Finite Group Databasebeta

← #29834316

Subgroups — #29834316

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840204C211yesmaximal, characteristic, complemented
2C2×#16840199C211yesmaximal, characteristic
3#495×D61C211yesmaximal, characteristic, complemented
4C4×#14519800C211yesmaximal, characteristic, complemented
5#21447883C211yesmaximal, characteristic, complemented
6#21447881C211yesmaximal, characteristic, complemented
7#21447879C211yesmaximal, characteristic, complemented
8#21447877C211yesmaximal, characteristic, complemented
9#21447875C211yesmaximal, characteristic, complemented
10#21447873C211yesmaximal, characteristic, complemented
11#21447871C211yesmaximal, characteristic, complemented
12#21447573C211yesmaximal, characteristic, complemented
13#21447537C211yesmaximal, characteristic, complemented
14#21447533C211yesmaximal, characteristic, complemented
15C61×#2211C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18C2×SD16×D61—21no—
19C2×D8×D61C2211yescharacteristic
20#124×D61—21no—
21C4×#11758226—21no—
22#16840204—24no—
23#16840199—24no—
24C2×#14519723C2211yescharacteristic, complemented
25C2×#14519722C2211yescharacteristic, complemented
26C2×#14519720C2211yescharacteristic
27C2×#14519720C2211yescharacteristic, complemented
28C2×D976C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30C2×#14519716C2211yescharacteristic
31C2×#14519716C2211yescharacteristic
32#16840147C2211yescharacteristic
33#16840146C2211yescharacteristic
34#16840145C2211yescharacteristic
35#16840144—21no—
36#16840143—21no—
37Q16×Dic61C2211yescharacteristic, complemented
38C4×#11758201—21no—
39#16839847—21no—
40C4×D488C2211yescharacteristic, complemented
41#16839844C2211yescharacteristic, complemented
42#16839799C2211yescharacteristic
43#16839757—21no—
44C4×C8×D61C2211yescharacteristic
45C4⋊#11758185C2211yescharacteristic, complemented
46#16839580C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839566C2211yescharacteristic, complemented
50#16839565C2211yescharacteristic, complemented
51#16839564C2211yescharacteristic
52#16839563C2211yescharacteristic
53#16839562C2211yescharacteristic
54C2×Q32×C61C2211yescharacteristic, complemented
55C2×SD32×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic
57C4×Q16×C61C2211yescharacteristic, complemented
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic
61C2×#11758226—21no—
62C2×Q8×D61—21no—
63C2×D4×D61—21no—
64#14519800—22no—
65#14519800D414yes—
66C2×Dic244C2311yescharacteristic
67C2×#11758202—21no—
68Q16×D61—22no—
69Q16×D61—22no—
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74SD16×D61—44no—
75C2×D488C2311yescharacteristic
76C2×D488C2311yescharacteristic
77C2×#11758199—21no—
78D8×D61—22no—
79D8×D61—22no—
80C2×C8×D61D411yescharacteristic
81C2×C8×D61C2311yescharacteristic
82#14519766—21no—
83#14519765—21no—
84#14519764—21no—
85#14519763—41no—
86#14519763—41no—
87Q8×Dic61—21no—
88C4×D244—21no—
89C4×D244—41no—
90#14519739—21no—
91#14519739—41no—
92#14519735—21no—
93#14519735—41no—
94C22⋊C4×D61—41no—
95C22⋊C4×D61—41no—
96C42×D61—41no—
97C42×D61D411yescharacteristic
98#14519723—22no—
99#14519722—22no—
100#14519720—22no—
101#14519720—22no—
102D976—22no—
103#14519718—22no—
104#14519716—22no—
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108#14519711C2311yescharacteristic
109#14519708—21no—
110#14519701—21no—
111#14519699—21no—
112#14519697—21no—
113C4×#92418D411yescharacteristic
114C8×Dic61D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×Q16×C61C2311yescharacteristic
117C2×Q16×C61C2311yescharacteristic
118C2×SD16×C61—21no—
119C2×D8×C61C2311yescharacteristic
120#124×C61—21no—
121C4×Q8×C61—21no—
122Q32×C61—22no—
123SD32×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#103×C61—21no—
128C4×C8×C61C2311yescharacteristic
129C23×D61—41no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×D4×C61—21no—
133C2×C22⋊D61—41no—
134#11758226—24no—
135#11758226—42no—
136Q8×D61—42no—
137Q8×D61—42no—
138D4×D61—42no—
139D4×D61—42no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—41no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61D811yescharacteristic
