Finite Group Databasebeta

← #29836505

Subgroups — #29836505

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840201C211yesmaximal, characteristic, complemented
2C22×#14519721C211yesmaximal, characteristic, complemented
3#21457568C211yesmaximal, characteristic, complemented
4#21457504C211yesmaximal, characteristic
5#21448089C211yesmaximal, characteristic, complemented
6#21447931C211yesmaximal, characteristic, complemented
7#21447908C211yesmaximal, characteristic, complemented
8#21447880C211yesmaximal, characteristic, complemented
9#21447862C211yesmaximal, characteristic, complemented
10#21447760C211yesmaximal, characteristic, complemented
11#21447755C211yesmaximal, characteristic, complemented
12#21447734C211yesmaximal, characteristic, complemented
13#21447680C211yesmaximal, characteristic, complemented
14#21447641C211yesmaximal, characteristic, complemented
15C61×#2187C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C2×Q16×D61C2211yescharacteristic, complemented
19#16840425—21no—
20#16840406—21no—
21#16840201—24no—
22C2×Dic488C2211yescharacteristic, complemented
23C2×Dic488C2211yescharacteristic, complemented
24C2×#14519723C2211yescharacteristic, complemented
25C2×#14519721—24no—
26C2×#14519721C2211yescharacteristic
27C2×#14519721C2211yescharacteristic
28C2×#14519721C2211yescharacteristic
29C2×#14519721C2211yescharacteristic
30C2×#14519716C2211yescharacteristic, complemented
31C22×#11758185C2211yescharacteristic
32#16840145C2211yescharacteristic
33#16840140C2211yescharacteristic, complemented
34#16840004C2211yescharacteristic, complemented
35#16840003C2211yescharacteristic
36#16839995C2211yescharacteristic
37#16839975—21no—
38#16839971—21no—
39#16839919—21no—
40#16839917—21no—
41#16839913—21no—
42#16839904—21no—
43#16839856C2211yescharacteristic, complemented
44D8×Dic61C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839567C2211yescharacteristic, complemented
50#16839566C2211yescharacteristic, complemented
51#16839565C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839537C2211yescharacteristic, complemented
54C22×D8×C61C2211yescharacteristic
55C2×SD32×C61C2211yescharacteristic
56C2×SD32×C61C2211yescharacteristic
57C61×#467C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C61×#357C2211yescharacteristic
60C61×C22⋊C16C2211yescharacteristic
61C2×#11758224—21no—
62C22×Dic122—21no—
63C2×Q8×D61—21no—
64#14519796—22no—
65#14519796—41no—
66C2×Dic244—22no—
67C2×Dic244—21no—
68C2×Dic244—41no—
69C2×Dic244C2311yescharacteristic
70C2×Dic244C2311yescharacteristic
71C2×Dic244C2311yescharacteristic
72C2×#11758202—21no—
73C2×#11758202—21no—
74Q16×D61—22no—
75Q16×D61—41no—
76C2×#11758200—21no—
77C2×#11758200—21no—
78C2×C8×D61C2311yescharacteristic
79#14519776—41no—
80#14519775—21no—
81#14519764—21no—
82#14519758—41no—
83#14519757—21no—
84#14519756—21no—
85#14519755—41no—
86#14519754—41no—
87#14519749—41no—
88#14519748—21no—
89#14519746—41no—
90#14519742—21no—
91D4×Dic61—21no—
92C4×C22⋊D61D411yescharacteristic
93#14519737—41no—
94C2×C22⋊Dic61—41no—
95Dic488—22no—
96Dic488—41no—
97#14519723—22no—
98#14519721—22no—
99#14519721—22no—
100#14519721—22no—
101#14519721D414yescomplemented
102#14519716—22no—
103C2×#11758185—22no—
104C2×#11758185C2311yescharacteristic
105C2×#11758185C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519702—21no—
109#14519700—21no—
110#14519700—21no—
111#14519698—21no—
112#14519698—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C22×D4×C61—21no—
117C2×Q16×C61C2311yescharacteristic
118C2×D8×C61—22no—
119C2×D8×C61—21no—
120C2×D8×C61—41no—
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125SD32×C61—22no—
126SD32×C61—41no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#103×C61—21no—
131C24×C61—41no—
132C2×Q8×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—42no—
136C2×D4×C61—41no—
137C2×D4×C61—81no—
138C22×C4×C61C2×D411yescharacteristic
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61D811yescharacteristic
142C22×Dic61—41no—
143C22×Dic61—41no—
144C22×Dic61—41no—
145Q8×D61—42no—
146Q8×D61—81no—
147#11758224—42no—
148#11758224—81no—
149C2×C4×D61—41no—
150C2×C4×D61C2×D411yescharacteristic
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—42no—
155C2×Dic122—41no—
156C2×Dic122—41no—
157C2×Dic122—81no—
