Finite Group Databasebeta

← #29738937

Subgroups — #29738937

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#21456353C211yesmaximal, characteristic, complemented
2#21455680C211yesmaximal, characteristic
3#21455666C211yesmaximal, characteristic
4#21454413C211yesmaximal, characteristic
5#21447332C211yesmaximal, characteristic, complemented
6#21447328C211yesmaximal, characteristic, complemented
7#21447163C211yesmaximal, characteristic, complemented
8#21446596C211yesmaximal, characteristic, complemented
9#21446571C211yesmaximal, characteristic, complemented
10#21446051C211yesmaximal, characteristic
11#21446046C211yesmaximal, characteristic
12#21445988C211yesmaximal, characteristic, complemented
13#21445882C211yesmaximal, characteristic
14#21445660C211yesmaximal, characteristic
15C61×#2061C211yesmaximal, characteristic, complemented
16C2×#14519785C2211yescharacteristic
17#16840390C2211yescharacteristic
18#16840243C2211yescharacteristic
19#16840242C2211yescharacteristic
20#16840172—21no—
21#16840168—21no—
22#16840165—21no—
23#16840161—21no—
24#16840068—21no—
25#16840066—21no—
26#16840065—21no—
27#16840059—21no—
28#16840054—21no—
29#16840051—21no—
30#16840046—21no—
31#16840039—21no—
32C2×#14519712C2211yescharacteristic
33#16839805C2211yescharacteristic
34#16839805C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839803—21no—
37#16839802C2211yescharacteristic
38#16839801—21no—
39#16839799—21no—
40#16839798C2211yescharacteristic
41C2×#14519710C2211yescharacteristic
42C2×#14519710C2211yescharacteristic
43C2×#14519708C2211yescharacteristic
44C2×#14519708C2211yescharacteristic
45#106×D61—21no—
46#16839790C2211yescharacteristic
47#16839789C2211yescharacteristic
48C2×#14519705C2211yescharacteristic
49#16839764C2211yescharacteristic
50#16839761C2211yescharacteristic
51#16839756C2211yescharacteristic
52#16839718C2211yescharacteristic
53C2×#14519688C2211yescharacteristic
54#16839632C2211yescharacteristic
55#16839531C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839526C2211yescharacteristic
59#16839522C2211yescharacteristic
60C61×#544C2211yescharacteristic
61C61×#428C2211yescharacteristic
62C61×#428C2211yescharacteristic
63C61×#382C2211yescharacteristic
64C61×#343C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#339C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519785—22no—
69#14519785Q814yescomplemented
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758189—21no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C2×C8×D61—21no—
76#14519777—21no—
77#14519776—21no—
78#14519769—21no—
79#14519768—21no—
80#14519754—41no—
81C4×Dic122—21no—
82#14519742—21no—
83C4×D244—21no—
84#14519739—21no—
85C4×C22⋊D61—41no—
86#14519737—41no—
87#14519736C2311yescharacteristic
88#14519735—21no—
89#14519735—21no—
90C2×C4⋊Dic61—41no—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C4⋊C4×D61—21no—
94C4⋊C4×D61—21no—
95#14519712—22no—
96#14519711—21no—
97#14519710—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710D411yescharacteristic
105#14519710D411yescharacteristic
106#14519709—21no—
107#14519708—21no—
108#14519708—21no—
109#14519708—21no—
110#14519708—21no—
111#14519708—21no—
112#14519708—21no—
113#14519708—21no—
114#14519708D411yescharacteristic
115#14519708D411yescharacteristic
116C4⋊#92418—21no—
117C4⋊#92418—21no—
118#14519706—21no—
119#14519706—21no—
120#14519705Q814yescomplemented
121#14519702—21no—
122#14519700—21no—
123#14519700D411yescharacteristic
124#14519700D411yescharacteristic
125#14519699—21no—
126#14519697—21no—
127#14519697D411yescharacteristic
128#14519697D411yescharacteristic
129#14519688—21no—
130#14519688—22no—
131C4×#92418—21no—
132C2×M16×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134C4⋊Q8×C61—21no—
