacteristic
310C2×C61#306111yescharacteristic
311C2×C61C2×#47211yescharacteristic
312D61—161no—
313#532—611no—
314#532C2×D6111yescharacteristic
315#529C2×D6111yescharacteristic
316#500—611no—
317#493—611no—
318#486—611no—
319#486—611no—
320#478—21no—
321#478—1221no—
322#477—611no—
323#477C2×D6111yescharacteristic
324#476—611no—
325#476C2×D6111yescharacteristic
326#466—21no—
327#466—1221no—
328#466—1221no—
329#466—1221no—
330#465—21no—
331#465—1221no—
332#463—21no—
333#463—1221no—
334#460—611no—
335#460C2×D6111yescharacteristic
336#459—611no—
337#459C2×D6111yescharacteristic
338#446—611no—
339C8×Q8—611no—
340C4×D8—1221no—
341C4×D8—1221no—
342#436—611no—
343#435—1221no—
344C8×D4—611no—
345C8×D4—1221no—
346#433—1221no—
347#432—611no—
348C2×C4⋊C8—611no—
349C2×C4⋊C8C2×D6111yescharacteristic
350#405—1221no—
351C2×C4×C8—611no—
352#335—611no—
353#335—611no—
354#335—611no—
355#335—611no—
356#333—611no—
357#333—611no—
358#333—611no—
359#333—611no—
360#331—611no—
361#331—611no—
362#331—611no—
363#331—611no—
364C61#1922211yescharacteristic, complemented, Sylow
365C2×#41—21no—
366C2×#41—1221no—
367C2×SD16—21no—
368C2×SD16—1221no—
369C2×D8—21no—
370C2×D8—1221no—
371C2×D8—1221no—
372C2×D8—1221no—
373C2×M16—1221no—
374C22×C8—21no—
375C22×C8—611no—
376C22×C8—611no—
377C22×C8—1221no—
378C22×C8—1221no—
379C4⋊Q8—611no—
380C4⋊Q8C22×D6111yescharacteristic
381#127—611no—
382#127C22×D6111yescharacteristic
383#126—21no—
384#125C22⋊D6111yescharacteristic
385#124—21no—
386#124—1221no—
387#124C22⋊D6111yescharacteristic
388#123—21no—
389C22⋊Q8C22⋊D6111yescharacteristic
390#121—21no—
391#121—21no—
392#121—1221no—
393#121—1221no—
394#121C22⋊D6111yescharacteristic
395C4×Q8C22×D6111yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4—1221no—
400C4×D4—1221no—
401C4×D4C22×D6111yescharacteristic
402#117—21no—
403C2×C42C22×D6111yescharacteristic
404#107—21no—
405#107—1221no—
406#107—1221no—
407#107—1221no—
408#106—21no—
409#106—1221no—
410C4⋊C8—21no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8C22×D6111yescharacteristic
419C4⋊C8C22×D6111yescharacteristic
420#103—21no—
421#103—1221no—
422#102—21no—
423#102—21no—
424#102—21no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435#97—1221no—
436#97—1221no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—1221no—
442#41—82no—
443#41—4882no—
444C2×Q8—21no—
445C2×Q8—1221no—
446C2×Q8C2×C22⋊D6111yescharacteristic
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—41no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—2441no—
456C2×D4C2×C22⋊D6111yescharacteristic
457C22×C4—21no—
458C22×C4—41no—
459C22×C4—2441no—
460C22×C4D4×D6111yescharacteristic
461C22×C4#1175822411yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463SD16—81no—
464SD16—4881no—
465D8—81no—
466D8—4881no—
467D8—2442no—
468D8—2442no—
469M16—2442no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—81no—
473C2×C8—611no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—1222no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—4881no—
483C2×C8—2442no—
484C2×C8—2441no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4C2×C22⋊D6111yescharacteristic
494C4⋊C4C2×C22⋊D6111yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C42—21no—
505C42D4×D6111yescharacteristic
506C42#1175822411yescharacteristic
507C42C2×C22⋊D6111yescharacteristic
508C42C23×D6111yescharacteristic
509C23—21no—
510C23—41no—
511C23—2441no—
512C23#1451974911yescharacteristic
513Q8—21no—
514Q8—21no—
515Q8—81no—
516Q8—4881no—
517D4—21no—
518D4—21no—
519D4—42no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—81no—
525D4—82no—
526D4—4881no—
527D4—4882no—
528D4—2441no—
529D4—2441no—
530D4—2441no—
531D4—2441no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—82no—
544C2×C4—4882no—
545C2×C4—2441no—
546C2×C4#1451974911yescharacteristic
547C2×C4#1451974911yescharacteristic
548C2×C4#1451974911yescharacteristic
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4C2×#1175822411yescharacteristic
552C8—41no—
553C8—41no—
554C8—612no—
555C8—1221no—
556C8—1221no—
557C8—1221no—
558C8—2441no—
559C8—2441no—
560C8—2441no—
561C8—2441no—
562C22—21no—
563C22—21no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—4881no—
571C22—4881no—
572C22—4881no—
573C22#1684003012yes—
574C22C2×#1451974911yescharacteristic
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—4881no—
584C4#135×D6111yescharacteristic
585C4#1684055911yescharacteristic
586C4#1684056511yescharacteristic
587C4#1684057711yescharacteristic
588C2—22no—
589C2—81no—
590C2—161no—
591C2—9761no—
592C2#2145527011yescharacteristic
593C2#2145548611yescharacteristic
594C2C2×#1684003011yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3247C2
socleC22×C61C2×#121
hypercenter#3247D61
hypocenterC61#19222
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3247

Complements

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

NH ≅ G/Ntype
1. #21457145C2semidirect
2. #21456394C2semidirect
3. #21454795C2semidirect
4. #21445361C2semidirect
5. #21445357C2semidirect
6. #21445354C2semidirect
7. #21445350C2semidirect
8. #21445221C2semidirect
9. #21445216C2semidirect
10. #21445177C2semidirect
11. #21445172C2semidirect
12. C2×#16839514C2semidirect
13. #21444514C2semidirect
14. #21444512C2semidirect
15. C61×#3247C2semidirect
292. #3247D61semidirect
364. C61#19222semidirect

Maximal subgroups

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

subgroupindex
#214571452
#214563942
#214547952
#214453612
#214453572
#214453542
#214453502
#214452212
#214452162
#214451772
#214451722
C2×#168395142
#214445142
#214445122
C61×#32472
#1922261