158Q16×C61—21no—
159Q16×C61—21no—
160D8×C61—21no—
161D8×C61—21no—
162D8×C61—21no—
163D8×C61—42no—
164D8×C61C2×D412yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C22⋊Dic61—41no—
173C22⋊Dic61—41no—
174C22⋊Dic61—41no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61D811yescharacteristic
178#11758202—42no—
179#11758202—81no—
180#11758200—42no—
181#11758200—81no—
182#11758198—41no—
183#11758198D811yescharacteristic
184C4⋊Dic61—21no—
185C4⋊Dic61—41no—
186#11758196—41no—
187#11758196—41no—
188#11758196—41no—
189#11758196—41no—
190#11758196D811yescharacteristic
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195Dic244—21no—
196Dic244—21no—
197Dic244—21no—
198Dic244—21no—
199Dic244—21no—
200Dic244—42no—
201Dic244C2×D412yes—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—82no—
211C23×C61—82no—
212C23×C61—161no—
213C23×C61C2×D811yescharacteristic
214C22×D61C2×D811yescharacteristic
215Q8×C61—41no—
216Q8×C61—41no—
217D4×C61—22no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—82no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C2×D811yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C22⋊D61—22no—
229C22⋊D61—82no—
230C22⋊D61—161no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—82no—
236C2×Dic61—82no—
237C2×Dic61—161no—
238C2×Dic61C2×D811yescharacteristic
239C2×Dic61C2×D811yescharacteristic
240C2×Dic61C2×D811yescharacteristic
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—82no—
244Dic122—22no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—82no—
251Dic122—82no—
252C8×C61—21no—
253C8×C61C22×D411yescharacteristic
254C8×C61H(5,𝔽2)11yescharacteristic
255#92418—21no—
256#92418—21no—
257#33832—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—42no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—164no—
267C22×C61#51012yescomplemented
268C22×C61C22×D811yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C4×C61—21no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61D4211yescharacteristic
276C4×C61#57611yescharacteristic
277Dic61—21no—
278Dic61—21no—
279Dic61—21no—
280Dic61—42no—
281Dic61—81no—
282Dic61—81no—
283Dic61—81no—
284C2×#510—611no—
285C22×SD32—611no—
286D4×Q16—611no—
287#3280—611no—
288#2251—611no—
289#2220—611no—
290#2216—611no—
291#2211—611no—
292#2208—611no—
293#2189—611no—
294#2188—611no—
295#2187D6111yescharacteristic, complemented
296#2185—611no—
297#2175—611no—
298#2168—611no—
299C2×C61—22no—
300C2×C61—82no—
301C2×C61—161no—
302C2×C61D4×D811yescharacteristic
303C2×C61C2×#51011yescharacteristic
304C2×C61#341611yescharacteristic
305D61—41no—
306C2×#135—611no—
307C22×Q16—611no—
308C22×Q16—611no—
309C22×D8C2×D6111yescharacteristic
310D4×Q8—1221no—
311#546—1221no—
312#510—1224no—
313C2×Q32—611no—
314C2×Q32—611no—
315C2×Q32—611no—
316C2×SD32—611no—
317C2×SD32—611no—
318C2×SD32—611no—
319C2×SD32—611no—
320C2×SD32—1224no—
321C2×SD32C2×D6111yescharacteristic
322C2×SD32C2×D6111yescharacteristic
323C2×M32—611no—
324C22×C16—611no—
325#495—611no—
326#494—611no—
327#467—611no—
328#467—611no—
329#467C2×D6111yescharacteristic
330#466—611no—
331#463—1221no—
332#462—1221no—
333#451—1221no—
334#451—1221no—
335#450—1221no—
336#449—1221no—
337C4×Q16—611no—
338C4×D8—611no—
339C8×D4—611no—
340#366—611no—
341C4⋊C16—611no—
342#358—611no—
343#358—611no—
344#358—611no—
345#358—611no—
346#357C2×D6111yescharacteristic
347#357C2×D6111yescharacteristic
348C22⋊C16—611no—
349C22⋊C16C2×D6111yescharacteristic
350#346—611no—
351C61#3383211yescharacteristic, complemented, Sylow
352C2×#41—1221no—
353C22×Q8—1221no—
354C22×Q8—1221no—
355C22×D4—21no—
356#135—1222no—
357#135—2441no—
358C2×Q16—611no—
359C2×Q16—611no—
360C2×Q16—611no—
361C2×Q16—1222no—
362C2×Q16—1221no—
363C2×Q16—1221no—
364C2×Q16—1222no—
365C2×Q16—1221no—
366C2×Q16—2441no—
367C2×Q16—2441no—
368C2×Q16C22×D6111yescharacteristic
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8—22no—
372C2×D8—21no—
373C2×D8—41no—
374C2×D8C22×D6111yescharacteristic
375C2×D8C22×D6111yescharacteristic
376C22×C8—611no—
377C22×C8C22×D6111yescharacteristic
378C4⋊Q8—1221no—
379C4⋊Q8—2441no—
380#124—1221no—
381#123—2441no—
382C22⋊Q8—21no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385C22⋊Q8—2441no—
386C22⋊Q8—2441no—
387#121—1221no—
388#121—2441no—
389#120—2441no—
390C4×Q8—1221no—
