Finite Group Databasebeta

← #29670681

Subgroups — #29670681

Order 15616.

644 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458544C211yesmaximal, characteristic, complemented
2#21458495C211yesmaximal, characteristic, complemented
3#21458486C211yesmaximal, characteristic, complemented
4#21445457C211yesmaximal, characteristic, complemented
5#21445357C211yesmaximal, characteristic, complemented
6#21445270C211yesmaximal, characteristic, complemented
7#21445261C211yesmaximal, characteristic, complemented
8#21445176C211yesmaximal, characteristic, complemented
9#21445012C211yesmaximal, characteristic, complemented
10#21444946C211yesmaximal, characteristic, complemented
11#21444913C211yesmaximal, characteristic, complemented
12#21444896C211yesmaximal, characteristic, complemented
13#21444670C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3287C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840490C2211yescharacteristic
18#16840134C2211yescharacteristic
19#16840126—21no—
20#16840113C2211yescharacteristic
21#16840111—21no—
22#16840108—21no—
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840108C2211yescharacteristic
26#16840062—21no—
27#16840057C2211yescharacteristic
28#16840056—21no—
29#16840052—21no—
30#16840052C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16839972C2211yescharacteristic
33#16839970—21no—
34#16839959—21no—
35#16839959—21no—
36#16839959C2211yescharacteristic
37#16839959C2211yescharacteristic
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839862—21no—
42#16839854—21no—
43C4×#11758200—21no—
44C4×#11758200C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839506C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C2×#135×C61—21no—
57C61×#550C2211yescharacteristic
58C61×#550C2211yescharacteristic
59C61×#493C2211yescharacteristic
60C61×#466—21no—
61C61×#460C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#449—21no—
64C61×#449—21no—
65C4×Q16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—41no—
71C2×#11758200D411yescharacteristic
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519776—21no—
76#14519776—41no—
77#14519775—21no—
78#14519769—21no—
79#14519767—41no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122—21no—
83C4×Dic122C2311yescharacteristic
84#14519742—21no—
85#14519710—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708D411yescharacteristic
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519702—21no—
96#14519702—21no—
97#14519701—21no—
98#14519700—21no—
99#14519700D411yescharacteristic
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110#14519688—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C2×#41×C61—21no—
117#135×C61—44no—
118C2×Q16×C61D411yescharacteristic
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×D8×C61—21no—
122C2×D8×C61—21no—
123C22×C8×C61—21no—
124#127×C61—21no—
125#127×C61—21no—
126#127×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#121×C61—21no—
129#121×C61—21no—
130#121×C61—41no—
131#121×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61D411yescharacteristic
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—42no—
147#41×C61—81no—
148#41×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61—41no—
160C2×D4×C61—41no—
161C2×D4×C61C2×D411yescharacteristic
162C22×C4×C61—21no—
163C22×C4×C61—41no—
164C22×C4×C61—41no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168C2×Dic122C2×D411yescharacteristic
169Q16×C61—22no—
170SD16×C61—42no—
171SD16×C61—42no—
172D8×C61—42no—
173D8×C61—81no—
174C2×C8×C61—21no—
175C2×C8×C61—42no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C2411yescharacteristic
189#11758202—42no—
190#11758200—22no—
191#11758200—42no—
192#11758200—42no—
193#11758200—82no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—21no—
201#11758196—41no—
202#11758196—41no—
203#11758196—41no—
204#11758196—41no—
205C4×Dic61—21no—
206C4×Dic61—41no—
207C4×Dic61—41no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C23×C61—21no—
215C23×C61—41no—
216C23×C61—41no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61AGL(1,ℤ/8)12yes—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—82no—
232D4×C61—81no—
233D4×C61—81no—
234D4×C61—82no—
235D4×C61—161no—
236D4×C61—161no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—82no—
245C2×C4×C61—82no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253Dic122—21no—
254Dic122—41no—
255Dic122—41no—
256Dic122—81no—
257Dic122—161no—
258Dic122#13512yescomplemented
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—41no—
262#92418—21no—
263#92418—21no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#19843—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61D4211yescharacteristic
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61C2×#13511yescharacteristic
286C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C4×C61#57711yescharacteristic
288C4×C61#57811yescharacteristic
289Dic61—22no—
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293#3368—611no—
294#3363—611no—
295#3362—611no—
296#3287D6111yescharacteristic
297#1717—611no—
298#1698—611no—
299#1683—611no—
300#1682—611no—
301#1668—611no—
302#1641—611no—
303#1631—611no—
304#1626—611no—
305#1623—611no—
306#1584—611no—
307SD16⋊C8—611no—
308C2×C61—81no—