148C2×C4×D61D811yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×Dic122—21no—
151C2×Dic122—41no—
152Q16×C61—21no—
153Q16×C61—21no—
154Q16×C61—21no—
155Q16×C61C2×D412yes—
156SD16×C61—42no—
157D8×C61—21no—
158D8×C61—21no—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2411yescharacteristic
161C4⋊C4×C61—21no—
162C4⋊C4×C61—41no—
163C22⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C42×C61—41no—
166C42×C61C2×D411yescharacteristic
167C22⋊Dic61—41no—
168C22⋊Dic61—41no—
169#11758202—42no—
170#11758201—24no—
171#11758201—42no—
172#11758200—42no—
173#11758199—42no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181C4⋊Dic61—21no—
182C4⋊Dic61—41no—
183C4×Dic61—41no—
184C4×Dic61—41no—
185C4×Dic61D811yescharacteristic
186C4×Dic61D811yescharacteristic
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192D488—21no—
193D488—21no—
194D488—21no—
195D488C2×D412yes—
196#11758190—42no—
197C8×D61—22no—
198C8×D61C2×D412yes—
199C8×D61#4112yes—
200C16×C61—21no—
201C16×C61—21no—
202#11758185—21no—
203#11758185—21no—
204C23×C61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—82no—
208C22×D61—82no—
209C22×D61—82no—
210C22×D61C2×D811yescharacteristic
211Q8×C61—22no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215D4×C61—41no—
216D4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61C2×D811yescharacteristic
221C2×C4×C61C2×D811yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C22⋊D61—82no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61C2×D811yescharacteristic
227C2×Dic61C2×D811yescharacteristic
228C2×Dic61C2×D811yescharacteristic
229D244—22no—
230D244—44no—
231D244—41no—
232D244—41no—
233D244—41no—
234C4×D61—41no—
235C4×D61—41no—
236C4×D61—44no—
237C4×D61—82no—
238C4×D61—82no—
239C4×D61—82no—
240C4×D61#12112yes—
241C4×D61AGL(1,ℤ/8)12yes—
242Dic122—41no—
243Dic122—41no—
244Dic122—82no—
245C8×C61—21no—
246C8×C61C22×D411yescharacteristic
247C8×C61C2×#4111yescharacteristic
248#92418—21no—
249#92418#12112yes—
250#33584—611nomaximal, Sylow
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61C22×D811yescharacteristic
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—42no—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—164no—
264C2×D61#45912yes—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—42no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61C2×#12111yescharacteristic
271C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
272Dic61—21no—
273Dic61—21no—
274Dic61—81no—
275Dic61—81no—
276Dic61#45912yes—
277C2×#510—611no—
278C2×#509—611no—
279C2×#495—611no—
280C4×#135—611no—
281#2211—611no—
282#2211—611no—
283#2211—611no—
284#2211D6111yescharacteristic, complemented
285#2210—611no—
286#2210—611no—
287#2210—611no—
288#2210—611no—
289#2150—611no—
290#2141—611no—
291#2140—611no—
292C2×C61—161no—
293C2×C61C2×#45911yescharacteristic
294C2×C61#341411yescharacteristic
295C2×C61#341411yescharacteristic
296D61—22no—
297D61—82no—
298D61—161no—
299C2×#135—611no—
300C22×Q16—611no—
301C22×SD16—1221no—
302C22×D8—611no—
303C2×#124—1221no—
304C4×#41—1221no—
305#510—1224no—
306#509—1224no—
307C2×Q32—611no—
308C2×Q32C2×D6111yescharacteristic, complemented
309C2×SD32—611no—
310C2×SD32—611no—
311C2×SD32—611no—
312C2×SD32C2×D6111yescharacteristic
313C2×D16—611no—
314C2×D16—611no—
315C2×M32—611no—
316C2×M32—611no—
317#495—611no—
318#495—611no—
319#495—611no—
320#495—1221no—
321#495—1221no—
322#495C2×D6111yescharacteristic
323C4×Q16—611no—
324C4×Q16C2×D6111yescharacteristic, complemented
325C4×SD16—1221no—
326C4×SD16—1221no—
327C4×D8—611no—
328C4×D8—611no—
329#427—611no—
330#416—1221no—
331C2×C4×C8—611no—
332C4⋊C16—611no—
333C4⋊C16—611no—
334C4⋊C16—611no—
335C4⋊C16C2×D6111yescharacteristic, complemented
336#358—611no—
337#358—611no—
338#358—611no—
339#358C2×D6111yescharacteristic, complemented
340#357—611no—
341#357—611no—
342#357—611no—
343#357C2×D6111yescharacteristic
344C61#3358411yescharacteristic, complemented, Sylow
345C2×#41—1221no—
346C22×Q8—1221no—
