Finite Group Databasebeta

← #29834548

Subgroups — #29834548

Order 15616.

694 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×#16840199C211yesmaximal, characteristic, complemented
2C2×#16840180C211yesmaximal, characteristic, complemented
3#21455490C211yesmaximal, characteristic, complemented
4#21455462C211yesmaximal, characteristic, complemented
5#21447935C211yesmaximal, characteristic, complemented
6#21447922C211yesmaximal, characteristic, complemented
7#21447921C211yesmaximal, characteristic, complemented
8#21447917C211yesmaximal, characteristic, complemented
9#21447912C211yesmaximal, characteristic, complemented
10#21447906C211yesmaximal, characteristic, complemented
11#21447903C211yesmaximal, characteristic, complemented
12#21447604C211yesmaximal, characteristic, complemented
13#21447525C211yesmaximal, characteristic, complemented
14#357×D61C211yesmaximal, characteristic, complemented
15C61×#2220C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18C2×#14519783C2211yescharacteristic
19#16840342—21no—
20#16840328—21no—
21#16840199—24no—
22C2×#14519720C2211yescharacteristic
23C2×D976C2211yescharacteristic
24C2×#14519718C2211yescharacteristic
25#16840180—24no—
26C2×#14519716C2211yescharacteristic
27C2×#14519716C2211yescharacteristic, complemented
28C22×#11758185C2211yescharacteristic, complemented
29C2×C16×D61C2211yescharacteristic
30C2×C16×D61C2211yescharacteristic, complemented
31#16840014—21no—
32#16840013—21no—
33#16840011—21no—
34#16840010—21no—
35#16840000C2211yescharacteristic, complemented
36#16839999C2211yescharacteristic
37#16839999C2211yescharacteristic, complemented
38#16839998C2211yescharacteristic
39#16839998C2211yescharacteristic, complemented
40#16839995C2211yescharacteristic
41#16839799C2211yescharacteristic
42#107×D61C2211yescharacteristic
43#16839765—21no—
44#16839762—21no—
45#16839590C2211yescharacteristic, complemented
46#16839589C2211yescharacteristic, complemented
47#16839587C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic
50#16839564C2211yescharacteristic, complemented
51#16839563C2211yescharacteristic
52#16839562C2211yescharacteristic
53#16839562C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C2×M32×C61C2211yescharacteristic, complemented
56C61×#467C2211yescharacteristic, complemented
57C61×#466C2211yescharacteristic
58C61×#366C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic
61C2×#11758226—21no—
62C2×#11758222D411yescharacteristic
63C2×D4×D61—21no—
64#14519800—22no—
65#14519800—22no—
66C2×#11758201—21no—
67C2×D488C2311yescharacteristic
68C2×D488C2311yescharacteristic
69C2×#11758199—21no—
70D8×D61—22no—
71D8×D61—22no—
72#14519783—22no—
73#14519783—22no—
74C2×#11758194D411yescharacteristic
75C2×#11758189D411yescharacteristic
76C2×#11758189D411yescharacteristic
77C2×C8×D61C2311yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519773—41no—
80#14519763—41no—
81#14519755—41no—
82#14519753—21no—
83#14519752—21no—
84#14519752—21no—
85#14519752—41no—
86#14519751—21no—
87#14519751—21no—
88#14519750—41no—
89#14519750—41no—
90#14519748—21no—
91#14519747—41no—
92#14519745—41no—
93#14519735—21no—
94C4⋊C4×D61—21no—
95#14519720—22no—
96D976—22no—
97#14519718—22no—
98#14519716—22no—
99#14519716D414yescomplemented
100C2×#11758185—22no—
101C2×#11758185C2311yescharacteristic
102C2×#11758185C2311yescharacteristic
103C16×D61—22no—
104C16×D61—22no—
