Finite Group Databasebeta

← #29738965

Subgroups — #29738965

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#21456347C211yesmaximal, characteristic, complemented
2#21455686C211yesmaximal, characteristic
3#21455675C211yesmaximal, characteristic
4#21454415C211yesmaximal, characteristic
5#21447330C211yesmaximal, characteristic, complemented
6#21447325C211yesmaximal, characteristic, complemented
7#21447169C211yesmaximal, characteristic, complemented
8#21447164C211yesmaximal, characteristic, complemented
9#21446602C211yesmaximal, characteristic, complemented
10#21446572C211yesmaximal, characteristic, complemented
11#21446051C211yesmaximal, characteristic
12#21446044C211yesmaximal, characteristic
13#21445879C211yesmaximal, characteristic
14#21445663C211yesmaximal, characteristic
15C61×#1847C211yesmaximal, characteristic, complemented
16C2×#14519783C2211yescharacteristic
17#16840393C2211yescharacteristic
18#16840393C2211yescharacteristic
19#16840243C2211yescharacteristic
20#16840243C2211yescharacteristic
21#16840169—21no—
22#16840169—21no—
23#16840162—21no—
24#16840159—21no—
25#16840072—21no—
26#16840068—21no—
27#16840065—21no—
28#16840061—21no—
29#16840056—21no—
30#16840051—21no—
31#16840046—21no—
32#16840041—21no—
33C2×#14519713C2211yescharacteristic
34#16839803—21no—
35#16839803—21no—
36#16839802C2211yescharacteristic
37#16839802C2211yescharacteristic
38#16839802C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839802C2211yescharacteristic
41#16839802C2211yescharacteristic
42#16839800C2211yescharacteristic
43#16839799—21no—
44#107×D61—21no—
45#16839788C2211yescharacteristic
46#16839788C2211yescharacteristic
47C2×#14519705C2211yescharacteristic
48#16839761C2211yescharacteristic
49#16839761C2211yescharacteristic
50#16839759C2211yescharacteristic
51#16839717C2211yescharacteristic
52#16839531C2211yescharacteristic
53#16839531C2211yescharacteristic
54#16839531C2211yescharacteristic
55#16839531C2211yescharacteristic
56#16839527C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839522C2211yescharacteristic
60C2×#107×C61C2211yescharacteristic
61C2×#107×C61C2211yescharacteristic
62C2×#106×C61C2211yescharacteristic
63C2×#106×C61C2211yescharacteristic
64C2×#97×C61C2211yescharacteristic
65C61×#384C2211yescharacteristic
66C61×#382C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519783—22no—
69#14519783Q814yescomplemented
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C2×#11758189—21no—
75C2×C8×D61—21no—
76#14519776—21no—
77#14519776—21no—
78#14519776—21no—
79#14519776—21no—
80#14519768—21no—
81#14519768—21no—
82#14519768—21no—
83#14519768—21no—
84#14519756—41no—
85#14519756—41no—
86#14519742—21no—
87#14519742—21no—
88#14519739—21no—
89#14519739—21no—
90C4×C22⋊D61—41no—
91C4×C22⋊D61—41no—
92#14519736C2311yescharacteristic
93#14519736C2311yescharacteristic
94#14519736C2311yescharacteristic
95#14519735—21no—
96#14519735—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99#14519713—22no—
100#14519711D411yescharacteristic
101#14519711D411yescharacteristic
102#14519710—21no—
103#14519710—21no—
104#14519710—21no—
105#14519710—21no—
106#14519710—21no—
107#14519710—21no—
108#14519709D411yescharacteristic
109#14519709D411yescharacteristic
110#14519708—21no—
111#14519708—21no—
112#14519708—21no—
113#14519708—21no—
114#14519708—21no—
115#14519708—21no—
116#14519706—21no—
117#14519706—21no—
118#14519706—21no—
119#14519706—21no—
120#14519705Q814yescomplemented
121#14519702D411yescharacteristic
122#14519702D411yescharacteristic
123#14519700—21no—
124#14519700—21no—
125#14519699D411yescharacteristic
126#14519699D411yescharacteristic
127#14519697—21no—
128#14519697—21no—
129#14519687—21no—
130C8×Dic61—21no—
131C22×C8×C61C2311yescharacteristic
