Finite Group Databasebeta

← #29662906

Subgroups — #29662906

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456190C211yesmaximal, characteristic
2#21455860C211yesmaximal, characteristic, complemented
3#460×D61C211yesmaximal, characteristic
4Q8×#11758189C211yesmaximal, characteristic, complemented
5#21445388C211yesmaximal, characteristic, complemented
6#21445386C211yesmaximal, characteristic, complemented
7#21445362C211yesmaximal, characteristic, complemented
8#21445356C211yesmaximal, characteristic, complemented
9#21445263C211yesmaximal, characteristic, complemented
10#21445261C211yesmaximal, characteristic, complemented
11#21445237C211yesmaximal, characteristic, complemented
12#21444568C211yesmaximal, characteristic, complemented
13#21444506C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic
15C61×#1678C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17#16840317C2211yescharacteristic
18#127×D61C2211yescharacteristic
19C4×Q8×D61C2211yescharacteristic
20#16840125—21no—
21#16840125—21no—
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840099—21no—
26#16840054C2211yescharacteristic, complemented
27#16839973C2211yescharacteristic, complemented
28#16839972C2211yescharacteristic, complemented
29#16839963C2211yescharacteristic
30#16839962C2211yescharacteristic
31#16839961—21no—
32#16839960—21no—
33#16839959C2211yescharacteristic
34#16839919—21no—
35#16839892—21no—
36#16839892C2211yescharacteristic, complemented
37#16839889—21no—
38#16839889C2211yescharacteristic, complemented
39Q8×#92418C2211yescharacteristic, complemented
40#16839856—21no—
41#16839854—21no—
42#16839814C2211yescharacteristic
43#16839784—21no—
44#16839784C2211yescharacteristic
45C4⋊C8×D61C2211yescharacteristic
46#16839765—21no—
47#16839765—21no—
48#102×D61—21no—
49C4×#11758189—21no—
50C4×#11758189C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839507C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#477C2211yescharacteristic, complemented
62C61×#460C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×Q8×D61D411yescharacteristic
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×Dic244—21no—
72C2×#11758202—21no—
73C2×#11758190—21no—
74C2×#11758200—21no—
75C2×#11758201—21no—
76SD16×D61—44no—
77C2×#11758189—21no—
78C2×#11758189D411yescharacteristic
79C2×#11758189D411yescharacteristic
80C2×C8×D61—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519773—21no—
84#14519773—21no—
85#14519773—21no—
86#14519768—21no—
87#14519757—21no—
88#14519753—21no—
89Q8×Dic61—21no—
90Q8×Dic61D411yescharacteristic
91C4×Dic122—21no—
92C4×Dic122C2311yescharacteristic
93#14519742—21no—
94#14519742—21no—
95#14519742D411yescharacteristic
96#14519735—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61D411yescharacteristic
99C42×D61—21no—
100C42×D61C2311yescharacteristic
101#14519710—21no—
102C4⋊#92418—21no—
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519702—21no—
111#14519700—21no—
112#14519700—21no—
113#14519699—21no—
114#14519699—21no—
115#14519699—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519697—21no—
120#14519697—21no—
121#14519697—21no—
122#14519689—21no—
123#14519687—21no—
124#14519687—21no—
125#14519687D411yescharacteristic
126#14519687D411yescharacteristic
127C4×#92418—21no—
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C2×SD16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134#107×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×C22⋊D61—41no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154Q8×D61—22no—
155Q8×D61—41no—
156D4×D61—42no—
157D4×D61—42no—
158D4×D61—42no—
159D4×D61—42no—
160D4×D61—82no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61#4111yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122C2×D411yescharacteristic
172SD16×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×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C42×C61—21no—
183C42×C61C2411yescharacteristic
184#11758202—42no—
185#11758201—42no—
186#11758200—42no—
187#11758198—41no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—41no—
