Finite Group Databasebeta

← #29738946

Subgroups — #29738946

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456355C211yesmaximal, characteristic, complemented
2#21455685C211yesmaximal, characteristic
3#21455674C211yesmaximal, characteristic
4#21454414C211yesmaximal, characteristic
5#21447333C211yesmaximal, characteristic, complemented
6#21447328C211yesmaximal, characteristic, complemented
7#21447170C211yesmaximal, characteristic, complemented
8#21447165C211yesmaximal, characteristic, complemented
9#21446600C211yesmaximal, characteristic, complemented
10#21446570C211yesmaximal, characteristic, complemented
11#21446050C211yesmaximal, characteristic
12#21446043C211yesmaximal, characteristic
13#21445986C211yesmaximal, characteristic, complemented
14#21445661C211yesmaximal, characteristic
15C61×#1816C211yesmaximal, characteristic
16#16840393C2211yescharacteristic
17#16840393C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840170—21no—
21#16840170—21no—
22#16840166—21no—
23#16840163—21no—
24#16840071—21no—
25#16840067—21no—
26#16840064—21no—
27#16840060—21no—
28#16840055—21no—
29#16840050—21no—
30#16840045—21no—
31#16840040—21no—
32#16839804—21no—
33#16839804—21no—
34#16839804C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839802C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839800—21no—
41#16839800C2211yescharacteristic
42C2×#14519710C2211yescharacteristic
43C2×#14519709—21no—
44C2×#14519709C2211yescharacteristic
45C2×#14519708C2211yescharacteristic
46#16839790C2211yescharacteristic
47#16839790C2211yescharacteristic
48C2×#14519704C2211yescharacteristic
49#16839758C2211yescharacteristic
50#16839717C2211yescharacteristic
51C2×#14519687C2211yescharacteristic
52#16839631C2211yescharacteristic
53#16839627C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839526C2211yescharacteristic
57#16839523C2211yescharacteristic
58#16839523C2211yescharacteristic
59#16839523C2211yescharacteristic
60C2×#131×C61C2211yescharacteristic
61C2×#108×C61C2211yescharacteristic
62C2×#107×C61C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C61×#417C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C2×#11758194C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—21no—
70#14519776—21no—
71#14519776—21no—
72#14519776—21no—
73#14519768—21no—
74#14519768—21no—
75#14519768—21no—
76#14519768—21no—
77#14519756—41no—
78#14519756—41no—
79Q8×Dic61—21no—
80Q8×Dic61—21no—
81D4×Dic61—21no—
82D4×Dic61—21no—
83D4×Dic61—41no—
84D4×Dic61—41no—
85#14519736—21no—
86#14519736—21no—
87#14519736C2311yescharacteristic
88#14519736C2311yescharacteristic
89#14519736C2311yescharacteristic
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×#11758196—41no—
95C2×C4×Dic61C2311yescharacteristic
96#14519711—21no—
97#14519711—21no—
98#14519711—21no—
99#14519711—21no—
100#14519711D411yescharacteristic
101#14519711D411yescharacteristic
102#14519710—21no—
103#14519710—21no—
104#14519710—21no—
105#14519710—21no—
106#14519709—21no—
107#14519709—21no—
108#14519709—21no—
109#14519709—21no—
110#14519709—21no—
111#14519709—21no—
112#14519709D411yescharacteristic
113#14519709D411yescharacteristic
114#14519708—21no—
115#14519708—21no—
116#14519708—21no—
117#14519708—21no—
118#14519706—21no—
119#14519706—21no—
120#14519706—21no—
121#14519706—21no—
122#14519704Q814yescomplemented
123#14519701D411yescharacteristic
124#14519701D411yescharacteristic
125#14519698D411yescharacteristic
126#14519698D411yescharacteristic
127#14519687—21no—
128#14519687—22no—
129C8×Dic61—21no—
130#14519684—41no—
131C2×#41×C61C2311yescharacteristic
132#131×C61—22no—
133#131×C61Q814yescomplemented
134C2×M16×C61—21no—
135C2×M16×C61C2311yescharacteristic
136C2×M16×C61C2311yescharacteristic
137C2×M16×C61C2311yescharacteristic
138C22×C8×C61—21no—
139C22×C8×C61C2311yescharacteristic
140C2×C4⋊C4×C61C2311yescharacteristic
141C2×C4⋊C4×C61C2311yescharacteristic
142#108×C61—22no—
143#107×C61—21no—
144#107×C61—21no—
145#103×C61—21no—
146#103×C61—21no—
147#102×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×C61C2411yescharacteristic
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—41no—
