Finite Group Databasebeta

← #29732166

Subgroups — #29732166

Order 15616.

648 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#21456369C211yesmaximal, characteristic, complemented
2#21455613C211yesmaximal, characteristic
3C2×#16840060C211yesmaximal, characteristic
4#21454763C211yesmaximal, characteristic, complemented
5#21447332C211yesmaximal, characteristic, complemented
6#21447326C211yesmaximal, characteristic, complemented
7#21447179C211yesmaximal, characteristic
8#21447177C211yesmaximal, characteristic
9#21446810C211yesmaximal, characteristic, complemented
10#21446604C211yesmaximal, characteristic, complemented
11#21446598C211yesmaximal, characteristic, complemented
12#21446593C211yesmaximal, characteristic, complemented
13#21446483C211yesmaximal, characteristic
14#100×Dic61C211yesmaximal, characteristic
15C61×#1931C211yesmaximal, characteristic
16#16840315C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19#16840170—21no—
20#16840169—21no—
21#16840163—21no—
22#16840162—21no—
23#16840064—21no—
24#16840060—21no—
25#16840060—21no—
26#16840060—21no—
27#16840060—21no—
28#16840060—21no—
29#16840050—21no—
30#16840040—21no—
31#16839892—21no—
32#16839890—21no—
33#16839890—21no—
34C8×Dic122—21no—
35C2×#14519713C2211yescharacteristic
36C2×#14519712C2211yescharacteristic
37#16839805C2211yescharacteristic
38#16839805C2211yescharacteristic
39#16839804C2211yescharacteristic
40#16839804C2211yescharacteristic
41#16839802C2211yescharacteristic
42C2×#14519711C2211yescharacteristic
43C2×#14519708C2211yescharacteristic
44#16839789C2211yescharacteristic
45#16839788C2211yescharacteristic
46#16839785C2211yescharacteristic
47C2×#14519706C2211yescharacteristic
48#16839767C2211yescharacteristic
49C2×#14519698C2211yescharacteristic
50C2×#14519694C2211yescharacteristic
51#16839720C2211yescharacteristic
52M16×Dic61C2211yescharacteristic
53M16×Dic61C2211yescharacteristic
54#16839531C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839526C2211yescharacteristic
60C61×#428C2211yescharacteristic
61C61×#418C2211yescharacteristic
62C2×#102×C61C2211yescharacteristic
63C2×#100×C61C2211yescharacteristic
64C61×#405C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#341C2211yescharacteristic
67C23×Dic61—21no—
68C2×#11758194C2311yescharacteristic
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519776D411yescharacteristic
73#14519776D411yescharacteristic
74#14519768—21no—
75#14519768—21no—
76#14519768—21no—
77#14519768—21no—
78#14519768D411yescharacteristic
79#14519768D411yescharacteristic
80#14519754—21no—
81#14519754—21no—
82C4×Dic122—21no—
83C4×Dic122—21no—
84D4×Dic61—21no—
85D4×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88#14519736C2311yescharacteristic
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×C22⋊Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95#14519713—22no—
96#14519712—22no—
97#14519711—21no—
98#14519711—21no—
99#14519711—21no—
100#14519711—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519709—21no—
105#14519709—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708—21no—
109#14519708—21no—
110#14519708—21no—
111C4⋊#92418—21no—
112C4⋊#92418—21no—
113#14519706—21no—
114#14519706—21no—
115#14519706—21no—
116#14519706—21no—
117#14519706D411yescharacteristic
118#14519706D411yescharacteristic
119#14519706D411yescharacteristic
120#14519706D411yescharacteristic
121#14519698—21no—
122#14519698—21no—
123#14519698—21no—
124#14519698—21no—
125#14519694—22no—
126#14519687—21no—
127#14519687—21no—
128#14519687—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C8×Dic61—21no—
132C22×D4×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134C2×M16×C61C2311yescharacteristic
135C2×M16×C61C2311yescharacteristic
136C22×C8×C61C2311yescharacteristic
137#117×C61C2311yescharacteristic
138C2×C4⋊C4×C61C2311yescharacteristic
139#107×C61—21no—
140#106×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143#102×C61—21no—
144#102×C61—21no—
145#100×C61Q814yescomplemented
146#97×C61—21no—
