Finite Group Databasebeta

← #29663360

Subgroups — #29663360

Order 15616.

630 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456791C211yesmaximal, characteristic, complemented
2#21456394C211yesmaximal, characteristic, complemented
3#21454795C211yesmaximal, characteristic, complemented
4#21445389C211yesmaximal, characteristic, complemented
5#21445385C211yesmaximal, characteristic, complemented
6#21445382C211yesmaximal, characteristic, complemented
7#21445378C211yesmaximal, characteristic, complemented
8#21445211C211yesmaximal, characteristic, complemented
9#21445206C211yesmaximal, characteristic, complemented
10#21445177C211yesmaximal, characteristic, complemented
11#21445172C211yesmaximal, characteristic, complemented
12#21444565C211yesmaximal, characteristic, complemented
13#21444524C211yesmaximal, characteristic, complemented
14#21444522C211yesmaximal, characteristic, complemented
15C61×#3212C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840144—21no—
19#16840138C2211yescharacteristic
20#16840119—21no—
21#16840110C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16839996—21no—
24#16839996—21no—
25#16839972C2211yescharacteristic
26#16839967C2211yescharacteristic
27#16839962C2211yescharacteristic
28#16839957C2211yescharacteristic
29#16839922—21no—
30#16839905—21no—
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
41#16839815C2211yescharacteristic
42#16839718—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839506C2211yescharacteristic
53#16839506C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#495—21no—
59C61×#488—21no—
60C61×#462C2211yescharacteristic
61C61×#461C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#459C2211yescharacteristic
64C61×#452—21no—
65C61×#449—21no—
66C61×#431C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C2×#11758199—21no—
73C2×#11758199—21no—
74C2×#11758194—21no—
75C22×#92418—21no—
76C22×#92418D411yescharacteristic
77C22×#92418D411yescharacteristic
78#14519777C2311yescharacteristic
79#14519774C2311yescharacteristic
80#14519765—21no—
81#14519751—21no—
82#14519751—21no—
83C4×D244—21no—
84C4×D244—21no—
85#14519710—21no—
86#14519710—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519700—21no—
94#14519700—21no—
95#14519698—21no—
96#14519698—21no—
97#14519697—21no—
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106#14519688—21no—
107C4×#92418—21no—
108C4×#92418—21no—
109C4×#92418D411yescharacteristic
110C4×#92418D411yescharacteristic
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×Q16×C61—21no—
115C2×SD16×C61—21no—
116C2×SD16×C61—21no—
117C2×D8×C61—21no—
118C4⋊Q8×C61C2311yescharacteristic
119#127×C61C2311yescharacteristic
120#126×C61—21no—
121#125×C61D411yescharacteristic
122#124×C61—21no—
123#124×C61D411yescharacteristic
124#123×C61—21no—
125C22⋊Q8×C61D411yescharacteristic
126#121×C61—21no—
127#121×C61—21no—
128#121×C61D411yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61C2311yescharacteristic
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61—21no—
144#41×C61—81no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61#4111yescharacteristic
159C2×C22⋊D61—41no—
160#11758222—81no—
161#11758222—81no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×C4×D61—41no—
166C2×Dic122—21no—
167Q16×C61—81no—
168SD16×C61—81no—
169SD16×C61—81no—
170D8×C61—81no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61#4111yescharacteristic
192C42×C61C2411yescharacteristic
193#11758202—81no—
194#11758201—81no—
195#11758200—81no—
196#11758199—42no—
197#11758199—42no—
198#11758199—81no—
199#11758198—41no—
200#11758198—41no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203#11758194—42no—
204C2×#92418—22no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—41no—
210C2×#92418—42no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C23×C61—21no—
216C23×C61—41no—
217C23×C61#12111yescharacteristic
218C22×D61—41no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—41no—
222Q8×C61—41no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—42no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—81no—
231D4×C61—82no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—82no—
244C2×C4×C61#12111yescharacteristic
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61#12111yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C22⋊D61—82no—
251C2×Dic61—41no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—81no—
257C4×D61—82no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—41no—
261C8×C61—41no—
262#92418—21no—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#92418#13512yes—
270#19219—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61C2×#12111yescharacteristic
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—81no—
292C4×C61C2×#13511yescharacteristic
293C4×C61#57511yescharacteristic
294C4×C61#57611yescharacteristic
295C4×C61#57611yescharacteristic
296Dic61—81no—
297#3212—611no—
298#3212D6111yescharacteristic, complemented
299#3169—611no—
300#2966—611no—
301#1702—611no—
302#1702—611no—
303#1701—611no—
304#1701—611no—
305#1674—611no—
306#1673—611no—
307#1668—611no—
308#1667—611no—
309#1564—611no—
310#1553—611no—
311#1553—611no—
312C2×C61—41no—
313C2×C61—81no—
314C2×C61—161no—
315C2×C61#303511yescharacteristic
316C2×C61#305811yescharacteristic
317C2×C61#306711yescharacteristic
318D61—161no—
319#532—611no—
320#532C2×D6111yescharacteristic