192C4⋊Dic61C2×D411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196#11758196—41no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—41no—
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61#4111yescharacteristic
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209Dic244—42no—
210#11758190—42no—
211#11758189—22no—
212#11758189—22no—
213#11758189—42no—
214C8×D61—42no—
215C23×C61—41no—
216C22×D61—41no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61#12111yescharacteristic
221Q8×C61—21no—
222Q8×C61—81no—
223Q8×C61AGL(1,ℤ/8)12yes—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—81no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—82no—
236C22⋊D61—82no—
237C22⋊D61—82no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241C2×Dic61#12111yescharacteristic
242C2×Dic61#12111yescharacteristic
243C2×Dic61#12111yescharacteristic
244D244—41no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—81no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—41no—
252C4×D61—42no—
253C4×D61—81no—
254C4×D61#13512yes—
255C4×D61#13712yes—
256Dic122—21no—
257Dic122—21no—
258Dic122—42no—
259Dic122—41no—
260Dic122—41no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264C8×C61—41no—
265C8×C61—41no—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#19174—611nomaximal, Sylow
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61C2×#12111yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—164no—
282C2×D61#46412yes—
283C4×C61—22no—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—81no—
287C4×C61C2×#13511yescharacteristic
288C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
289C4×C61C2×#13711yescharacteristic
290C4×C61#57511yescharacteristic
291Dic61—21no—
292Dic61—21no—
293Dic61—41no—
294Dic61—41no—
295Dic61—81no—
296Dic61#46412yes—
297#3138—611no—
298#3104—611no—
299C2×#460—611no—
300Q8×M16—611no—
301#1702—611no—
302#1702—611no—
303#1698—611no—
304#1698—611no—
305#1682—611no—
306#1682—611no—
307#1678—611no—
308#1678D6111yescharacteristic, complemented
309#1565—611no—
310#1550—611no—
311#1545—611no—
312C2×C61—161no—
313C2×C61C2×#46411yescharacteristic
314C2×C61#305511yescharacteristic
315C2×C61#306811yescharacteristic
316D61—22no—
317D61—161no—
318C22×SD16—1221no—
319#533—611no—
320C2×#127—611no—
321C2×C4×Q8—611no—
322#490—1221no—
323#490—1221no—
324#486—611no—
325#486—611no—
326#486—611no—
327#486C2×D6111yescharacteristic
328#484—1221no—
329#477—611no—
330#477C2×D6111yescharacteristic, complemented
331#462—611no—
332#462—611no—
333#460—611no—
334#460—611no—
335#460—611no—
336#460—1221no—
337#460—1221no—
338#460C2×D6111yescharacteristic
339#451—1221no—
340#446—611no—
341#446—611no—
342#446—1221no—
343#446—1221no—
344C8×Q8—611no—
345C8×Q8C2×D6111yescharacteristic, complemented
346C4×Q16—1221no—
347C4×Q16—1221no—
348#431—611no—
349#423—611no—
350#423—1221no—
351C2×C4⋊C8—611no—
352#418—1221no—
353#418—1221no—
354C2×#102—1221no—
355C4×M16—611no—
356C4×M16—1221no—
357#335—611no—
358#335—611no—
359#334—611no—
360#334C2×D6111yescharacteristic, complemented
361#333—611no—
362#333C2×D6111yescharacteristic, complemented
363#332—611no—
364#332—611no—
365#331—611no—
366#331—611no—
367#331—611no—
368#331C2×D6111yescharacteristic
369C61#1917411yescharacteristic, complemented, Sylow
370C22×Q8—611no—
371C22×D4—1221no—
372C22×D4—1221no—
373C22×D4—1221no—
374C2×Q16—1221no—
375C2×Q16—1221no—
376C2×SD16—21no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—2444no—
381C2×M16—611no—
382C2×M16—611no—
383C2×M16—1221no—
384C22×C8—1221no—
385C4⋊Q8—611no—
386C4⋊Q8C22×D6111yescharacteristic
387#127—611no—
388#127—1221no—
389#127—1221no—
390#127—1221no—
391#127C22×D6111yescharacteristic
392#125—1221no—
393C22⋊Q8—1221no—
394C22⋊Q8—1221no—
395C4×Q8—611no—
396C4×Q8—611no—
397C4×Q8—611no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8C22×D6111yescharacteristic
403#117—1221no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1221no—
406C2×C42—611no—
407C2×C42—1221no—
408#107—21no—
409#107—1221no—
410C4⋊C8—21no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8C22×D6111yescharacteristic