147C4×C8×C61—21no—
148C24×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61C2×Q812yes—
155C22×C4×C61—21no—
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—41no—
160C22×Dic61—41no—
161C22×Dic61—82no—
162C22×Dic61#4111yescharacteristic
163C22×Dic61#4111yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—21no—
166M16×C61—21no—
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61C2×Q812yes—
171M16×C61C2×Q812yes—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C42×C61—21no—
185C22⋊Dic61—21no—
186C22⋊Dic61—41no—
187C22⋊Dic61—41no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—41no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—21no—
199#11758196—21no—
200#11758196—21no—
201#11758196—21no—
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—21no—
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209#11758194—21no—
210#11758194—21no—
211#11758194—21no—
212#11758194—21no—
213#11758194—21no—
214#11758194#4112yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—81no—
226C23×C61—81no—
227C23×C61C2×#4111yescharacteristic
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—81no—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×Q811yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C612−1+411yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—82no—
249C2×Dic61—82no—
250C2×Dic61—81no—
251C2×Dic61—81no—
252C2×Dic61#12111yescharacteristic
253C2×Dic61#12111yescharacteristic
254C2×Dic61#12111yescharacteristic
255C2×Dic61#12111yescharacteristic
256Dic122—42no—
257Dic122—81no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—22no—
261C8×C61—22no—
262C8×C61—41no—
263#92418—21no—
264#92418—21no—
265#92418—22no—
266#92418—41no—
267#92418—41no—
268#25493—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—42no—
272C22×C61—82no—
273C22×C61—82no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61C2×#12111yescharacteristic
277C22×C61#53311yescharacteristic
278C22×C61#54111yescharacteristic
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—81no—
282C4×C61#54711yescharacteristic
283C4×C61#54811yescharacteristic
284C4×C61D4×Q811yescharacteristic
285C4×C61D4×Q811yescharacteristic
286Dic61—21no—
287Dic61—21no—
288Dic61—21no—
289Dic61—21no—
290Dic61—41no—
291Dic61—41no—
292Dic61—81no—
293Dic61—81no—
294#3166—611no—
295#3077—611no—
296C2×#478—611no—
297#2962—611no—
298#2090—611no—
299#2089—611no—
300#2063—611no—
301#2063—611no—
302#1998—611no—
303#1952—611no—
304#1951—611no—
305#1950—611no—
306#1931—611no—
307#1931D6111yescharacteristic
308C4×#100—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—82no—
312C2×C61#242911yescharacteristic
313C2×C61#306711yescharacteristic
314C2×C61#306811yescharacteristic
315#533—611no—
316C2×#125—611no—
317C2×C4×D4—611no—
318#501—1221no—
319#501—1221no—
320#499—1221no—
321#499—1221no—
322#478—1221no—
323#478—1221no—
324#478—1221no—
325#478—1221no—
326#478—1221no—
327#478—1221no—
328#476—1221no—
329#474—1221no—
330#446—1221no—
331#446—1221no—
332#446—1221no—
333C8×Q8—1221no—
334C2×#108—611no—
335C2×#108—611no—
336#428—611no—
337#428—611no—
338#428—611no—
339#428—611no—
340#428—611no—
341#428C2×D6111yescharacteristic
342C2×#107—611no—
343C2×#106—611no—
344#424—611no—
345#424—611no—
346#423—611no—
347C2×C4⋊C8—611no—
348#418—611no—
349#418C2×D6111yescharacteristic
350C2×#102—611no—
351C2×#102C2×D6111yescharacteristic
352C2×#100—611no—
353C2×#100C2×D6111yescharacteristic
354#405—611no—
355#405C2×D6111yescharacteristic
356C4×M16—611no—
357C4×M16—611no—
358#343—611no—
359#343—611no—
360#343—611no—
361#343—611no—
362#343—611no—
363#343C2×D6111yescharacteristic
364#341—611no—
365#341C2×D6111yescharacteristic
366C61#2549311yescharacteristic, complemented, Sylow
367C22×D4C22×D6111yescharacteristic
368C23×C4—1221no—
369C2×M16—611no—
370C2×M16—611no—
371C2×M16—611no—
372C2×M16C22×D6111yescharacteristic
373C2×M16C22×D6111yescharacteristic
374C2×M16C22×D6111yescharacteristic
375C22×C8—611no—
376C22×C8C22×D6111yescharacteristic
