Finite Group Databasebeta

← #29738944

Subgroups — #29738944

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#21456358C211yesmaximal, characteristic, complemented
2#21455679C211yesmaximal, characteristic
3#21455668C211yesmaximal, characteristic
4#21454414C211yesmaximal, characteristic
5#21447331C211yesmaximal, characteristic, complemented
6#21447326C211yesmaximal, characteristic, complemented
7#21447168C211yesmaximal, characteristic, complemented
8#21447163C211yesmaximal, characteristic, complemented
9#21446599C211yesmaximal, characteristic, complemented
10#21446572C211yesmaximal, characteristic, complemented
11#21446054C211yesmaximal, characteristic
12#21446047C211yesmaximal, characteristic
13#21445987C211yesmaximal, characteristic, complemented
14#21445883C211yesmaximal, characteristic
15C61×#1765C211yesmaximal, characteristic
16#16840390C2211yescharacteristic
17#16840390C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840171—21no—
21#16840171—21no—
22#16840167—21no—
23#16840164—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—
32C2×#14519713C2211yescharacteristic
33#16839805C2211yescharacteristic
34#16839805C2211yescharacteristic
35#16839804—21no—
36#16839804—21no—
37#16839804C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839804C2211yescharacteristic
40#16839804C2211yescharacteristic
41#16839800—21no—
42#16839800C2211yescharacteristic
43C2×#14519711C2211yescharacteristic
44C2×#14519711C2211yescharacteristic
45C2×#14519709—21no—
46C2×#14519709C2211yescharacteristic
47C2×#14519709C2211yescharacteristic
48#16839788C2211yescharacteristic
49#16839788C2211yescharacteristic
50#16839763C2211yescharacteristic
51#16839763C2211yescharacteristic
52#16839632C2211yescharacteristic
53#16839628C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839522C2211yescharacteristic
60C2×#131×C61C2211yescharacteristic
61C2×#104×C61C2211yescharacteristic
62C61×#416C2211yescharacteristic
63C61×#405C2211yescharacteristic
64C2×#97×C61C2211yescharacteristic
65C61×#341C2211yescharacteristic
66C61×#339C2211yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758194C2311yescharacteristic
69#14519777—21no—
70#14519777—21no—
71#14519776—21no—
72#14519776—21no—
73#14519769—21no—
74#14519769—21no—
75#14519768—21no—
76#14519768—21no—
77#14519754—41no—
78#14519754—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
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—41no—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96#14519713—22no—
97#14519711—21no—
98#14519711—21no—
99#14519711—21no—
100#14519711—21no—
101#14519711—21no—
102#14519711—21no—
103#14519711—21no—
104#14519711—21no—
105#14519711D411yescharacteristic
106#14519711D411yescharacteristic
107#14519710—21no—
108#14519710—21no—
109#14519709—21no—
110#14519709—21no—
111#14519709—21no—
112#14519709—21no—
113#14519709—21no—
114#14519709—21no—
115#14519709—21no—
116#14519709—21no—
117#14519709D411yescharacteristic
118#14519709D411yescharacteristic
119#14519708—21no—
120#14519708—21no—
121#14519706—21no—
122#14519706—21no—
123#14519706—21no—
124#14519706—21no—
125#14519701—21no—
126#14519701—21no—
127#14519698—21no—
128#14519698—21no—
129#14519684—41no—
130C2×#41×C61C2311yescharacteristic
131#131×C61—22no—
132#131×C61Q814yescomplemented
133C2×M16×C61—21no—
134C2×M16×C61C2311yescharacteristic
135C2×M16×C61C2311yescharacteristic
136C22×C8×C61—21no—
137C22×C8×C61C2311yescharacteristic
138C22×C8×C61C2311yescharacteristic
139#117×C61C2311yescharacteristic
140C2×C42×C61C2311yescharacteristic
141#104×C61Q814yescomplemented
142#103×C61D411yescharacteristic
143#103×C61D411yescharacteristic
144#102×C61D411yescharacteristic
145#102×C61D411yescharacteristic
146#97×C61—21no—
147#97×C61—22no—
148C4×C8×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×C61C2411yescharacteristic
159C22×Dic61—21no—
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—21no—
169M16×C61—24no—
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—24no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×Q812yes—
182C2×C8×C61#4112yes—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C2×Q812yes—
189C22⋊Dic61—21no—
190C22⋊Dic61—21no—
191C22⋊Dic61—41no—
192C22⋊Dic61—41no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4⋊Dic61—82no—
206C4⋊Dic61C2×D411yescharacteristic
207C4⋊Dic61C2×D411yescharacteristic
208C4⋊Dic61C2×D411yescharacteristic
209C4⋊Dic61C2×D411yescharacteristic
210#11758196—41no—
211#11758196—41no—
212#11758196—41no—
213#11758196—41no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—41no—
217C4×Dic61—41no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222C2×#92418—21no—
223C2×#92418—21no—
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—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×Q811yescharacteristic
243C2×C4×C61C22×Q811yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C612−1+411yescharacteristic
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—81no—
255C2×Dic61—81no—
256C2×Dic61—81no—
257C2×Dic61—81no—
258C2×Dic61—161no—
259C2×Dic61—161no—
260Dic122—161no—
261Dic122—161no—
262C8×C61—22no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—22no—
266C8×C61C22⋊Q812yes—
267C8×C61C22⋊Q812yes—
268#92418—41no—
269#92418—41no—