420C4⋊C8C22×D6111yescharacteristic
421C4⋊C8C22×D6111yescharacteristic
422#103—21no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#102—21no—
427#102—21no—
428#102—21no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438C22⋊C8—1221no—
439#97—611no—
440#97—611no—
441#97—1221no—
442#97—1221no—
443C4×C8—21no—
444C4×C8—611no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8C22×D6111yescharacteristic
448C24—2441no—
449C2×Q8—21no—
450C2×Q8—611no—
451C2×Q8—1221no—
452C2×Q8—1222no—
453C2×Q8—1221no—
454C2×Q8—2441no—
455C2×Q8D4×D6111yescharacteristic
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—4882no—
463C2×D4—2442no—
464C2×D4—2442no—
465C2×D4—2441no—
466C2×D4—2441no—
467C2×D4—2441no—
468C2×D4—2442no—
469C2×D4—2442no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—1221no—
474C22×C4—1221no—
475Q16—2442no—
476Q16—2442no—
477SD16—42no—
478SD16—2442no—
479SD16—2442no—
480SD16—2442no—
481M16—1222no—
482M16—1222no—
483M16—2442no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—611no—
488C2×C8—611no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—2442no—
493C2×C8D4×D6111yescharacteristic
494C2×C8D4×D6111yescharacteristic
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—611no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—2441no—
506C4⋊C4—2441no—
507C4⋊C4D4×D6111yescharacteristic
508C22⋊C4—2441no—
509C42—21no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—1221no—
514C42—1221no—
515C42—1221no—
516C42—2441no—
517C42C23×D6111yescharacteristic
518C23—41no—
519C23—611no—
520C23—4882no—
521C23—4882no—
522C23—4882no—
523C23—2441no—
524Q8—21no—
525Q8—81no—
526Q8—1221no—
527Q8—1221no—
528Q8—2442no—
529Q8—2441no—
530Q8—2441no—
531Q8#1451980412yes—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—81no—
537D4—4881no—
538D4—4882no—
539D4—4882no—
540D4—4882no—
541D4—2441no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—611no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—4881no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4—2441no—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—1221no—
570C8—1221no—
571C8—2441no—
572C8—2441no—
573C8—2441no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—612no—
578C22—1221no—
579C22—1221no—
580C22—9764no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22#121×D6111yescharacteristic
585C4—22no—
586C4—21no—
587C4—41no—
588C4—81no—
589C4—612no—
590C4—1221no—
591C4—1221no—
592C4—4881no—
593C4—2441no—
594C4—2441no—
595C4#135×D6111yescharacteristic
596C4C2×#1451980411yescharacteristic
597C4#137×D6111yescharacteristic
598C4#1684055911yescharacteristic
599C2—161no—
600C2—1222no—
601C2—9761no—
602C2#464×D6111yescharacteristic
603C2#2145543111yescharacteristic
604C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1678C2
socleC22×C61C2×#121
prime radical#127×D61C22
hypercenter#1678D61
hypocenterC61#19174
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1678

Complements

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

NH ≅ G/Ntype
2. #21455860C2semidirect
4. Q8×#11758189C2semidirect
5. #21445388C2semidirect
6. #21445386C2semidirect
7. #21445362C2semidirect
8. #21445356C2semidirect
9. #21445263C2semidirect
10. #21445261C2semidirect
11. #21445237C2semidirect
12. #21444568C2semidirect
13. #21444506C2semidirect
15. C61×#1678C2semidirect
26. #16840054C22semidirect
27. #16839973C22semidirect
28. #16839972C22semidirect
36. #16839892C22semidirect
38. #16839889C22semidirect
39. Q8×#92418C22semidirect
51. #16839515C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839507C22semidirect
61. C61×#477C22semidirect
63. C8×Q8×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#333C22semidirect
308. #1678D61semidirect
330. #477C2×D61semidirect
345. C8×Q8C2×D61semidirect
360. #334C2×D61semidirect
362. #333C2×D61semidirect
369. C61#19174semidirect

Maximal subgroups

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

subgroupindex
#214561902
#214558602
#460×D612
Q8×#117581892
#214453882
#214453862
#214453622
#214453562
#214452632
#214452612
#214452372
#214445682
#214445062
#214444852
C61×#16782
#1917461