Finite Group Databasebeta

← #29662590

Subgroups — #29662590

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456138C211yesmaximal, characteristic, complemented
2#21455829C211yesmaximal, characteristic, complemented
3#21455818C211yesmaximal, characteristic
4#436×D61C211yesmaximal, characteristic, complemented
5#21445397C211yesmaximal, characteristic, complemented
6#21445396C211yesmaximal, characteristic, complemented
7#21445379C211yesmaximal, characteristic, complemented
8#21445378C211yesmaximal, characteristic, complemented
9#21445367C211yesmaximal, characteristic, complemented
10#21445366C211yesmaximal, characteristic, complemented
11#21445349C211yesmaximal, characteristic, complemented
12#21444566C211yesmaximal, characteristic, complemented
13#21444473C211yesmaximal, characteristic, complemented
14#21444465C211yesmaximal, characteristic, complemented
15C61×#1697C211yesmaximal, characteristic, complemented
16C2×M16×D61—21no—
17#16840317C2211yescharacteristic
18#16840310C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840115—21no—
21#16840115—21no—
22#16840114C2211yescharacteristic, complemented
23#16840113C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic, complemented
25#16840094—21no—
26#16840074—21no—
27#16840049C2211yescharacteristic
28#16840043C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16839968C2211yescharacteristic
31#16839967C2211yescharacteristic
32#16839954C2211yescharacteristic
33#16839914—21no—
34#16839894—21no—
35#16839853—21no—
36#16839849—21no—
37D8×Dic61—21no—
38#16839842—21no—
39#16839841C2211yescharacteristic
40#16839840C2211yescharacteristic
41#16839839C2211yescharacteristic
42#16839838—21no—
43#16839837—21no—
44#16839814C2211yescharacteristic
45#16839814C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839755—21no—
48#16839755—21no—
49C22⋊C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839509C2211yescharacteristic, complemented
57#16839507C2211yescharacteristic, complemented
58#16839507C2211yescharacteristic, complemented
59#16839506C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#436C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×#11758199—21no—
73C2×#11758194—21no—
74C2×#11758189—21no—
75C2×#11758189—21no—
76M16×D61—44no—
77C2×C8×D61—21no—
78C2×C8×D61D411yescharacteristic
79C2×C8×D61D411yescharacteristic
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519768—21no—
83#14519764—21no—
84#14519757—21no—
85#14519753—21no—
86#14519748—21no—
87#14519748—21no—
88Q8×Dic61—21no—
89#14519742—21no—
90C4×D244C2311yescharacteristic
91D4×Dic61—21no—
92D4×Dic61—21no—
93D4×Dic61D411yescharacteristic
94#14519739D411yescharacteristic
95C4×C22⋊D61—21no—
96#14519737—21no—
97#14519735—21no—
98C4⋊C4×D61D411yescharacteristic
99C22⋊C4×D61—21no—
100C42×D61C2311yescharacteristic
101#14519711—21no—
102#14519710—21no—
103#14519708—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706—21no—
108#14519702—21no—
109#14519702—21no—
110#14519702—21no—
111#14519700—21no—
112#14519700—21no—
113#14519700—21no—
114#14519699—21no—
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519697—21no—
120C22⋊#92418—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×D8×C61—21no—
131C2×M16×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133#127×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—41no—
154C22×Dic61—21no—
155C22×Dic61—41no—
156#11758224—42no—
157#11758224—42no—
158D4×D61—41no—
159D4×D61—41no—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61#4111yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171D8×C61—42no—
172M16×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C42×C61C2411yescharacteristic
183C22⋊Dic61—21no—
184C22⋊Dic61—41no—
185C22⋊Dic61—41no—
186#11758200—42no—
187#11758199—42no—
188#11758198—21no—
189#11758198—21no—
190#11758198—41no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—41no—
199C4×Dic61—41no—
200C4×Dic61C2×D411yescharacteristic