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379#125—611no—
380#125—611no—
381#125—1221no—
382#125—1221no—
383#125—1221no—
384#125—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×D4—1221no—
390C4×D4—1221no—
391C4×D4—1221no—
392C4×D4—1221no—
393#117—611no—
394#117C22×D6111yescharacteristic
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—1221no—
401C2×C42—611no—
402#108—1222no—
403#108—1222no—
404#107—21no—
405#107—1221no—
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412#106—21no—
413#106—1221no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417#106—1221no—
418#106—1221no—
419#106—1221no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—1221no—
425C4⋊C8—1221no—
426C4⋊C8—1221no—
427C4⋊C8—1221no—
428C4⋊C8—1221no—
429C4⋊C8—1221no—
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#100—1222no—
439#100Dic12214yescomplemented
440#97—21no—
441#97—1221no—
442#97—1221no—
443#97—1221no—
444C4×C8—21no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8—1221no—
448C24—21no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4C2×Dic12212yes—
457C22×C4—21no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—4882no—
463C22×C4—2441no—
464C22×C4—2441no—
465C22×C4C23×D6111yescharacteristic
466M16—21no—
467M16—21no—
468M16—21no—
469M16—21no—
470M16—612no—
471M16—1221no—
472M16—1221no—
473M16—1221no—
474M16—1221no—
475M16—1221no—
476M16C2×Dic12212yes—
477M16C2×Dic12212yes—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8D4×D6111yescharacteristic
491C2×C8D4×D6111yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—611no—
495C4⋊C4—611no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4D4×D6111yescharacteristic
511C4⋊C4D4×D6111yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—1221no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C42—21no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—1221no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—81no—
526C23—81no—
527C23#41×D6111yescharacteristic
528Q8—4881no—
529Q8—2442no—
530D4—22no—
531D4—22no—
532D4—41no—
533D4—41no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—81no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—4882no—
545C2×C4—4882no—
546C2×C4—4881no—
547C2×C4—4881no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C22×Dic12211yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4#1451982411yescharacteristic
557C2×C4#1451982411yescharacteristic
558C8—21no—
559C8—21no—
560C8—22no—
561C8—22no—
562C8—41no—
563C8—1221no—
564C8—1221no—
565C8—1222no—
566C8—2441no—
567C8—2441no—
568C22—21no—
569C22—21no—
570C22—42no—
571C22—82no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22#121×D6111yescharacteristic
576C22#1684031511yescharacteristic
577C22#1684036911yescharacteristic
578C4—41no—
579C4—41no—
580C4—81no—
581C4—1221no—
582C4—1221no—
583C4—1221no—
584C4—1221no—
585C4—4881no—
586C4—4881no—
587C4—2441no—
588C4—2441no—
589C4#1684041011yescharacteristic
590C4#1684042011yescharacteristic
591C4D4×Dic12211yescharacteristic
592C4D4×Dic12211yescharacteristic
593C2—21no—
594C2—21no—
595C2—82no—
596C2#2144941511yescharacteristic
597C2#2145553711yescharacteristic
598C2#2145554211yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1931C2
socleC22×C61C2×#121
prime radicalC22×D4×C61C23
hypercenter#1931D61
hypocenterC61#25493
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1931

Complements

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

NH ≅ G/Ntype
1. #21456369C2semidirect
4. #21454763C2semidirect
5. #21447332C2semidirect
6. #21447326C2semidirect
9. #21446810C2semidirect
10. #21446604C2semidirect
11. #21446598C2semidirect
12. #21446593C2semidirect
145. #100×C61Q8semidirect
366. C61#25493semidirect
439. #100Dic122semidirect

Maximal subgroups

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

subgroupindex
#214563692
#214556132
C2×#168400602
#214547632
#214473322
#214473262
#214471792
#214471772
#214468102
#214466042
#214465982
#214465932
#214464832
#100×Dic612
C61×#19312
#2549361