135C4⋊Q8×C61—21no—
136#125×C61—21no—
137#125×C61—21no—
138C22⋊Q8×C61—41no—
139#117×C61C2311yescharacteristic
140#117×C61C2311yescharacteristic
141C2×C4⋊C4×C61C2311yescharacteristic
142C2×C4⋊C4×C61C2311yescharacteristic
143C2×C42×C61C2311yescharacteristic
144#107×C61—21no—
145#107×C61—21no—
146#106×C61—21no—
147#106×C61—21no—
148#95×C61—41no—
149C2×Q8×C61—41no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157#11758222—21no—
158#11758222—21no—
159#11758222C2×Q812yes—
160#11758222C2×Q812yes—
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—41no—
164C2×C4×D61—41no—
165C2×Dic122—41no—
166C2×Dic122C2×D411yescharacteristic
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—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61C2×Q812yes—
190C22⋊Dic61—21no—
191#11758198—41no—
192#11758198—41no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—41no—
198C4⋊Dic61—82no—
199C4⋊Dic61C2×D411yescharacteristic
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—41no—
202#11758196—41no—
203#11758196—41no—
204#11758196—41no—
205C4×Dic61—21no—
206C4×Dic61—41no—
207C4×Dic61—41no—
208#11758194—21no—
209#11758194—21no—
210#11758194—21no—
211#11758194C2×Q812yes—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×Q812yes—
221C2×#92418#4112yes—
222#11758189—24no—
223C8×D61—24no—
224C23×C61H(5,𝔽2)11yescharacteristic
225C22×D61—21no—
226Q8×C61—161no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—81no—
237C2×C4×C61—81no—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×Q811yescharacteristic
241C2×C4×C61C22×Q811yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C22⋊D61—22no—
245C22⋊D61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—81no—
252C2×Dic61—81no—
253D244—21no—
254D244C4⋊Q812yes—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—161no—
258C4×D61—161no—
259Dic122—21no—
260Dic122—161no—
261Dic122C4⋊Q812yes—
262C8×C61—22no—
263C8×C61—41no—
264#92418—21no—
265#92418—21no—
266#92418—22no—
267#92418—41no—
268#92418C22⋊Q812yes—
269#92418C22⋊Q812yes—
270#26084—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61#54411yescharacteristic
274C22×C61D4211yescharacteristic
275C22×C61#55411yescharacteristic
276C2×D61—22no—
277C2×D61—41no—
278C2×D61—41no—
279C4×C61—42no—
280C4×C61—41no—
281C4×C61—41no—
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—22no—
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#2061D6111yescharacteristic, complemented
302#2060—611no—
303#1951—611no—
304#1946—611no—
305#1860—611no—
306#1859—611no—
307#1847—611no—
308#1816—611no—
309#1765—611no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61#244011yescharacteristic
313C2×C61#328811yescharacteristic
314C2×C61#328911yescharacteristic
315D61—42no—
316C2×#131—611no—
317#544—611no—
318#544C2×D6111yescharacteristic
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—1221no—
339#428—1221no—
340#428C2×D6111yescharacteristic
341#428C2×D6111yescharacteristic
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×#104—611no—
352#417—611no—
353#417—611no—
354#416—611no—
355#405—611no—
356C2×#97—611no—
357#384—611no—
358#382C2×D6111yescharacteristic
359#343—611no—
360#343—611no—
361#343—611no—
362#343C2×D6111yescharacteristic
363#341—611no—
364#339—611no—
365#339C2×D6111yescharacteristic
366#339C2×D6111yescharacteristic
367C61#2608411yescharacteristic, complemented, Sylow
368C2×#41—611no—
369#131—614no—
370#131—1222no—
371C2×M16—611no—
372C2×M16—611no—
373C2×M16—1221no—
374C2×M16C22×D6111yescharacteristic
375C2×M16C22×D6111yescharacteristic
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—1221no—
379C4⋊Q8—21no—
380C4⋊Q8—21no—
381C4⋊Q8—1221no—
382C4⋊Q8—1221no—
383#125—21no—