321#529C2×D6111yescharacteristic
322#500—611no—
323#495—21no—
324#495—1221no—
325#493—611no—
326#488—21no—
327#488—1221no—
328#486—611no—
329#486—611no—
330#466—1221no—
331#466—1221no—
332#462—611no—
333#462C2×D6111yescharacteristic
334#461—611no—
335#461C2×D6111yescharacteristic
336#460—611no—
337#460C2×D6111yescharacteristic
338#459—611no—
339#459C2×D6111yescharacteristic
340#452—21no—
341#452—1221no—
342#449—21no—
343#449—1221no—
344#446—611no—
345C8×Q8—611no—
346C4×D8—1221no—
347C4×D8—1221no—
348#436—611no—
349#435—1221no—
350C8×D4—611no—
351C8×D4—1221no—
352#433—1221no—
353#432—611no—
354#431—611no—
355#431C2×D6111yescharacteristic
356#405—1221no—
357C2×C4×C8—611no—
358#335—611no—
359#335—611no—
360#334—611no—
361#334—611no—
362#332—611no—
363#332—611no—
364#332—611no—
365#332—611no—
366#331—611no—
367#331—611no—
368#331—611no—
369#331—611no—
370C61#1921911yescharacteristic, complemented, Sylow
371C2×#41—21no—
372C2×#41—1221no—
373C2×Q16—21no—
374C2×Q16—1221no—
375C2×SD16—21no—
376C2×SD16—21no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×D8—21no—
380C2×D8—1221no—
381C2×D8—1221no—
382C2×D8—1221no—
383C2×M16—1221no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8—1221no—
387C4⋊Q8—611no—
388C4⋊Q8C22×D6111yescharacteristic
389#127—611no—
390#127C22×D6111yescharacteristic
391#126—21no—
392#125C22⋊D6111yescharacteristic
393#124—21no—
394#124—1221no—
395#124C22⋊D6111yescharacteristic
396#123—21no—
397C22⋊Q8C22⋊D6111yescharacteristic
398#121—21no—
399#121—21no—
400#121—1221no—
401#121—1221no—
402#121C22⋊D6111yescharacteristic
403C4×Q8C22×D6111yescharacteristic
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—1221no—
408C4×D4—1221no—
409C4×D4C22×D6111yescharacteristic
410#117—21no—
411C2×C42C22×D6111yescharacteristic
412#107—1221no—
413#107—1221no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421C4⋊C8C22×D6111yescharacteristic
422#103—21no—
423#103—21no—
424#103—1221no—
425#103—1221no—
426#102—21no—
427#102—21no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434C22⋊C8—21no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439#97—1221no—
440#97—1221no—
441C4×C8—21no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—1221no—
447C4×C8—1221no—
448#41—81no—
449#41—81no—
450#41—4881no—
451#41—4881no—
452C2×Q8—21no—
453C2×Q8—1221no—
454C2×Q8C2×C22⋊D6111yescharacteristic
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—41no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—2441no—
464C2×D4C2×C22⋊D6111yescharacteristic
465C22×C4—21no—
466C22×C4—41no—
467C22×C4—2441no—
468C22×C4D4×D6111yescharacteristic
469C22×C4#1175822411yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471Q16—81no—
472Q16—4881no—
473SD16—81no—
474SD16—81no—
475SD16—4881no—
476SD16—4881no—
477D8—81no—
478D8—4881no—
479D8—2442no—
480D8—2442no—
481M16—2442no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—1222no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—2442no—
494C2×C8—2441no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C4⋊C4C2×C22⋊D6111yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C42—21no—
515C42D4×D6111yescharacteristic
516C42#1175822411yescharacteristic
517C42C2×C22⋊D6111yescharacteristic
518C42C23×D6111yescharacteristic
519C23—21no—
520C23—41no—
521C23—2441no—
522C23#1451974911yescharacteristic
523Q8—21no—
524Q8—21no—
525Q8—41no—
526Q8—41no—
527Q8—2441no—
528Q8—2441no—
529D4—21no—
530D4—21no—
531D4—42no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—81no—
537D4—82no—
538D4—4881no—
539D4—4882no—
540D4—2441no—
541D4—2441no—
542D4—2441no—
543D4—2441no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—82no—
556C2×C4—4882no—
557C2×C4—2441no—
558C2×C4#1451974911yescharacteristic
559C2×C4#1451974911yescharacteristic
560C2×C4#1451974911yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4C2×#1175822411yescharacteristic
564C8—41no—
565C8—41no—
566C8—612no—
567C8—1221no—
568C8—1221no—
569C8—1221no—
570C8—2441no—
571C8—2441no—
572C8—2441no—
573C8—2441no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22C2×#1451974911yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—4881no—
596C4#135×D6111yescharacteristic
597C4#1684055911yescharacteristic
598C4#1684056511yescharacteristic
599C4#1684056511yescharacteristic
600C2—41no—
601C2—81no—
602C2—161no—
603C2—9761no—
604C2#2145527011yescharacteristic
605C2#2145546411yescharacteristic
606C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3212C2
socleC22×C61C2×#121
hypercenter#3212D61
hypocenterC61#19219
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3212

Complements

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

NH ≅ G/Ntype
1. #21456791C2semidirect
2. #21456394C2semidirect
3. #21454795C2semidirect
4. #21445389C2semidirect
5. #21445385C2semidirect
6. #21445382C2semidirect
7. #21445378C2semidirect
8. #21445211C2semidirect
9. #21445206C2semidirect
10. #21445177C2semidirect
11. #21445172C2semidirect
12. #21444565C2semidirect
13. #21444524C2semidirect
14. #21444522C2semidirect
15. C61×#3212C2semidirect
298. #3212D61semidirect
370. C61#19219semidirect

Maximal subgroups

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

subgroupindex
#214567912
#214563942
#214547952
#214453892
#214453852
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C61×#32122
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