164C22×Dic61—41no—
165C2×Dic122—41no—
166C2×Dic122—41no—
167M16×C61—21no—
168M16×C61—24no—
169M16×C61—21no—
170M16×C61—21no—
171M16×C61—21no—
172M16×C61—21no—
173M16×C61C2×Q812yes—
174C2×C8×C61—21no—
175C2×C8×C61—24no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×Q812yes—
181C2×C8×C61#4112yes—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C22⋊Dic61—21no—
187C22⋊Dic61—21no—
188C22⋊Dic61—21no—
189C22⋊Dic61—41no—
190C22⋊Dic61—41no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—41no—
196C4⋊Dic61—41no—
197C4⋊Dic61—41no—
198C4⋊Dic61—41no—
199C4⋊Dic61C2×D411yescharacteristic
200C4⋊Dic61C2×D411yescharacteristic
201C4⋊Dic61C2×D411yescharacteristic
202C4⋊Dic61C2×D411yescharacteristic
203C4⋊Dic61C2×D411yescharacteristic
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—41no—
206#11758196—41no—
207#11758196—41no—
208#11758196—41no—
209#11758196—82no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—41no—
215C4×Dic61—41no—
216C4×Dic61C2×Q812yes—
217#11758194—21no—
218#11758194C2×Q812yes—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C2×D411yescharacteristic
223C2×#92418C2×D411yescharacteristic
224C23×C61—21no—
225C23×C61H(5,𝔽2)11yescharacteristic
226Q8×C61—21no—
227Q8×C61C4⋊Q812yes—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—21no—
231D4×C61C4⋊Q812yes—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—81no—
238C2×C4×C61—81no—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×Q811yescharacteristic
242C2×C4×C61C22×Q811yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C612−1+411yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—81no—
257C2×Dic61—81no—
258C2×Dic61—161no—
259C2×Dic61—161no—
260Dic122—161no—
261Dic122—161no—
262C8×C61—22no—
263C8×C61—41no—
264C8×C61—41no—
265C8×C61C22⋊Q812yes—
266C8×C61C22⋊Q812yes—
267#92418—21no—
268#92418—21no—
269#92418—22no—
270#26084—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61#54411yescharacteristic
277C22×C61D4211yescharacteristic
278C22×C61#55411yescharacteristic
279C4×C61—22no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61C2×C22⋊Q811yescharacteristic
283C4×C61C2×C4⋊Q811yescharacteristic
284C4×C61#54711yescharacteristic
285C4×C61D4×Q811yescharacteristic
286Dic61—42no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292Dic61—41no—
293Dic61—81no—
294Dic61—81no—
295#3164—611no—
296#3084—611no—
297#3083—611no—
298#2912—611no—
299#2090—611no—
300#2089—611no—
301#2061—611no—
302#2060—611no—
303#1951—611no—
304#1946—611no—
305#1860—611no—
306#1859—611no—
307#1847—611no—
308#1816D6111yescharacteristic
309#1765—611no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61—42no—
313C2×C61#244011yescharacteristic
314C2×C61#328811yescharacteristic
315C2×C61#328911yescharacteristic
316C2×#131C2×D6111yescharacteristic
317#544—611no—
318#544—611no—
319#520—611no—
320#520—611no—
321#501—1221no—
322#501—1221no—
323#500—1221no—
324#499—1221no—
325#479—1221no—
326#479—1221no—
327#478—1221no—
328#478—1221no—
329#477—1221no—
330#476—1221no—
331#475—1221no—
332#474—1221no—
333C2×#108C2×D6111yescharacteristic
334#428—611no—
335#428—611no—
336#428—611no—
337#428—611no—
338#428—611no—
339#428—611no—
340#428—1221no—
341#428—1221no—
342#427—611no—
343#427—1221no—
344C2×#107—611no—
345C2×#107C2×D6111yescharacteristic
346C2×#106—611no—
347C2×#106—611no—
348C2×#106—1221no—
349#424—611no—
350#424—611no—
351C2×#104—611no—
352#417C2×D6111yescharacteristic
353#417C2×D6111yescharacteristic
354#416—611no—
355#405—611no—
356C2×#97—611no—
357#384—611no—
358#382—611no—
359#343—611no—
360#343—611no—
361#343C2×D6111yescharacteristic
362#343C2×D6111yescharacteristic
363#341—611no—
364#339—611no—
365#339—611no—
366#339—611no—
367C61#2608411yescharacteristic, complemented, Sylow
368C2×#41C22×D6111yescharacteristic
369#131—22no—
370#131Dic12214yescomplemented
371C2×M16—21no—
372C2×M16—611no—
373C2×M16C22×D6111yescharacteristic
374C2×M16C22×D6111yescharacteristic
375C2×M16C22×D6111yescharacteristic
376C22×C8—21no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C4⋊Q8—1221no—
380C4⋊Q8—1221no—
381C4⋊Q8—1221no—
382C4⋊Q8—1221no—
383#125—1221no—
384#125—1221no—