132C22×C8×C61C2311yescharacteristic
133C2×C4⋊C4×C61—41no—
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C42×C61C2311yescharacteristic
139#107×C61—21no—
140#107×C61—21no—
141#107×C61—21no—
142#107×C61—21no—
143#106×C61—21no—
144#106×C61—21no—
145#106×C61—21no—
146#106×C61—21no—
147#97×C61—22no—
148#95×C61—41no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—21no—
156#11758222—21no—
157#11758222—21no—
158#11758222C2×Q812yes—
159#11758222C2×Q812yes—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—41no—
163C2×C4×D61—41no—
164C2×Dic122—41no—
165C2×Dic122—41no—
166C2×Dic122C2×D411yescharacteristic
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×Q812yes—
174C2×C8×C61#4112yes—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—82no—
184C42×C61—21no—
185C42×C61C2×Q812yes—
186C22⋊Dic61—21no—
187C22⋊Dic61—21no—
188C22⋊Dic61—21no—
189#11758198—41no—
190#11758198—41no—
191C4⋊Dic61—41no—
192C4⋊Dic61—41no—
193C4⋊Dic61C2×D411yescharacteristic
194C4⋊Dic61C2×D411yescharacteristic
195C4⋊Dic61C2×D411yescharacteristic
196C4⋊Dic61C2×D411yescharacteristic
197C4⋊Dic61C2×D411yescharacteristic
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—41no—
200#11758196—41no—
201#11758196—41no—
202#11758196—41no—
203#11758196—41no—
204#11758196—41no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—41no—
209C4×Dic61—41no—
210#11758194—21no—
211#11758194—21no—
212#11758194—21no—
213#11758194—21no—
214#11758194—21no—
215#11758194—21no—
216#11758194C2×Q812yes—
217#11758194C2×Q812yes—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222#11758189—24no—
223C8×D61—24no—
224C23×C61H(5,𝔽2)11yescharacteristic
225C22×D61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—81no—
234C2×C4×C61—81no—
235C2×C4×C61—81no—
236C2×C4×C61—81no—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×Q811yescharacteristic
240C2×C4×C61C22×Q811yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C612−1+411yescharacteristic
243C22⋊D61—22no—
244C22⋊D61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252D244—21no—
253D244C4⋊Q812yes—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—161no—
257C4×D61—161no—
258Dic122—21no—
259Dic122—161no—
260Dic122—161no—
261Dic122C4⋊Q812yes—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61C22⋊Q812yes—
265C8×C61C22⋊Q812yes—
266#92418—22no—
267#92418—22no—
268#92418—41no—
269#92418—41no—
270#26085—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—81no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61C2×C22⋊Q811yescharacteristic
285C4×C61C2×C4⋊Q811yescharacteristic
286C4×C61#54711yescharacteristic
287C4×C61D4×Q811yescharacteristic
288Dic61—22no—
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292Dic61—41no—
293Dic61—41no—
294Dic61—41no—
295#3163—611no—
296#3084—611no—
297#3083—611no—
298#2913—611no—
299#2090—611no—
300#2089—611no—
301#2061—611no—
302#2060—611no—
303#1952—611no—
304#1946—611no—
305#1860—611no—
306#1859—611no—
307#1847D6111yescharacteristic, complemented
308#1815—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#544—611no—
319#520—611no—
320#520—611no—
321#501—1221no—
322#501—1221no—
323#499—1221no—
324#498—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—1221no—
345C2×#107C2×D6111yescharacteristic
346C2×#107C2×D6111yescharacteristic
347C2×#106C2×D6111yescharacteristic
348C2×#106C2×D6111yescharacteristic
349#424—611no—
350#424—611no—
351C2×#104—611no—
352#417—611no—
353#417—611no—
354#416—611no—
355#405—611no—
356C2×#97C2×D6111yescharacteristic
357#384C2×D6111yescharacteristic
358#382C2×D6111yescharacteristic
359#343—611no—
360#343—611no—
361#343—611no—
362#343—611no—
363#341—611no—
364#339—611no—
365#339—611no—
366#339—611no—
367C61#2608511yescharacteristic, complemented, Sylow
