Finite Group Databasebeta

← #29834287

Subgroups — #29834287

Order 15616.

770 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840199C211yesmaximal, characteristic, complemented
2C2×#16840196C211yesmaximal, characteristic, complemented
3#21456244C211yesmaximal, characteristic, complemented
4C4×D8×D61C211yesmaximal, characteristic, complemented
5#21447875C211yesmaximal, characteristic, complemented
6#21447874C211yesmaximal, characteristic, complemented
7#21447871C211yesmaximal, characteristic, complemented
8#21447863C211yesmaximal, characteristic, complemented
9#21447862C211yesmaximal, characteristic, complemented
10#21447853C211yesmaximal, characteristic, complemented
11#21447852C211yesmaximal, characteristic, complemented
12#21447573C211yesmaximal, characteristic, complemented
13#21447530C211yesmaximal, characteristic, complemented
14#21447528C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519798—21no—
18C2×#14519796C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840311—21no—
21C4×D4×D61—21no—
22#16840199—24no—
23#16840196—24no—
24C2×#14519722C2211yescharacteristic
25C2×#14519721C2211yescharacteristic
26C2×#14519720C2211yescharacteristic
27C2×D976C2211yescharacteristic
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30C2×#14519716C2211yescharacteristic
31C2×#14519716C2211yescharacteristic
32#16840146C2211yescharacteristic, complemented
33#16840144—21no—
34#16840140C2211yescharacteristic, complemented
35#16840137C2211yescharacteristic, complemented
36#16840133—21no—
37C4×D488C2211yescharacteristic
38D8×Dic61C2211yescharacteristic
39#16839844C2211yescharacteristic
40C4×#11758199—21no—
41#16839842—21no—
42#107×D61C2211yescharacteristic
43#102×D61—21no—
44C4×C8×D61C2211yescharacteristic
45C4⋊#11758185C2211yescharacteristic, complemented
46#16839580C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839566C2211yescharacteristic, complemented
50#16839565C2211yescharacteristic, complemented
51#16839564C2211yescharacteristic, complemented
52#16839563C2211yescharacteristic, complemented
53#16839562C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C2×D16×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×#11758224—21no—
63C2×D4×D61—21no—
64C22×C4×D61—41no—
65#14519800—22no—
66#14519800—22no—
67#14519798—44no—
68#14519796—22no—
69#14519796—22no—
70C2×Dic244C2311yescharacteristic
71C2×#11758202—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×D488C2311yescharacteristic
76C2×D488C2311yescharacteristic
77C2×#11758199—21no—
78C2×#11758199—21no—
79D8×D61—22no—
80D8×D61D414yescomplemented
81C2×C8×D61D411yescharacteristic
82C2×C8×D61C2311yescharacteristic
83#14519775—21no—
84#14519773—21no—
85#14519765—21no—
86#14519750—41no—
87#14519750—41no—
88C4×D244—21no—
89D4×Dic61—21no—
90#14519739—21no—
91C4×C22⋊D61—41no—
92#14519737—41no—
93#14519737—41no—
94#14519737—41no—
95C4⋊C4×D61—21no—
96C22⋊C4×D61—41no—
97C42×D61D411yescharacteristic
98#14519722—22no—
99#14519721—22no—
100#14519720—22no—
101D976—22no—
102#14519718—22no—
103#14519718—22no—
104#14519716—22no—
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108#14519711C2311yescharacteristic
109#14519710—21no—
110#14519699—21no—
111#14519698—21no—
112#14519697—21no—
113C4×#92418D411yescharacteristic