347C22×D4—1221no—
348#135—614no—
349#135—1222no—
350C2×Q16—611no—
351C2×Q16—1222no—
352C2×Q16—1222no—
353C2×Q16—1221no—
354C2×Q16C22×D6111yescharacteristic
355C2×Q16C22×D6111yescharacteristic
356C2×SD16—21no—
357C2×SD16—1221no—
358C2×SD16—1221no—
359C2×SD16—1221no—
360C2×SD16—1221no—
361C2×SD16—2444no—
362C2×D8—611no—
363C2×D8—611no—
364C2×D8—1222no—
365C2×D8—1222no—
366C2×D8—1221no—
367C2×D8C22×D6111yescharacteristic
368C22×C8—611no—
369C22×C8—611no—
370#124—21no—
371#124—1221no—
372#124—1221no—
373#124—1221no—
374#124—2441no—
375#124—2441no—
376C4×Q8—21no—
377C4×Q8—1221no—
378C4×D4—1221no—
379C4×D4—1221no—
380C4×D4—2441no—
381C4×D4—2441no—
382#117—1221no—
383#117—2441no—
384C2×C22⋊C4—2441no—
385C2×C22⋊C4—2441no—
386C2×C42—611no—
387C2×C42—2441no—
388Q32—22no—
389Q32—1222no—
390SD32—22no—
391SD32—1222no—
392SD32—1222no—
393SD32—1222no—
394D16—1222no—
395D16—1222no—
396M32—1222no—
397M32—1222no—
398C2×C16—611no—
399C2×C16—611no—
400C2×C16C22×D6111yescharacteristic
401C2×C16C22×D6111yescharacteristic
402#107—611no—
403#107C22×D6111yescharacteristic
404#106—1221no—
405#103—21no—
406#103—1221no—
407#102—1221no—
408#102—1221no—
409C4×C8—611no—
410C4×C8—611no—
411C4×C8—611no—
412C4×C8C22×D6111yescharacteristic
413C24—2441no—
414#41—1224no—
415#41—2442no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—1221no—
419C2×Q8—2441no—
420C2×Q8—2442no—
421C2×Q8—2442no—
422C2×D4—21no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—2441no—
426C2×D4—2442no—
427C2×D4—2442no—
428C2×D4—2441no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—2441no—
433C22×C4—2441no—
434C22×C4—2441no—
435C22×C4—2441no—
436Q16—21no—
437Q16—21no—
438Q16—21no—
439Q16—1221no—
440Q16—1221no—
441Q16—2442no—
442Q16D4×D6112yes—
443SD16—42no—
444SD16—1224no—
445SD16—2442no—
446SD16—2442no—
447SD16—2442no—
448D8—21no—
449D8—21no—
450D8—612no—
451D8—1221no—
452D8—1221no—
453D8—1221no—
454D8—2442no—
455C2×C8—611no—
456C2×C8—612no—
457C2×C8—612no—
458C2×C8—611no—
459C2×C8—1222no—
460C2×C8D4×D6111yescharacteristic
461C2×C8C23×D6111yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—1221no—
465C4⋊C4—2441no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—2441no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C22⋊C4—2441no—
475C22⋊C4—2441no—
476C22⋊C4—2441no—
477C42—41no—
478C42—611no—
479C42—611no—
480C42—611no—
481C42—2441no—
482C42—2441no—
483C42D4×D6111yescharacteristic
484C16—21no—
485C16—21no—
486C16—1221no—
487C16—1221no—
488C23—41no—
489C23—611no—
490C23—4882no—
491C23—4882no—
492C23—4882no—
493C23—2441no—
494C23—2441no—
495Q8—22no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—4882no—
500Q8—2441no—
501Q8—2441no—
502D4—41no—
503D4—41no—
504D4—1222no—
505D4—4882no—
506D4—2444no—
507D4—2441no—
508D4—2441no—
509D4—2441no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—611no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—611no—
517C2×C4—611no—
518C2×C4—4882no—
519C2×C4—4882no—
520C2×C4—4882no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2444no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4D8×D6111yescharacteristic
527C2×C4D8×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C8—21no—
530C8—612no—
531C8—1221no—
532C8C2×D4×D6111yescharacteristic
533C8#41×D6111yescharacteristic
534C22—81no—
535C22—81no—
536C22—81no—
537C22—612no—
538C22—1221no—
539C22—1221no—
540C22—9764no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22—4881no—
545C22—4881no—
546C22—2442no—
547C22C2×D8×D6111yescharacteristic
548C4—21no—
549C4—21no—
550C4—42no—
551C4—81no—
552C4—81no—
553C4—612no—
554C4—1221no—
555C4—1221no—
556C4—4881no—
557C4—4881no—
558C4#121×D6111yescharacteristic
559C4AGL(1,ℤ/8)×D6111yescharacteristic
560C2—161no—
561C2—1222no—
562C2—9761no—
563C2—4882no—
564C2#459×D6111yescharacteristic
565C2#2145895211yescharacteristic
566C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2211C2
socleC22×C61C22×D8
prime radicalC2×#16840199C2
hypercenter#2211D61
hypocenterC61#33584
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2211