368C2×#41—611no—
369#131—614no—
370#131—1222no—
371C2×M16—611no—
372C2×M16—611no—
373C2×M16—611no—
374C2×M16—611no—
375C2×M16—1221no—
376C22×C8—1221no—
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—611no—
398#117—1221no—
399#117—1221no—
400C2×C4⋊C4—41no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408#108—1222no—
409#107—21no—
410#107—21no—
411#107—21no—
412#107—21no—
413#107—611no—
414#107—611no—
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—21no—
424#106—21no—
425#106—611no—
426#106—611no—
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—22no—
447#97—1221no—
448C4×C8—1221no—
449#95—41no—
450#41—612no—
451#41—612no—
452#41—1221no—
453#41—1221no—
454C2×Q8—611no—
455C2×Q8—2441no—
456C2×Q8—2441no—
457C2×D4—611no—
458C2×D4—1221no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4C23×D6111yescharacteristic
468M16—612no—
469M16—612no—
470M16—1221no—
471M16—1221no—
472M16—1224no—
473M16—1221no—
474M16—1221no—
475M16—1221no—
476M16—1221no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1224no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8#1175822212yes—
485C2×C8D4×D6111yescharacteristic
486C2×C8D4×D6111yescharacteristic
487C2×C8D4×D6111yescharacteristic
488C2×C8D4×D6111yescharacteristic
489C2×C8Q8×D6112yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—82no—
499C4⋊C4—611no—
500C4⋊C4—611no—
501C4⋊C4—611no—
502C4⋊C4—611no—
503C4⋊C4—611no—
504C4⋊C4—611no—
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—21no—
519C42—1221no—
520C42—1221no—
521C42—1221no—
522C42—2441no—
523C42—2441no—
524C42Q8×D6112yes—
525C23—1221no—
526C23#1451982211yescharacteristic
527Q8—612no—
528Q8—1221no—
529Q8—9761no—
530Q8—9761no—
531D4—612no—
532D4—1222no—
533D4—1222no—
534D4—1221no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—1221no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—9761no—
556C2×C4—9761no—
557C2×C4C2×D4×D6111yescharacteristic
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×Q8×D6111yescharacteristic
560C2×C4C2×Q8×D6111yescharacteristic
561C2×C4C2×#1175822211yescharacteristic
562C2×C4#1451982511yescharacteristic
563C8—21no—
564C8—21no—
565C8—1222no—
566C8—1222no—
567C8—2441no—
568C8—2441no—
569C8#1451975512yes—
570C8#1451975512yes—
571C22—21no—
572C22—21no—
573C22—1222no—
574C22—2441no—
575C22—2441no—
576C22#1684038911yescharacteristic
577C22#1684039711yescharacteristic
578C22#1684046711yescharacteristic
579C4—42no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—1222no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4—2441no—
590C4—2441no—
591C4C2×#1451975511yescharacteristic
592C4C4⋊Q8×D6111yescharacteristic
593C4#1684041311yescharacteristic
594C4#1684042711yescharacteristic
595C2—21no—
596C2—21no—
597C2—2442no—
598C2#2144952411yescharacteristic
599C2#2145758511yescharacteristic
600C2#2145759411yescharacteristic
subgroupG/H
centerC22#16840397
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1847C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#1847D61
hypocenterC61#26085
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1847

Complements

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

NH ≅ G/Ntype
1. #21456347C2semidirect
5. #21447330C2semidirect
6. #21447325C2semidirect
7. #21447169C2semidirect
8. #21447164C2semidirect
9. #21446602C2semidirect
10. #21446572C2semidirect
15. C61×#1847C2semidirect
69. #14519783Q8semidirect
120. #14519705Q8semidirect
307. #1847D61semidirect
367. C61#26085semidirect

Maximal subgroups

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

subgroupindex
#214563472
#214556862
#214556752
#214544152
#214473302
#214473252
#214471692
#214471642
#214466022
#214465722
#214460512
#214460442
#214458792
#214456632
C61×#18472
#2608561