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61#4111yescharacteristic
203#11758194—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758190—42no—
210#11758189—42no—
211#11758189—42no—
212C8×D61—22no—
213C8×D61—22no—
214C8×D61—42no—
215C23×C61—21no—
216C23×C61—41no—
217C22×D61—21no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61#12111yescharacteristic
222Q8×C61—81no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—81no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C22⋊D61—42no—
238C22⋊D61—82no—
239C2×Dic61—21no—
240C2×Dic61—42no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—82no—
244C2×Dic61#12111yescharacteristic
245C2×Dic61#12111yescharacteristic
246C2×Dic61#12111yescharacteristic
247D244—21no—
248D244—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—41no—
252C4×D61—42no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—42no—
256C4×D61#13512yes—
257C4×D61AGL(1,ℤ/8)12yes—
258Dic122—41no—
259Dic122—41no—
260Dic122—41no—
261Dic122—41no—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61—41no—
267#92418—21no—
268#92418—21no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#19142—611nomaximal, Sylow
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61C2×#12111yescharacteristic
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—84no—
286C2×D61#46312yes—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—81no—
290C4×C61C2×#13511yescharacteristic
291C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C61#57511yescharacteristic
293C4×C61#57511yescharacteristic
294Dic61—21no—
295Dic61—21no—
296Dic61—41no—
297Dic61—81no—
298Dic61—81no—
299Dic61#46312yes—
300#3131—611no—
301#3101—611no—
302#3100—611no—
303C2×#436—611no—
304#1703—611no—
305#1703—611no—
306#1701—611no—
307#1701—611no—
308#1699—611no—
309#1699—611no—
310#1697—611no—
311#1697D6111yescharacteristic, complemented
312#1564—611no—
313#1542—611no—
314#1540—611no—
315C2×C61—81no—
316C2×C61—161no—
317C2×C61C2×#46311yescharacteristic
318C2×C61#305411yescharacteristic
319C2×C61#306711yescharacteristic
320D61—22no—
321D61—81no—
322C22×M16—1221no—
323#533—611no—
324#532—611no—
325C2×C4×D4—611no—
326#488—1221no—
327#488—1221no—
328#487—611no—
329#487—611no—
330#486—611no—
331#486C2×D6111yescharacteristic, complemented
332#483—1221no—
333#480—1221no—
334#476—611no—
335#476C2×D6111yescharacteristic
336#474—611no—
337#474—611no—
338#461—611no—
339#461—611no—
340#459—611no—
341#459C2×D6111yescharacteristic
342#450—1221no—
343#447—1221no—
344C4×SD16—1221no—
345C4×SD16—1221no—
346C4×D8—1221no—
347C4×D8—1221no—
348#436—611no—
349#436—611no—
350#436—611no—
351#436—1221no—
352#436—1221no—
353#436C2×D6111yescharacteristic
354#431—611no—
355#431—611no—
356C2×C4⋊C8—611no—
357#416—1221no—
358#416—1221no—
359C2×C22⋊C8—1221no—
360C2×C4×C8—611no—
361#335—611no—
362#335C2×D6111yescharacteristic, complemented
363#334—611no—
364#334—611no—
365#333—611no—
366#333C2×D6111yescharacteristic, complemented
367#332—611no—
368#332—611no—
369#332—611no—
370#332—611no—
371#331—611no—
372#331C2×D6111yescharacteristic, complemented
373C61#1914211yescharacteristic, complemented, Sylow
374C2×#41—1221no—
375C22×D4—611no—
376C23×C4—1221no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×D8—21no—
380C2×D8—1221no—
381C2×M16—21no—
382C2×M16—1221no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—2444no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8—1221no—
389C4⋊Q8—611no—
390C4⋊Q8—611no—
391C4⋊Q8C22×D6111yescharacteristic
392#127C22×D6111yescharacteristic
393#125—1221no—
394#124—1221no—
395C22⋊Q8—1221no—
396C22⋊Q8—1221no—
397#121—1221no—
398#121—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×D4—611no—
402C4×D4—611no—
403C4×D4—611no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4C22×D6111yescharacteristic
409#117—1221no—
410C2×C4⋊C4—611no—
411C2×C22⋊C4—1221no—
412C2×C42—611no—
413#107—1221no—
414#107—1221no—
415#106—21no—
416#106—1221no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422C4⋊C8C22×D6111yescharacteristic