105#14519711C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519710—21no—
108#14519708—21no—
109#14519701—21no—
110#14519699—21no—
111#14519699—21no—
112#14519698—21no—
113#14519697—21no—
114#14519697—21no—
115C2×Q16×C61C2311yescharacteristic
116C2×D8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22⋊Q8×C61—21no—
119#121×C61—21no—
120SD32×C61—22no—
121M32×C61—22no—
122C2×C16×C61C2311yescharacteristic
123C2×C16×C61C2311yescharacteristic
124#107×C61C2311yescharacteristic
125#107×C61C2311yescharacteristic
126#103×C61—21no—
127#102×C61—21no—
128C23×D61—41no—
129C2×Q8×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—41no—
132C22×C4×C61C2×D411yescharacteristic
133C2×C22⋊D61—41no—
134C2×C22⋊D61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61D811yescharacteristic
138C2×C22⋊D61D811yescharacteristic
139#11758226—42no—
140#11758226—42no—
141D4×D61—42no—
142D4×D61—42no—
143#11758222—22no—
144#11758222—22no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—41no—
148C2×D244—41no—
149C2×D244—41no—
150C2×D244D811yescharacteristic
151C2×C4×D61—41no—
152C2×C4×D61—41no—
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×Dic122D811yescharacteristic
156Q16×C61—21no—
157Q16×C61—21no—
158D8×C61—21no—
159D8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2411yescharacteristic
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—41no—
167C22⋊C4×C61—41no—
168C22⋊C4×C61—41no—
169C22⋊Dic61—41no—
170#11758201—42no—
171#11758199—42no—
172#11758198—41no—
173#11758198—41no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177#11758198—41no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180#11758196—41no—
181#11758196—41no—
182C4×Dic61—41no—
183#11758194—22no—
184C2×#92418C2×D411yescharacteristic
185C2×#92418C2×D411yescharacteristic
186D488—21no—
187D488—21no—
188D488—21no—
189D488—21no—
190#11758189—22no—
191#11758189—22no—
192C8×D61—21no—
193C8×D61—21no—
194C8×D61C2×D412yes—
195C8×D61#4112yes—
196C16×C61—21no—
197C16×C61C2×D412yes—
198#11758185—21no—
199#11758185C2×D412yes—
200C23×C61—41no—
201C23×C61C2×D811yescharacteristic
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—82no—
205C22×D61—161no—
206C22×D61—161no—
207C22×D61C2×D811yescharacteristic
208C22×D61C2×D811yescharacteristic
209Q8×C61—41no—
210Q8×C61—41no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—161no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61C2×D811yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C22⋊D61—22no—
222C22⋊D61—22no—
223C22⋊D61—82no—
224C22⋊D61—161no—
225C22⋊D61—161no—
226C22⋊D61—161no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61C2×D811yescharacteristic
230C2×Dic61C2×D811yescharacteristic
231D244—22no—
232D244—41no—
233D244—41no—
234D244—41no—
235D244—41no—
236D244—82no—
237D244—161no—
238D244—161no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—82no—
242C4×D61—161no—
243C4×D61#12112yes—
244C4×D61#13512yes—
245Dic122—22no—
246C8×C61—21no—
247C8×C61C22×D411yescharacteristic
248C8×C61C2×#4111yescharacteristic
249#92418—21no—
250#92418#12112yes—
251#33612—611nomaximal, Sylow
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61C22×D811yescharacteristic
259C2×D61—21no—
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—164no—
271C2×D61#46612yes—
272C4×C61—21no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61C2×#12111yescharacteristic
277C4×C61C2×#13511yescharacteristic
278Dic61—21no—