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—42no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61C2×C22⋊Q811yescharacteristic
284C4×C61C2×C4⋊Q811yescharacteristic
285C4×C61#54711yescharacteristic
286C4×C61D4×Q811yescharacteristic
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
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#1816—611no—
309#1765D6111yescharacteristic
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×#108—611no—
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×#107—611no—
346C2×#106—611no—
347C2×#106—611no—
348C2×#106—1221no—
349#424—611no—
350#424—611no—
351C2×#104C2×D6111yescharacteristic
352#417—611no—
353#417—611no—
354#416C2×D6111yescharacteristic
355#405C2×D6111yescharacteristic
356C2×#97C2×D6111yescharacteristic
357#384—611no—
358#382—611no—
359#343—611no—
360#343—611no—
361#343—611no—
362#343—611no—
363#341C2×D6111yescharacteristic
364#339—611no—
365#339—611no—
366#339C2×D6111yescharacteristic
367C61#2608411yescharacteristic, complemented, Sylow
368C2×#41C22×D6111yescharacteristic
369#131—22no—
370#131Dic12214yescomplemented
371C2×M16—21no—
372C2×M16—611no—
373C2×M16—611no—
374C2×M16C22×D6111yescharacteristic
375C2×M16C22×D6111yescharacteristic
376C22×C8—21no—
377C22×C8C22×D6111yescharacteristic
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—1221no—
398#117—1221no—
399#117C22×D6111yescharacteristic
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—2441no—
407C2×C42C22×D6111yescharacteristic
408#108—1222no—
409#107—611no—
410#107—611no—
411#107—1221no—
412#107—1221no—
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#104Dic12214yescomplemented
438#103—1221no—
439#103—1221no—
440#103D24411yescharacteristic
441#103D24411yescharacteristic
442#102—1221no—
443#102—1221no—
444#102D24411yescharacteristic
445#102D24411yescharacteristic
446#97—21no—
447#97—22no—
448C4×C8—21no—
449#95—2441no—
450#41—21no—
451#41—21no—
452#41C2×Dic12212yes—
453#41C2×Dic12212yes—
454C2×Q8—2441no—
455C2×Q8—2441no—
456C2×Q8C2×D24411yescharacteristic
457C2×D4—21no—
458C2×D4C2×D24411yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4C23×D6111yescharacteristic
468M16—21no—
469M16—21no—
470M16—24no—
471M16—1221no—
472M16—1221no—
473M16—1221no—
474M16—1221no—
475M16C2×Dic12212yes—
476M16C2×Dic12212yes—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—24no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8C2×Dic12212yes—
485C2×C8C2×D24411yescharacteristic
486C2×C8C2×D24411yescharacteristic
487C2×C8D4×D6111yescharacteristic
488C2×C8D4×D6111yescharacteristic
489C2×C8#1175822412yes—
490C4⋊C4—611no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—611no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—4882no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4—2441no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4C2×D24411yescharacteristic
512C4⋊C4C2×D24411yescharacteristic
513C22⋊C4—21no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—2441no—
517C22⋊C4—2441no—
518C42—21no—
519C42—21no—
520C42—1221no—
521C42—1221no—
522C42—2441no—
523C42—2441no—
524C42C2×Dic12212yes—
525C23—21no—
526C23#1451982311yescharacteristic
527Q8—21no—
528Q8—9761no—
529Q8—9761no—
530Q8#1451977712yes—
531D4—22no—
532D4—22no—
533D4—21no—
534D4#1451977712yes—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—9761no—
548C2×C4—9761no—
549C2×C4—4881no—
550C2×C4—4881no—
551C2×C4—4881no—
552C2×C4—4881no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4C2×D4×D6111yescharacteristic
558C2×C4C22×D24411yescharacteristic
559C2×C4C22×Dic12211yescharacteristic
560C2×C4C22×Dic12211yescharacteristic
561C2×C4C2×#1175822411yescharacteristic
562C2×C4#1451982411yescharacteristic
563C8—22no—
564C8—21no—
565C8—21no—
566C8—22no—
567C8—2441no—
568C8—2441no—
569C8#1451975412yes—
570C8#1451975412yes—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22#1684039011yescharacteristic
577C22D4×D24411yescharacteristic
578C22#1684046911yescharacteristic
579C4—22no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—4881no—
584C4—4881no—
585C4—4881no—
586C4—4881no—
587C4—2441no—
588C4—2441no—
589C4—2441no—
590C4—2441no—
591C4C2×#1451975411yescharacteristic
592C4C2×#1451977711yescharacteristic
593C4#1684041611yescharacteristic
594C4D4×Dic12211yescharacteristic
595C2—21no—
596C2—21no—
597C2—42no—
598C2#2144952111yescharacteristic
599C2#2145758611yescharacteristic
600C2#2145759511yescharacteristic
subgroupG/H
centerC22D4×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1765C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#1765D61
hypocenterC61#26084
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1765

Complements

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

NH ≅ G/Ntype
1. #21456358C2semidirect
5. #21447331C2semidirect
6. #21447326C2semidirect
7. #21447168C2semidirect
8. #21447163C2semidirect
9. #21446599C2semidirect
10. #21446572C2semidirect
13. #21445987C2semidirect
132. #131×C61Q8semidirect
141. #104×C61Q8semidirect
367. C61#26084semidirect
370. #131Dic122semidirect
437. #104Dic122semidirect

Maximal subgroups

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

subgroupindex
#214563582
#214556792
#214556682
#214544142
#214473312
#214473262
#214471682
#214471632
#214465992
#214465722
#214460542
#214460472
#214459872
#214458832
C61×#17652
#2608461