Finite Group Databasebeta

← #29738910

Subgroups — #29738910

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456349C211yesmaximal, characteristic, complemented
2#21455685C211yesmaximal, characteristic
3#21455681C211yesmaximal, characteristic
4#21454412C211yesmaximal, characteristic
5#21447333C211yesmaximal, characteristic, complemented
6#21447332C211yesmaximal, characteristic, complemented
7#21447164C211yesmaximal, characteristic, complemented
8#21447163C211yesmaximal, characteristic, complemented
9#21446598C211yesmaximal, characteristic, complemented
10#21446566C211yesmaximal, characteristic, complemented
11#21446047C211yesmaximal, characteristic
12#21445975C211yesmaximal, characteristic, complemented
13#21445876C211yesmaximal, characteristic
14#21445657C211yesmaximal, characteristic
15C61×#1859C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840393C2211yescharacteristic
18#16840390C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840170—21no—
21#16840169—21no—
22#16840163—21no—
23#16840159—21no—
24#16840071—21no—
25#16840069—21no—
26#16840062—21no—
27#16840060—21no—
28#16840052—21no—
29#16840050—21no—
30#16840047—21no—
31#16840045—21no—
32C2×#14519712C2211yescharacteristic
33#16839805—21no—
34#16839805C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839802—21no—
38#16839802C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839798—21no—
41#16839798C2211yescharacteristic
42C2×#14519708—21no—
43C2×#14519708C2211yescharacteristic
44C2×#14519708C2211yescharacteristic
45C2×#14519708C2211yescharacteristic
46C2×#14519708C2211yescharacteristic
47#16839790C2211yescharacteristic
48#16839789C2211yescharacteristic
49#16839763C2211yescharacteristic
50#16839758C2211yescharacteristic
51#16839720C2211yescharacteristic
52C2×C4×#92418C2211yescharacteristic
53#16839632C2211yescharacteristic
54#16839531C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839526C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839522C2211yescharacteristic
60C61×#520C2211yescharacteristic
61C61×#428C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C61×#384C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#339C2211yescharacteristic
67#14519784—22no—
68#14519784Q814yescomplemented
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C22×#92418—21no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519777—21no—
76#14519776—21no—
77#14519776—21no—
78#14519776—21no—
79#14519769—21no—
80#14519768—21no—
81#14519768—21no—
82#14519768—21no—
83#14519756—41no—
84#14519754—41no—
85Q8×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61—41no—
88#14519736—21no—
89#14519736C2311yescharacteristic
90#14519736C2311yescharacteristic
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—41no—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95#14519712—22no—
96#14519711—21no—
97#14519711—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519710D411yescharacteristic
102#14519710D411yescharacteristic
103#14519709—21no—
104#14519709—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708—21no—
109#14519708—21no—
110#14519708—21no—
111#14519708—21no—
112#14519708—21no—
113#14519708—21no—
114#14519708—21no—
115#14519708—21no—
116#14519708D411yescharacteristic
117#14519708D411yescharacteristic
118C4⋊#92418—21no—
119C4⋊#92418—21no—
120#14519706—21no—
121#14519706—21no—
122#14519701—21no—
123#14519701D411yescharacteristic
124#14519701D411yescharacteristic
125#14519698—21no—
126#14519698D411yescharacteristic
127#14519698D411yescharacteristic
128#14519687—21no—
129C4×#92418—22no—
130C8×Dic61—21no—
131C2×#41×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134C4×Q8×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—41no—
137#117×C61—21no—
138#117×C61C2311yescharacteristic
139C2×C4⋊C4×C61—21no—
140C2×C4⋊C4×C61—41no—
141C2×C4⋊C4×C61C2311yescharacteristic
142C2×C4⋊C4×C61C2311yescharacteristic
143C2×C42×C61C2311yescharacteristic
144#107×C61—21no—
145#106×C61—21no—
146#104×C61Q814yescomplemented
147#103×C61—21no—
148#102×C61—21no—
149#41×C61—21no—
150#41×C61—21no—
151#41×C61C2×Q812yes—
152#41×C61C2×Q812yes—
153C2×Q8×C61C2×D411yescharacteristic