279Dic61—81no—
280Dic61—81no—
281Dic61#46612yes—
282C2×#509—611no—
283C2×#504—611no—
284#3061—611no—
285#3058—611no—
286#2220—611no—
287#2220D6111yescharacteristic, complemented
288#2218—611no—
289#2218—611no—
290#2217—611no—
291#2217—611no—
292#2215—611no—
293#2215—611no—
294#2159—611no—
295#2138—611no—
296C2×#357—611no—
297C2×C61—41no—
298C2×C61—161no—
299C2×C61C2×#46611yescharacteristic
300C2×C61#341511yescharacteristic
301C2×C61#341611yescharacteristic
302D61—22no—
303D61—41no—
304D61—161no—
305D61—161no—
306C2×#135—611no—
307C22×D8—611no—
308C2×#131—611no—
309#537—1221no—
310#535—1221no—
311#509—1224no—
312C2×SD32—611no—
313C2×SD32C2×D6111yescharacteristic
314C2×D16—611no—
315C2×D16—611no—
316#504—1224no—
317C2×M32—611no—
318C2×M32—611no—
319C2×M32C2×D6111yescharacteristic, complemented
320C22×C16—611no—
321C22×C16—611no—
322C22×C16—611no—
323#469—1221no—
324#469—1221no—
325#468—1221no—
326#468—1221no—
327#467—611no—
328#467C2×D6111yescharacteristic, complemented
329#466—611no—
330#466—611no—
331#466—611no—
332#466—611no—
333#466—611no—
334#466C2×D6111yescharacteristic
335#427—611no—
336C2×#107—611no—
337#418—1221no—
338#417—1221no—
339#367—611no—
340#367—611no—
341#366—611no—
342#366C2×D6111yescharacteristic, complemented
343#358—611no—
344#358C2×D6111yescharacteristic, complemented
345#357—611no—
346#357—611no—
347#357—611no—
348#357—611no—
349#357—611no—
350#357C2×D6111yescharacteristic
351C61#3361211yescharacteristic, complemented, Sylow
352C2×#41—611no—
353C2×#41—1221no—
354C22×D4—1221no—
355#135—1222no—
356#135—1222no—
357C2×Q16C22×D6111yescharacteristic
358C2×SD16—1221no—
359C2×D8—611no—
360C2×D8—611no—
361C2×D8—1222no—
362C2×D8—1222no—
363C2×D8—1221no—
364C2×D8C22×D6111yescharacteristic
365#131—1222no—
366#131—1222no—
367C2×M16—611no—
368C2×M16—611no—
369C2×M16—611no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8C22×D6111yescharacteristic
373#127—2441no—
374#124—2441no—
375C22⋊Q8—21no—
376C22⋊Q8—1221no—
377C22⋊Q8—2441no—
378#121—21no—
379#121—1221no—
380#121—1221no—
381#121—1221no—
382#121—1221no—
383#121—1221no—
384#121—2441no—
385#121—2441no—
386#121—2441no—
387#120—2441no—
388#120—2441no—
389#117—1221no—
390C2×C4⋊C4—1221no—
391SD32—22no—
392SD32—1222no—
393D16—1222no—
394D16—1222no—
395M32—22no—
396M32—614no—
397M32—1222no—
398C2×C16—611no—
399C2×C16—611no—
400C2×C16—1222no—
401C2×C16—1222no—
402C2×C16—1222no—
403C2×C16C22×D6111yescharacteristic
404C2×C16C22×D6111yescharacteristic
405#107—611no—
406#107—611no—
407#107—1221no—
408#107C22×D6111yescharacteristic
409#107C22×D6111yescharacteristic
410#106—1221no—
411#103—21no—
412#103—1221no—
413#102—21no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419C24—2441no—
420#41—1222no—
421#41—1222no—
422#41—2442no—
423#41—2442no—
424C2×Q8—21no—
425C2×Q8—611no—
426C2×D4—21no—
427C2×D4—41no—
428C2×D4—611no—
429C2×D4—611no—
430C2×D4—611no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—2441no—
434C2×D4—2441no—
435C2×D4—2441no—
436C2×D4—2442no—
437C2×D4—2442no—
438C2×D4—2441no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2441no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—2441no—
445C22×C4—2441no—
446C22×C4D4×D6111yescharacteristic
447Q16—21no—
448Q16—21no—
449SD16—2442no—
450D8—21no—
451D8—21no—
452D8—1221no—
453D8—1221no—
454D8—1221no—
455D8—1221no—
456D8—2442no—
457M16—1222no—
458M16—1222no—