423C4⋊C8C22×D6111yescharacteristic
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—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#102—1221no—
439#102—1221no—
440C22⋊C8—21no—
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—611no—
449C4×C8—1221no—
450C4×C8—1221no—
451C4×C8C22×D6111yescharacteristic
452C24—1221no—
453#41—2442no—
454#41—2442no—
455C2×Q8—21no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—611no—
462C2×D4—1221no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4—2441no—
466C2×D4D4×D6111yescharacteristic
467C22×C4—21no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4—2441no—
478SD16—2442no—
479SD16—2442no—
480D8—42no—
481D8—2442no—
482M16—42no—
483M16—2442no—
484M16—2442no—
485M16—2442no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—1222no—
492C2×C8—1222no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—2442no—
497C2×C8D4×D6111yescharacteristic
498C2×C8D4×D6111yescharacteristic
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—611no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4D4×D6111yescharacteristic
510C22⋊C4—21no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C22⋊C4—2441no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—2441no—
521C42C23×D6111yescharacteristic
522C23—21no—
523C23—41no—
524C23—611no—
525C23—1221no—
526C23—2442no—
527C23—2442no—
528C23—2442no—
529Q8—81no—
530Q8—2441no—
531Q8—2441no—
532Q8—2441no—
533Q8—2441no—
534D4—21no—
535D4—21no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—81no—
541D4—1221no—
542D4—1221no—
543D4—4882no—
544D4—2442no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—611no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—4882no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4#41×D6111yescharacteristic
569C8—21no—
570C8—21no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—1221no—
575C8—1221no—
576C8—2441no—
577C8—2441no—
578C8—2441no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—612no—
586C22—1221no—
587C22—1221no—
588C22—4884no—
589C22—2441no—
590C22—2441no—
591C22—2441no—
592C22#121×D6111yescharacteristic
593C4—21no—
594C4—41no—
595C4—81no—
596C4—612no—
597C4—1221no—
598C4—1221no—
599C4—4881no—
600C4—4881no—
601C4—2441no—
602C4#135×D6111yescharacteristic
603C4AGL(1,ℤ/8)×D6111yescharacteristic
604C4#1684055811yescharacteristic
605C4#1684056111yescharacteristic
606C2—81no—
607C2—161no—
608C2—1222no—
609C2—4881no—
610C2#463×D6111yescharacteristic
611C2#2145542311yescharacteristic
612C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1697C2
socleC22×C61C2×#121
prime radical#21455818C2
hypercenter#1697D61
hypocenterC61#19142
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1697

Complements

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

NH ≅ G/Ntype
1. #21456138C2semidirect
2. #21455829C2semidirect
4. #436×D61C2semidirect
5. #21445397C2semidirect
6. #21445396C2semidirect
7. #21445379C2semidirect
8. #21445378C2semidirect
9. #21445367C2semidirect
10. #21445366C2semidirect
11. #21445349C2semidirect
12. #21444566C2semidirect
13. #21444473C2semidirect
14. #21444465C2semidirect
15. C61×#1697C2semidirect
22. #16840114C22semidirect
23. #16840113C22semidirect
24. #16840108C22semidirect
51. #16839515C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839509C22semidirect
57. #16839507C22semidirect
58. #16839507C22semidirect
59. #16839506C22semidirect
60. C61×#486C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
66. C61×#331C22semidirect
311. #1697D61semidirect
331. #486C2×D61semidirect
362. #335C2×D61semidirect
366. #333C2×D61semidirect
372. #331C2×D61semidirect
373. C61#19142semidirect

Maximal subgroups

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

subgroupindex
#214561382
#214558292
#214558182
#436×D612
#214453972
#214453962
#214453792
#214453782
#214453672
#214453662
#214453492
#214445662
#214444732
#214444652
C61×#16972
#1914261