309C2×C61—161no—
310C2×C61—161no—
311C2×C61#326511yescharacteristic
312C2×C61#328511yescharacteristic
313C2×C61#328811yescharacteristic
314C2×#135—21no—
315Q82—611no—
316#557—611no—
317#550C2×D6111yescharacteristic
318#550C2×D6111yescharacteristic
319#493C2×D6111yescharacteristic
320#492—611no—
321#490—1221no—
322#487—611no—
323#487—1221no—
324#486—611no—
325#486—611no—
326#486—611no—
327#486—1221no—
328#478—1221no—
329#477—611no—
330#477—1221no—
331#476—611no—
332#476—1221no—
333#474—611no—
334#466—21no—
335#462—611no—
336#462—1221no—
337#460—611no—
338#460—611no—
339#460—1221no—
340#460—1221no—
341#460C2×D6111yescharacteristic
342#460C2×D6111yescharacteristic
343#449—21no—
344#449—21no—
345#446—611no—
346C8×Q8—611no—
347C8×Q8—611no—
348#440—1221no—
349C4×Q16—1221no—
350C4×Q16C2×D6111yescharacteristic
351C4×SD16—611no—
352C4×SD16—1221no—
353C8×D4C2×D6111yescharacteristic
354#335—611no—
355#335—611no—
356#333—611no—
357#332—611no—
358#331—611no—
359#331—611no—
360Q8⋊C8—611no—
361Q8⋊C8—611no—
362Q8⋊C8—611no—
363D4⋊C8—611no—
364#322—611no—
365C61#1984311yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C2×#41—21no—
368#135—44no—
369C2×Q16—1221no—
370C2×Q16C22⋊D6111yescharacteristic
371C2×SD16—21no—
372C2×SD16—21no—
373C2×SD16—611no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—2441no—
377C2×D8—21no—
378C2×D8—21no—
379C22×C8—21no—
380C4⋊Q8—611no—
381C4⋊Q8—611no—
382C4⋊Q8—1221no—
383C4⋊Q8—1221no—
384C4⋊Q8—1221no—
385C4⋊Q8—2441no—
386#127—21no—
387#127—21no—
388#127C22×D6111yescharacteristic
389#127C22×D6111yescharacteristic
390#125—1221no—
391#125—2441no—
392#121—21no—
393#121—21no—
394#121—41no—
395#121—41no—
396C4×Q8—611no—
397C4×Q8—1221no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8C22×D6111yescharacteristic
402C4×Q8C22×D6111yescharacteristic
403C4×D4—21no—
404C4×D4—21no—
405C4×D4C22×D6111yescharacteristic
406#107—1221no—
407#107—1221no—
408#107C22⋊D6111yescharacteristic
409#106—611no—
410#106—1221no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8C22×D6111yescharacteristic
418#103—611no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103C22⋊D6111yescharacteristic
424#103C22⋊D6111yescharacteristic
425#102—21no—
426#102—21no—
427#102—611no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—2441no—
437C22⋊C8—21no—
438#97—1221no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—1221no—
442C4×C8—1221no—
443C4×C8C22×D6111yescharacteristic
444#41—42no—
445#41—42no—
446#41—81no—
447#41—81no—
448C2×Q8—611no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—2441no—
452C2×Q8C2×C22⋊D6111yescharacteristic
453C2×Q8C2×C22⋊D6111yescharacteristic
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4D4×D6111yescharacteristic
465C22×C4—21no—
466C22×C4—41no—
467C22×C4—41no—
468Q16—22no—
469Q16—2442no—
470SD16—42no—
471SD16—42no—
472SD16—1222no—
473SD16—4882no—
474SD16—2442no—
475SD16—2442no—
476D8—42no—
477D8—81no—
478C2×C8—21no—
479C2×C8—42no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8C2×C22⋊D6111yescharacteristic
487C2×C8C2×C22⋊D6111yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—611no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—2441no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C4⋊C4D4×D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C42—21no—
508C42—21no—
509C42—1221no—
510C42—2441no—
511C42—2441no—
512C42C23×D6111yescharacteristic
513C23—21no—
514C23—41no—
515C23—41no—
516Q8—21no—
517Q8—21no—
518Q8—21no—
519Q8—612no—
520Q8—1221no—
521Q8—9761no—
522Q8—4881no—
523Q8—2441no—
524Q8—2441no—
525Q8#1451980312yes—
526D4—21no—
527D4—21no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—82no—
537D4—81no—
538D4—81no—
539D4—82no—
540D4—161no—
541D4—161no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4#1451982211yescharacteristic
558C8—21no—
559C8—21no—
560C8—41no—
561C8—1221no—
562C8—1221no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22D4×C22⋊D6111yescharacteristic
577C4—22no—
578C4—21no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—1222no—
584C4—4881no—
585C4—4881no—
586C4—2441no—
587C4#135×D6111yescharacteristic
588C4C2×#1451980311yescharacteristic
589C4#1684057311yescharacteristic
590C4#1684057611yescharacteristic
591C2—81no—
592C2—161no—
593C2—161no—
594C2#2145731011yescharacteristic
595C2#2145755411yescharacteristic
596C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3287C2
socleC22×C61D42
hypercenter#3287D61
hypocenterC61#19843
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3287

Complements

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

NH ≅ G/Ntype
1. #21458544C2semidirect
2. #21458495C2semidirect
3. #21458486C2semidirect
4. #21445457C2semidirect
5. #21445357C2semidirect
6. #21445270C2semidirect
7. #21445261C2semidirect
8. #21445176C2semidirect
9. #21445012C2semidirect
10. #21444946C2semidirect
11. #21444913C2semidirect
12. #21444896C2semidirect
13. #21444670C2semidirect
14. #21444631C2semidirect
258. Dic122#135semidirect
365. C61#19843semidirect

Maximal subgroups

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

subgroupindex
#214585442
#214584952
#214584862
#214454572
#214453572
#214452702
#214452612
#214451762
#214450122
#214449462
#214449132
#214448962
#214446702
#214446312
C61×#32872
#1984361