459M16—1222no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—611no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—611no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8D4×D6111yescharacteristic
469C2×C8C23×D6111yescharacteristic
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—41no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—2441no—
480C22⋊C4—2441no—
481C22⋊C4—2441no—
482C22⋊C4—2441no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C42—2441no—
487C16—21no—
488C16—612no—
489C16—1221no—
490C16D4×D6112yes—
491C23—41no—
492C23—611no—
493C23—611no—
494C23—9761no—
495C23—9761no—
496C23—4882no—
497C23—2441no—
498C23—2441no—
499C23D8×D6111yescharacteristic
500Q8—41no—
501Q8—41no—
502Q8—1222no—
503D4—41no—
504D4—41no—
505D4—161no—
506D4—1222no—
507D4—1222no—
508D4—1222no—
509D4—9761no—
510D4—9761no—
511D4—9761no—
512D4—9761no—
513D4—9761no—
514D4—4882no—
515D4—4882no—
516D4—2441no—
517D4—2441no—
518D4—2441no—
519D4—2441no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—611no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—611no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—9761no—
532C2×C4—4882no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4D8×D6111yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C8—21no—
538C8—612no—
539C8—1221no—
540C8C2×D4×D6111yescharacteristic
541C8#41×D6111yescharacteristic
542C22—21no—
543C22—21no—
544C22—21no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—612no—
549C22—1221no—
550C22—1221no—
551C22—1221no—
552C22—1221no—
553C22—1221no—
554C22—9764no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22C2×D8×D6111yescharacteristic
562C4—21no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—612no—
567C4—1221no—
568C4—4881no—
569C4—4881no—
570C4#121×D6111yescharacteristic
571C4#135×D6111yescharacteristic
572C2—41no—
573C2—161no—
574C2—1222no—
575C2—9761no—
576C2—9761no—
577C2—2441no—
578C2#466×D6111yescharacteristic
579C2#2145895911yescharacteristic
580C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2220C2
socleC22×C61C22×D8
hypercenter#2220D61
hypocenterC61#33612
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2220

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. C2×#16840180C2semidirect
3. #21455490C2semidirect
4. #21455462C2semidirect
5. #21447935C2semidirect
6. #21447922C2semidirect
7. #21447921C2semidirect
8. #21447917C2semidirect
9. #21447912C2semidirect
10. #21447906C2semidirect
11. #21447903C2semidirect
12. #21447604C2semidirect
13. #21447525C2semidirect
14. #357×D61C2semidirect
15. C61×#2220C2semidirect
27. C2×#14519716C22semidirect
28. C22×#11758185C22semidirect
30. C2×C16×D61C22semidirect
35. #16840000C22semidirect
37. #16839999C22semidirect
39. #16839998C22semidirect
45. #16839590C22semidirect
46. #16839589C22semidirect
47. #16839587C22semidirect
48. #16839566C22semidirect
50. #16839564C22semidirect
53. #16839562C22semidirect
55. C2×M32×C61C22semidirect
56. C61×#467C22semidirect
58. C61×#366C22semidirect
59. C61×#358C22semidirect
99. #14519716D4semidirect
287. #2220D61semidirect
319. C2×M32C2×D61semidirect
328. #467C2×D61semidirect
342. #366C2×D61semidirect
344. #358C2×D61semidirect
351. C61#33612semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×#168401802
#214554902
#214554622
#214479352
#214479222
#214479212
#214479172
#214479122
#214479062
#214479032
#214476042
#214475252
#357×D612
C61×#22202
#3361261