Complements

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

NH ≅ G/Ntype
1. C2×#16840204C2semidirect
3. #495×D61C2semidirect
4. C4×#14519800C2semidirect
5. #21447883C2semidirect
6. #21447881C2semidirect
7. #21447879C2semidirect
8. #21447877C2semidirect
9. #21447875C2semidirect
10. #21447873C2semidirect
11. #21447871C2semidirect
12. #21447573C2semidirect
13. #21447537C2semidirect
14. #21447533C2semidirect
15. C61×#2211C2semidirect
24. C2×#14519723C22semidirect
25. C2×#14519722C22semidirect
27. C2×#14519720C22semidirect
37. Q16×Dic61C22semidirect
40. C4×D488C22semidirect
41. #16839844C22semidirect
45. C4⋊#11758185C22semidirect
46. #16839580C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839566C22semidirect
50. #16839565C22semidirect
54. C2×Q32×C61C22semidirect
57. C4×Q16×C61C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
284. #2211D61semidirect
308. C2×Q32C2×D61semidirect
324. C4×Q16C2×D61semidirect
335. C4⋊C16C2×D61semidirect
339. #358C2×D61semidirect
344. C61#33584semidirect

Maximal subgroups

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

subgroupindex
C2×#168402042
C2×#168401992
#495×D612
C4×#145198002
#214478832
#214478812
#214478792
#214478772
#214478752
#214478732
#214478712
#214475732
#214475372
#214475332
C61×#22112
#3358461