154C2×D4×C61—21no—
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—41no—
165C2×Dic122—41no—
166C2×Dic122—41no—
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61C2×Q812yes—
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—41no—
178C4⋊C4×C61—82no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61C2×Q812yes—
187C22⋊Dic61—21no—
188C22⋊Dic61—21no—
189C22⋊Dic61—41no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—41no—
195C4⋊Dic61—41no—
196C4⋊Dic61—41no—
197C4⋊Dic61—82no—
198C4⋊Dic61C2×D411yescharacteristic
199C4⋊Dic61C2×D411yescharacteristic
200C4⋊Dic61C2×D411yescharacteristic
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—41no—
203#11758196—41no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—41no—
209#11758194—21no—
210#11758194—24no—
211#11758194—21no—
212#11758194—21no—
213#11758194C2×Q812yes—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—24no—
218C2×#92418—21no—
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222C2×#92418C2×D411yescharacteristic
223C2×#92418C2×Q812yes—
224C2×#92418#4112yes—
225C23×C61—21no—
226C23×C61H(5,𝔽2)11yescharacteristic
227Q8×C61—21no—
228Q8×C61C4⋊Q812yes—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—21no—
232D4×C61C4⋊Q812yes—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—81no—
244C2×C4×C61—81no—
245C2×C4×C61—161no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×Q811yescharacteristic
249C2×C4×C61C22×Q811yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—81no—
259C2×Dic61—81no—
260C2×Dic61—161no—
261Dic122—161no—
262Dic122—161no—
263C8×C61—22no—
264C8×C61—41no—
265#92418—22no—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418C22⋊Q812yes—
270#92418C22⋊Q812yes—
271#26081—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—22no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61#54411yescharacteristic
278C22×C61D4211yescharacteristic
279C22×C61#55411yescharacteristic
280C4×C61—24no—
281C4×C61—22no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61C2×C22⋊Q811yescharacteristic
287C4×C61C2×C4⋊Q811yescharacteristic
288C4×C61#54711yescharacteristic
289C4×C61D4×Q811yescharacteristic
290Dic61—41no—
291Dic61—41no—
292Dic61—41no—
293Dic61—41no—
294Dic61—81no—
295Dic61—81no—
296#3163—611no—
297#3084—611no—
298#3084—611no—
299#2912—611no—
300#2090—611no—
301#2090—611no—
302#2060—611no—
303#2060—611no—
304#1951—611no—
305#1945—611no—
306#1859—611no—
307#1859D6111yescharacteristic
308#1844—611no—
309#1815—611no—
310#1764—611no—
311C2×C61—21no—
312C2×C61—21no—
313C2×C61—42no—
314C2×C61#244011yescharacteristic
315C2×C61#329111yescharacteristic
316C2×C61#329111yescharacteristic
317C2×#131—611no—
318#544—611no—
319#544—611no—
320#520—611no—
321#520C2×D6111yescharacteristic
322#501—1221no—
323#501—1221no—
324#499—1221no—
325#498—1221no—
326#479—1221no—
327#479—1221no—
328#478—1221no—
329#478—1221no—
330#476—1221no—
331#476—1221no—
332#475—1221no—
333#475—1221no—
334C2×#108—611no—
335#428—611no—
336#428—611no—
337#428—611no—
338#428—611no—
339#428—611no—
340#428—1221no—
341#428—1221no—
342#428C2×D6111yescharacteristic
343#427—611no—
344#427—1221no—
345C2×#106—611no—
346C2×#106—611no—
347C2×#106—611no—
348C2×#106—611no—
349C2×#106—1221no—
350#424—611no—
351#424—611no—
352C2×#104C2×D6111yescharacteristic
353#417—611no—
354#417C2×D6111yescharacteristic
355#416—611no—
356#405—611no—
357C2×C4×C8—611no—
358#384—611no—
359#384C2×D6111yescharacteristic
360#343—611no—
361#343—611no—
362#343—611no—
363#343C2×D6111yescharacteristic
364#341—611no—
365#339—611no—
366#339—611no—
367#339C2×D6111yescharacteristic
368C61#2608111yescharacteristic, complemented, Sylow
369C2×#41C22×D6111yescharacteristic
370#131—614no—
371#131—1222no—
372C2×M16—611no—
373C2×M16—611no—
374C2×M16—1221no—
375C2×M16C22×D6111yescharacteristic
376C2×M16C22×D6111yescharacteristic
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—1221no—
380C4⋊Q8—1221no—
381C4⋊Q8—1221no—
382C4⋊Q8—1221no—
383C4⋊Q8—1221no—
384#125—1221no—
385#125—1221no—
386#125—1221no—
387#125—1221no—
388C22⋊Q8—2441no—
389C22⋊Q8—2441no—
390C4×Q8—21no—
391C4×Q8—1221no—
392C4×D4—21no—
393C4×D4—41no—