384#125—21no—
385#125—1221no—
386#125—1221no—
387C22⋊Q8—41no—
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—1221no—
397#117—1221no—
398#117C22×D6111yescharacteristic
399#117C22×D6111yescharacteristic
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×C42C22×D6111yescharacteristic
408#108—1222no—
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—21no—
422#106—21no—
423#106—611no—
424#106—611no—
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—611no—
439#103—611no—
440#103—1221no—
441#103—1221no—
442#102—611no—
443#102—611no—
444#102—1221no—
445#102—1221no—
446#97—1222no—
447#97—1221no—
448C4×C8—1221no—
449#95—41no—
450#41—612no—
451#41—612no—
452#41—1221no—
453#41—1221no—
454C2×Q8—41no—
455C2×Q8—611no—
456C2×Q8—2441no—
457C2×D4—611no—
458C2×D4—1221no—
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—21no—
470M16—21no—
471M16—612no—
472M16—1224no—
473M16—1221no—
474M16—1221no—
475M16—1221no—
476M16Q8×D6112yes—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—612no—
482C2×C8—612no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—1224no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—611no—
498C4⋊C4—611no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—4882no—
504C4⋊C4—2441no—
505C4⋊C4—2441no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4D4×D6111yescharacteristic
510C4⋊C4D4×D6111yescharacteristic
511C4⋊C4D4×D6111yescharacteristic
512C4⋊C4D4×D6111yescharacteristic
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—1221no—
516C22⋊C4—2441no—
517C22⋊C4—2441no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—1221no—
522C42—2441no—
523C42—2441no—
524C42Q8×D6112yes—
525C23—1221no—
526C23#1451982211yescharacteristic
527Q8—161no—
528Q8—612no—
529Q8—1221no—
530Q8—9761no—
531D4—612no—
532D4—1222no—
533D4—1222no—
534D4—1221no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—1221no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—9761no—
552C2×C4—9761no—
553C2×C4—4881no—
554C2×C4—4881no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4C2×D4×D6111yescharacteristic
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×Q8×D6111yescharacteristic
560C2×C4C2×Q8×D6111yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4#1451982511yescharacteristic
563C8—22no—
564C8—41no—
565C8—612no—
566C8—612no—
567C8—1221no—
568C8—1221no—
569C8—1222no—
570C8—2441no—
571C22—21no—
572C22—21no—
573C22—1222no—
574C22—2441no—
575C22—2441no—
576C22#1684038911yescharacteristic
577C22#1684039411yescharacteristic
578C22#1684046611yescharacteristic
579C4—42no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—1222no—
587C4—4881no—
588C4—4881no—
589C4—2441no—
590C4—2441no—
591C4C22⋊Q8×D6111yescharacteristic
592C4C4⋊Q8×D6111yescharacteristic
593C4#1684041411yescharacteristic
594C4#1684042711yescharacteristic
595C2—21no—
596C2—21no—
597C2—2442no—
598C2#2144951411yescharacteristic
599C2#2145758811yescharacteristic
600C2#2145759611yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2061C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#2061D61
hypocenterC61#26084
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
1. #21456353C2semidirect
5. #21447332C2semidirect
6. #21447328C2semidirect
7. #21447163C2semidirect
8. #21446596C2semidirect
9. #21446571C2semidirect
12. #21445988C2semidirect
15. C61×#2061C2semidirect
69. #14519785Q8semidirect
120. #14519705Q8semidirect
301. #2061D61semidirect
367. C61#26084semidirect

Maximal subgroups

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

subgroupindex
#214563532
#214556802
#214556662
#214544132
#214473322
#214473282
#214471632
#214465962
#214465712
#214460512
#214460462
#214459882
#214458822
#214456602
C61×#20612
#2608461