385#125—1221no—
386#125—1221no—
387C22⋊Q8—2441no—
388C22⋊Q8—2441no—
389C4×Q8—1221no—
390C4×Q8—1221no—
391C4×D4—1221no—
392C4×D4—1221no—
393C4×D4—2441no—
394C4×D4—2441no—
395#117—611no—
396#117—611no—
397#117—611no—
398#117—1221no—
399#117—1221no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—2441no—
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C42—611no—
408#108—22no—
409#107—21no—
410#107—21no—
411#107—611no—
412#107—611no—
413#107—1221no—
414#107—1221no—
415#107—1221no—
416#107—1221no—
417#107—1221no—
418#107—1221no—
419#107—1221no—
420#107—1221no—
421#106—611no—
422#106—611no—
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—
433C4⋊C8—1221no—
434C4⋊C8—1221no—
435C4⋊C8—1221no—
436C4⋊C8—1221no—
437#104—614no—
438#103—21no—
439#103—21no—
440#103—611no—
441#103—611no—
442#102—21no—
443#102—21no—
444#102—611no—
445#102—611no—
446#97—1221no—
447#97—1222no—
448C4×C8—1221no—
449#95—2441no—
450#41—21no—
451#41—21no—
452#41C2×Dic12212yes—
453#41Q8×D6112yes—
454C2×Q8—2441no—
455C2×Q8—2441no—
456C2×Q8D4×D6111yescharacteristic
457C2×D4—21no—
458C2×D4D4×D6111yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4C23×D6111yescharacteristic
468M16—21no—
469M16—24no—
470M16—21no—
471M16—21no—
472M16—21no—
473M16—21no—
474M16—612no—
475M16—1221no—
476M16C2×Dic12212yes—
477C2×C8—21no—
478C2×C8—24no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8C2×Dic12212yes—
487C2×C8D4×D6111yescharacteristic
488C2×C8D4×D6111yescharacteristic
489C2×C8#1175822412yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—611no—
495C4⋊C4—611no—
496C4⋊C4—611no—
497C4⋊C4—611no—
498C4⋊C4—611no—
499C4⋊C4—611no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—4882no—
505C4⋊C4—2441no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C4⋊C4—2441no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—2441no—
517C22⋊C4—2441no—
518C42—612no—
519C42—1221no—
520C42—1221no—
521C42—1221no—
522C42—1221no—
523C42—2441no—
524C42—2441no—
525C23—21no—
526C23#1451982311yescharacteristic
527Q8—21no—
528Q8—9761no—
529Q8—9761no—
530Q8#1451977612yes—
531D4—22no—
532D4—22no—
533D4—21no—
534D4#1451977612yes—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—9761no—
550C2×C4—9761no—
551C2×C4—4881no—
552C2×C4—4881no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4C2×D4×D6111yescharacteristic
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×Q8×D6111yescharacteristic
560C2×C4C22×Dic12211yescharacteristic
561C2×C4C2×#1175822411yescharacteristic
562C2×C4#1451982411yescharacteristic
563C8—22no—
564C8—41no—
565C8—41no—
566C8—1221no—
567C8—1221no—
568C8—1222no—
569C8#1451975412yes—
570C8#1451975412yes—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22#1684039011yescharacteristic
577C22#1684039611yescharacteristic
578C22#1684047011yescharacteristic
579C4—22no—
580C4—81no—
581C4—81no—
582C4—4881no—
583C4—4881no—
584C4—2442no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4—2441no—
590C4—2441no—
591C4C2×#1451975411yescharacteristic
592C4C2×#1451977611yescharacteristic
593C4#1684041211yescharacteristic
594C4#1684042511yescharacteristic
595C2—21no—
596C2—21no—
597C2—42no—
598C2#2144952211yescharacteristic
599C2#2145758211yescharacteristic
600C2#2145759311yescharacteristic
subgroupG/H
centerC22#16840396
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1816C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#1816D61
hypocenterC61#26084
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1816

Complements

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

NH ≅ G/Ntype
1. #21456355C2semidirect
5. #21447333C2semidirect
6. #21447328C2semidirect
7. #21447170C2semidirect
8. #21447165C2semidirect
9. #21446600C2semidirect
10. #21446570C2semidirect
13. #21445986C2semidirect
122. #14519704Q8semidirect
133. #131×C61Q8semidirect
367. C61#26084semidirect
370. #131Dic122semidirect

Maximal subgroups

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

subgroupindex
#214563552
#214556852
#214556742
#214544142
#214473332
#214473282
#214471702
#214471652
#214466002
#214465702
#214460502
#214460432
#214459862
#214456612
C61×#18162
#2608461