114C8×Dic61D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×Q16×C61C2311yescharacteristic
117C2×SD16×C61—21no—
118C2×D8×C61C2311yescharacteristic
119C2×D8×C61C2311yescharacteristic
120#124×C61—21no—
121C4×D4×C61—21no—
122SD32×C61—22no—
123D16×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#102×C61—21no—
128C4×C8×C61C2311yescharacteristic
129C23×D61—41no—
130C2×Q8×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C22×C4×C61—41no—
134C2×C22⋊D61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C22×Dic61—41no—
139C22×Dic61—41no—
140#11758226—42no—
141#11758226—42no—
142#11758224—42no—
143#11758224—42no—
144D4×D61—24no—
145D4×D61—42no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—41no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×C4×D61—41no—
153C2×C4×D61—84no—
154C2×C4×D61D811yescharacteristic
155C2×C4×D61D811yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122—21no—
158Q16×C61—21no—
159Q16×C61—21no—
160SD16×C61—42no—
161D8×C61—21no—
162D8×C61—21no—
163D8×C61—21no—
164D8×C61C2×D412yes—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2411yescharacteristic
167C4⋊C4×C61—21no—
168C22⋊C4×C61—41no—
169C22⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C42×C61C2×D411yescharacteristic
172C22⋊Dic61—41no—
173#11758202—42no—
174#11758201—42no—
175#11758200—42no—
176#11758199—24no—
177#11758199—42no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182C4⋊Dic61—21no—
183#11758196—41no—
184#11758196—41no—
185#11758196—41no—
186C4×Dic61D811yescharacteristic
187C4×Dic61D811yescharacteristic
188C4×Dic61C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190C2×#92418C2×D411yescharacteristic
191Dic244—21no—
192Dic244—21no—
193D488—21no—
194D488—21no—
195D488—21no—
196D488C2×D412yes—
197#11758190—42no—
198C8×D61—22no—
199C8×D61C2×D412yes—
200C8×D61#4112yes—
201C16×C61—21no—
202C16×C61—21no—
203#11758185—21no—
204#11758185—21no—
205C23×C61—41no—
206C23×C61—41no—
207C22×D61—41no—
208C22×D61—44no—
209C22×D61—41no—
210C22×D61—82no—
211C22×D61—82no—
212C22×D61C2×D811yescharacteristic
213Q8×C61—41no—
214Q8×C61—41no—
215D4×C61—22no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—82no—
222C2×C4×C61C2×D811yescharacteristic
223C2×C4×C61C2×D811yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C22⋊D61—44no—
226C22⋊D61—82no—
227C22⋊D61—82no—
228C22⋊D61—82no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—82no—
232C2×Dic61—82no—
233C2×Dic61C2×D811yescharacteristic
234C2×Dic61C2×D811yescharacteristic
235C2×Dic61C2×D811yescharacteristic
236D244—22no—
237D244—41no—
238D244—41no—
239D244—41no—
240D244—82no—
241C4×D61—22no—
242C4×D61—41no—
243C4×D61—82no—
244C4×D61—82no—
245C4×D61—82no—
246C4×D61—82no—
247C4×D61#12112yes—
248C4×D61AGL(1,ℤ/8)12yes—
249Dic122—41no—
250Dic122—41no—
251C8×C61—21no—
252C8×C61C22×D411yescharacteristic
253C8×C61C2×#4111yescharacteristic
254#92418—21no—
255#92418#12112yes—
256#33580—611nomaximal, Sylow
257C22×C61—42no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61C22×D811yescharacteristic
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—42no—
267C2×D61—88no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61#45912yes—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61C2×#12111yescharacteristic
279C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
280Dic61—21no—