394C4×D4—1221no—
395C4×D4—2441no—
396#117—21no—
397#117—611no—
398#117—611no—
399#117—1221no—
400#117C22×D6111yescharacteristic
401C2×C4⋊C4—21no—
402C2×C4⋊C4—41no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—2441no—
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C42C22×D6111yescharacteristic
410#108—1222no—
411#107—21no—
412#107—611no—
413#107—611no—
414#107—1221no—
415#107—1221no—
416#107—1221no—
417#107—1221no—
418#107—1221no—
419#106—21no—
420#106—611no—
421#106—611no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425#106—1221no—
426#106—1221no—
427#106—1221no—
428#106—1221no—
429#106—1221no—
430#106—1221no—
431#106—1221no—
432#106—1221no—
433#106—1221no—
434#106—1221no—
435C4⋊C8—1221no—
436C4⋊C8—1221no—
437C4⋊C8—1221no—
438C4⋊C8—1221no—
439#104Dic12214yescomplemented
440#103—21no—
441#103—611no—
442#103—611no—
443#103—1221no—
444#102—21no—
445#102—611no—
446#102—611no—
447#102—1221no—
448#97—1221no—
449C4×C8—1222no—
450C4×C8—1221no—
451#41—21no—
452#41—21no—
453#41C2×Dic12212yes—
454#41Q8×D6112yes—
455C2×Q8—2441no—
456C2×Q8—2441no—
457C2×Q8D4×D6111yescharacteristic
458C2×D4—21no—
459C2×D4D4×D6111yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—41no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—2441no—
468C22×C4C23×D6111yescharacteristic
469M16—21no—
470M16—21no—
471M16—21no—
472M16—612no—
473M16—1224no—
474M16—1221no—
475M16—1221no—
476M16—1221no—
477M16C2×Dic12212yes—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—612no—
483C2×C8—612no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—1224no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—82no—
497C4⋊C4—611no—
498C4⋊C4—611no—
499C4⋊C4—611no—
500C4⋊C4—611no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—4882no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C4⋊C4—2441no—
513C4⋊C4D4×D6111yescharacteristic
514C4⋊C4D4×D6111yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—41no—
517C22⋊C4—1221no—
518C22⋊C4—1221no—
519C22⋊C4—2441no—
520C42—21no—
521C42—21no—
522C42—41no—
523C42—1221no—
524C42—1221no—
525C42—2441no—
526C42C2×Dic12212yes—
527C23—21no—
528C23#1451982211yescharacteristic
529Q8—21no—
530Q8—9761no—
531Q8—9761no—
532Q8#1451977612yes—
533D4—22no—
534D4—22no—
535D4—21no—
536D4#1451977612yes—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—161no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—9761no—
555C2×C4—4881no—
556C2×C4—4881no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×Q8×D6111yescharacteristic
562C2×C4C22×Dic12211yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4#1451982411yescharacteristic
565C8—22no—
566C8—41no—
567C8—612no—
568C8—612no—
569C8—1222no—
570C8—1221no—
571C8—1221no—
572C8—2441no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22#1684039111yescharacteristic
579C22#1684039411yescharacteristic
580C22#1684047111yescharacteristic
581C4—24no—
582C4—22no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—4881no—
588C4—4881no—
589C4—2441no—
590C4—2441no—
591C4—2441no—
592C4—2441no—
593C4C22⋊Q8×D6111yescharacteristic
594C4C2×#1451977611yescharacteristic
595C4#1684041011yescharacteristic
596C4D4×Dic12211yescharacteristic
597C2—21no—
598C2—21no—
599C2—42no—
600C2#2144952011yescharacteristic
601C2#2145760911yescharacteristic
602C2#2145761411yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1859C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#1859D61
hypocenterC61#26081
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1859

Complements

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

NH ≅ G/Ntype
1. #21456349C2semidirect
5. #21447333C2semidirect
6. #21447332C2semidirect
7. #21447164C2semidirect
8. #21447163C2semidirect
9. #21446598C2semidirect
10. #21446566C2semidirect
12. #21445975C2semidirect
68. #14519784Q8semidirect
146. #104×C61Q8semidirect
368. C61#26081semidirect
439. #104Dic122semidirect

Maximal subgroups

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

subgroupindex
#214563492
#214556852
#214556812
#214544122
#214473332
#214473322
#214471642
#214471632
#214465982
#214465662
#214460472
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