391C4×D4—611no—
392C4×D4—1221no—
393C4×D4—2441no—
394C2×C22⋊C4—2441no—
395Q32—1222no—
396Q32—1222no—
397Q32—2441no—
398SD32—22no—
399SD32—41no—
400SD32—614no—
401SD32—1222no—
402SD32—1222no—
403SD32—1222no—
404M32—1222no—
405C2×C16—611no—
406C2×C16—611no—
407C2×C16—1222no—
408C2×C16C22×D6111yescharacteristic
409C2×C16C22×D6111yescharacteristic
410#107—611no—
411#107C22×D6111yescharacteristic
412C4⋊C8—611no—
413#103—21no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#102—1221no—
418#102—1221no—
419C22⋊C8—611no—
420C22⋊C8—611no—
421C4×C8—611no—
422C24—41no—
423#41—4881no—
424#41—2442no—
425C2×Q8—21no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—4881no—
430C2×Q8—4881no—
431C2×Q8—2442no—
432C2×Q8—2441no—
433C2×Q8—2442no—
434C2×Q8—2441no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—41no—
439C2×D4—81no—
440C2×D4—611no—
441C2×D4—2441no—
442C2×D4—2441no—
443C22×C4—611no—
444C22×C4—2441no—
445C22×C4—2441no—
446C22×C4—2441no—
447C22×C4—2441no—
448C22×C4D4×D6111yescharacteristic
449Q16—21no—
450Q16—21no—
451Q16—612no—
452Q16—1221no—
453Q16—1221no—
454Q16—1221no—
455Q16—1221no—
456Q16—1221no—
457Q16—4881no—
458Q16—2442no—
459Q16—2442no—
460SD16—4881no—
461SD16—2442no—
462D8—21no—
463D8—21no—
464D8—21no—
465D8—42no—
466D8D4×D6112yes—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8D4×D6111yescharacteristic
474C2×C8C23×D6111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—611no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C22⋊C4—41no—
485C22⋊C4—611no—
486C22⋊C4—611no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C42—611no—
494C42—2441no—
495C16—21no—
496C16—21no—
497C16—612no—
498C16—1221no—
499C23—41no—
500C23—41no—
501C23—82no—
502C23—82no—
503C23—161no—
504C23—611no—
505C23D8×D6111yescharacteristic
506Q8—41no—
507Q8—41no—
508Q8—1222no—
509Q8—4882no—
510Q8—4882no—
511Q8—2441no—
512Q8—2441no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516D4—22no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—82no—
521D4—1222no—
522D4—9761no—
523D4—4882no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—9761no—
534C2×C4—4882no—
535C2×C4—4882no—
536C2×C4—4882no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4D8×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C8—21no—
544C8—1221no—
545C8—1221no—
546C8C2×D4×D6111yescharacteristic
547C8#1451982211yescharacteristic
548C22—21no—
549C22—21no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—164no—
557C22—1221no—
558C22—1221no—
559C22—1221no—
560C22#1684020112yescomplemented
561C22C2×D8×D6111yescharacteristic
562C4—21no—
563C4—81no—
564C4—81no—
565C4—1221no—
566C4—1221no—
567C4—1221no—
568C4—4881no—
569C4—4881no—
570C4—4881no—
571C4—2442no—
572C4#1684039411yescharacteristic
573C4#1684056611yescharacteristic
574C2—22no—
575C2—82no—
576C2—161no—
577C2—2441no—
578C2#2145748911yescharacteristic
579C2C2×#1684020111yescharacteristic
580C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2187C2
socleC22×C61C22×D8
prime radical#21457504C2
hypercenter#2187D61
hypocenterC61#33832
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. C2×#16840201C2semidirect
2. C22×#14519721C2semidirect
3. #21457568C2semidirect
5. #21448089C2semidirect
6. #21447931C2semidirect
7. #21447908C2semidirect
8. #21447880C2semidirect
9. #21447862C2semidirect
10. #21447760C2semidirect
11. #21447755C2semidirect
12. #21447734C2semidirect
13. #21447680C2semidirect
14. #21447641C2semidirect
15. C61×#2187C2semidirect
18. C2×Q16×D61C22semidirect
22. C2×Dic488C22semidirect
23. C2×Dic488C22semidirect
24. C2×#14519723C22semidirect
30. C2×#14519716C22semidirect
33. #16840140C22semidirect
34. #16840004C22semidirect
43. #16839856C22semidirect
46. #16839587C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839567C22semidirect
50. #16839566C22semidirect
51. #16839565C22semidirect
52. #16839541C22semidirect
53. #16839537C22semidirect
101. #14519721D4semidirect
267. C22×C61#510semidirect
295. #2187D61semidirect
351. C61#33832semidirect
560. C22#16840201semidirect

Maximal subgroups

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

subgroupindex
C2×#168402012
C22×#145197212
#214575682
#214575042
#214480892
#214479312
#214479082
#214478802
#214478622
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