281Dic61—81no—
282Dic61—81no—
283Dic61#45912yes—
284Dic61#50912yescomplemented
285C2×#509—611no—
286C2×#509—611no—
287#3147—611no—
288C2×C4×D8—611no—
289#2210—611no—
290#2210—611no—
291#2210—611no—
292#2210D6111yescharacteristic, complemented
293#2208—611no—
294#2208—611no—
295#2206—611no—
296#2206—611no—
297#2150—611no—
298#2139—611no—
299#2139—611no—
300C2×C61—82no—
301C2×C61—161no—
302C2×C61C2×#45911yescharacteristic
303C2×C61C2×#50911yescharacteristic
304C2×C61#341411yescharacteristic
305D61—22no—
306D61—82no—
307D61—161no—
308C2×#135—611no—
309C2×#135—611no—
310C2×#135—1221no—
311C22×D8—611no—
312#532—1221no—
313C2×C4×D4—1221no—
314#509—1224no—
315#509—1224no—
316C2×SD32—611no—
317C2×SD32—611no—
318C2×SD32—611no—
319C2×SD32C2×D6111yescharacteristic
320C2×D16—611no—
321C2×D16—611no—
322C2×D16—611no—
323C2×D16C2×D6111yescharacteristic
324C2×M32—611no—
325C2×M32—611no—
326#495—611no—
327#495—1221no—
328#495C2×D6111yescharacteristic, complemented
329#494—611no—
330#493—611no—
331#492—1221no—
332C4×D8—611no—
333C4×D8—611no—
334C4×D8—611no—
335C4×D8—1221no—
336C4×D8—1221no—
337C4×D8C2×D6111yescharacteristic
338C2×#107—611no—
339C2×#102—1221no—
340C2×C4×C8—611no—
341C4⋊C16—611no—
342C4⋊C16—611no—
343C4⋊C16—611no—
344C4⋊C16C2×D6111yescharacteristic, complemented
345#358—611no—
346#358—611no—
347#358—611no—
348#358C2×D6111yescharacteristic, complemented
349#357—611no—
350#357—611no—
351#357—611no—
352#357C2×D6111yescharacteristic, complemented
353C61#3358011yescharacteristic, complemented, Sylow
354C2×#41—1221no—
355C2×#41—1221no—
356C22×D4—1221no—
357C23×C4—2441no—
358#135—1222no—
359#135—1222no—
360#135—1222no—
361#135—1222no—
362#135—2444no—
363C2×Q16—611no—
364C2×Q16—1221no—
365C2×Q16C22×D6111yescharacteristic
366C2×SD16—21no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C2×SD16—1221no—
370C2×D8—614no—
371C2×D8—611no—
372C2×D8—611no—
373C2×D8—1222no—
374C2×D8—1221no—
375C2×D8—1221no—
376C2×D8C22×D6111yescharacteristic
377C2×D8C22×D6111yescharacteristic
378C22×C8—611no—
379C22×C8—611no—
380C4⋊Q8—1221no—
381#127—1221no—
382#124—21no—
383#124—1221no—
384#121—2441no—
385#121—2441no—
386C4×D4—21no—
387C4×D4—1221no—
388C4×D4—1221no—
389C4×D4—1221no—
390C4×D4—2441no—
391C4×D4—2441no—
392C4×D4—2441no—
393C4×D4—2441no—
394C2×C4⋊C4—1221no—
395C2×C22⋊C4—2441no—
396C2×C42—611no—
397SD32—22no—
398SD32—1222no—
399SD32—1222no—
400SD32—1222no—
401D16—22no—
402D16—1222no—
403D16—1222no—
404D16—1222no—
405M32—1222no—
406M32—1222no—
407C2×C16—611no—
408C2×C16—611no—
409C2×C16C22×D6111yescharacteristic
410C2×C16C22×D6111yescharacteristic
411#107—611no—
412#107—1221no—
413#107C22×D6111yescharacteristic
414#102—21no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8C22×D6111yescharacteristic
422C24—2441no—
423#41—2442no—
424#41—2442no—
425#41—2442no—
426#41—2442no—
427C2×Q8—21no—
428C2×Q8—1221no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—1224no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—2441no—
435C2×D4—2442no—
436C2×D4—2441no—
437C2×D4—2441no—
438C2×D4—2441no—
439C2×D4—2441no—
440C22×C4—41no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—4884no—
445C22×C4—2441no—
446C22×C4—2441no—
447C22×C4—2441no—
448C22×C4—2441no—
449C22×C4—2441no—
450C22×C4—2441no—
451Q16—21no—
452Q16—21no—
453Q16—1221no—
454Q16—1221no—
455Q16—2442no—
456SD16—42no—
457SD16—2442no—
458SD16—2442no—
459SD16—2442no—
460D8—21no—
461D8—21no—
462D8—21no—
463D8—612no—
464D8—1224no—
465D8—1221no—
466D8—1221no—
467D8—1221no—
468D8—2442no—
469D8D4×D6112yes—
470C2×C8—611no—
471C2×C8—612no—
472C2×C8—612no—
473C2×C8—611no—
474C2×C8—1222no—
475C2×C8D4×D6111yescharacteristic
476C2×C8C23×D6111yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C42—611no—
491C42—611no—
492C42—611no—
493C42D4×D6111yescharacteristic
494C16—21no—
495C16—21no—
496C16—1221no—
497C16—1221no—
498C23—41no—
499C23—41no—
500C23—611no—
501C23—4882no—
502C23—4882no—
503C23—2444no—
504C23—2441no—
505C23—2441no—
506Q8—41no—
507Q8—41no—
508Q8—2441no—
509Q8—2441no—
510D4—22no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—1222no—
515D4—4882no—
516D4—4882no—
517D4—4882no—
518D4—4882no—
519D4—2444no—
520D4—2441no—
521D4—2441no—
522D4—2441no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—82no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—611no—
531C2×C4—1222no—
532C2×C4—4882no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4—4882no—
536C2×C4—4882no—
537C2×C4—4882no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4D8×D6111yescharacteristic
542C2×C4D8×D6111yescharacteristic
543C2×C4C2×D4×D6111yescharacteristic
544C8—21no—
545C8—612no—
546C8—1221no—
547C8C2×D4×D6111yescharacteristic
548C8#41×D6111yescharacteristic
549C22—42no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—612no—
556C22—1221no—
557C22—1221no—
558C22—4888no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22—4881no—
563C22—4881no—
564C22—2442no—
565C22C2×D8×D6111yescharacteristic
566C4—21no—
567C4—21no—
568C4—81no—
569C4—81no—
570C4—612no—
571C4—612no—
572C4—1221no—
573C4—4881no—
574C4—4881no—
575C4#121×D6111yescharacteristic
576C4AGL(1,ℤ/8)×D6111yescharacteristic
577C2—82no—
578C2—161no—
579C2—1222no—
580C2—9761no—
581C2—4882no—
582C2#459×D6111yescharacteristic
583C2#509×D6111yescharacteristic
584C2#2145895211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
hypercenter#2210D61
hypocenterC61#33580
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. C2×#16840196C2semidirect
3. #21456244C2semidirect
4. C4×D8×D61C2semidirect
5. #21447875C2semidirect
6. #21447874C2semidirect
7. #21447871C2semidirect
8. #21447863C2semidirect
9. #21447862C2semidirect
10. #21447853C2semidirect
11. #21447852C2semidirect
12. #21447573C2semidirect
13. #21447530C2semidirect
14. #21447528C2semidirect
15. C61×#2210C2semidirect
32. #16840146C22semidirect
34. #16840140C22semidirect
35. #16840137C22semidirect
45. C4⋊#11758185C22semidirect
46. #16839580C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839566C22semidirect
50. #16839565C22semidirect
51. #16839564C22semidirect
52. #16839563C22semidirect
53. #16839562C22semidirect
56. C61×#495C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
80. D8×D61D4semidirect
284. Dic61#509semidirect
292. #2210D61semidirect
328. #495C2×D61semidirect
344. C4⋊C16C2×D61semidirect
348. #358C2×D61semidirect
352. #357C2×D61semidirect
353. C61#33580semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×#168401962
#214562442
C4×D8×D612
#214478752
#214478742
#214478712
#214478632
#214478622
#214478532
#214478522
#214475732
#214475302
#214475282
C61×#22102
#3358061