Finite Group Databasebeta

← #29834498

Subgroups — #29834498

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840201C211yesmaximal, characteristic, complemented
2C22×#14519716C211yesmaximal, characteristic, complemented
3#21455398C211yesmaximal, characteristic
4#467×D61C211yesmaximal, characteristic, complemented
5#21447931C211yesmaximal, characteristic, complemented
6#21447929C211yesmaximal, characteristic, complemented
7#21447928C211yesmaximal, characteristic, complemented
8#21447926C211yesmaximal, characteristic, complemented
9#21447913C211yesmaximal, characteristic, complemented
10#21447910C211yesmaximal, characteristic, complemented
11#21447908C211yesmaximal, characteristic, complemented
12#21447606C211yesmaximal, characteristic, complemented
13#21447537C211yesmaximal, characteristic, complemented
14#21447528C211yesmaximal, characteristic, complemented
15C61×#2217C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19#16840272—21no—
20C22⋊Q8×D61—21no—
21#16840201—24no—
22C2×Dic488C2211yescharacteristic, complemented
23C2×#14519723C2211yescharacteristic, complemented
24C2×#14519721C2211yescharacteristic, complemented
25C2×#14519716—24no—
26C2×#14519716C2211yescharacteristic
27C2×#14519716C2211yescharacteristic
28C2×#14519716C2211yescharacteristic
29C2×#14519716C2211yescharacteristic
30C22×#11758185C2211yescharacteristic
31#16840004C2211yescharacteristic
32#16840004C2211yescharacteristic
33#16840003C2211yescharacteristic
34#16840003C2211yescharacteristic
35#16840002—21no—
36#16840001—21no—
37#16840000C2211yescharacteristic
38#16839995C2211yescharacteristic
39#16839990—21no—
40#16839989—21no—
41#16839799C2211yescharacteristic
42#107×D61C2211yescharacteristic
43#16839755—21no—
44#103×D61—21no—
45#16839589C2211yescharacteristic, complemented
46#16839588C2211yescharacteristic, complemented
47#16839587C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839567C2211yescharacteristic, complemented
50#16839566C2211yescharacteristic, complemented
51#16839565C2211yescharacteristic, complemented
52#16839565C2211yescharacteristic, complemented
53#16839563C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic, complemented
55C22×C16×C61C2211yescharacteristic
56C61×#467C2211yescharacteristic
57C61×#466C2211yescharacteristic
58C61×#367C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×Q8×D61—21no—
63C22×C4×D61D411yescharacteristic
64#14519796—22no—
65#14519796—22no—
66C2×Dic244C2311yescharacteristic
67C2×Dic244C2311yescharacteristic
68C2×#11758202—21no—
69Q16×D61—22no—
70Q16×D61—22no—
71C2×#11758200—21no—
72C22×#92418D411yescharacteristic
73C2×C8×D61—22no—
74C2×C8×D61—22no—
75C2×C8×D61D411yescharacteristic
76C2×C8×D61D411yescharacteristic
77C2×C8×D61C2311yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519759—41no—
80#14519757—21no—
81#14519757—21no—
82#14519756—21no—
83#14519756—21no—
84#14519755—41no—
85#14519754—41no—
86#14519753—21no—
87#14519748—21no—
88#14519745—41no—
89D4×Dic61—41no—
90#14519737—41no—
91#14519735—21no—
92C4⋊C4×D61—21no—
93C4⋊C4×D61—41no—
94C22⋊C4×D61—41no—
95Dic488—22no—
96#14519723—22no—
97#14519721—22no—
98#14519716—22no—
99#14519716—22no—
100#14519716—22no—
101#14519716D414yescomplemented
102C2×#11758185—22no—
103C2×#11758185C2311yescharacteristic
104C2×#11758185C2311yescharacteristic
105#14519711C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519710—21no—
108#14519708—21no—
109#14519702—21no—
110#14519702—21no—
111#14519701—21no—
112#14519700—21no—
113#14519700—21no—
114#14519698—21no—
115C2×Q16×C61C2311yescharacteristic
116C2×D8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22⋊Q8×C61—21no—
119#121×C61—21no—
120SD32×C61—22no—
121C2×C16×C61—22no—
122C2×C16×C61C2311yescharacteristic
123C2×C16×C61C2311yescharacteristic
124#107×C61C2311yescharacteristic
125#107×C61C2311yescharacteristic
126#103×C61—21no—
127#102×C61—21no—
128C23×D61D811yescharacteristic
129C2×Q8×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—41no—
132C22×C4×C61C2×D411yescharacteristic
133C2×C22⋊D61—41no—
134C2×C22⋊D61—41no—
135C22×Dic61—41no—
136C22×Dic61D811yescharacteristic
137Q8×D61—42no—
138Q8×D61—42no—
139#11758224—42no—
140#11758224—42no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C4×D61D811yescharacteristic
146C2×C4×D61D811yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—41no—
152Q16×C61—21no—
153Q16×C61—21no—
154D8×C61—21no—
155D8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2411yescharacteristic
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—41no—
163C22⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C22⋊Dic61—41no—
166C22⋊Dic61—41no—
167C22⋊Dic61—41no—
168#11758202—42no—
169#11758200—42no—
170#11758198—41no—
171#11758198—41no—
172#11758198—41no—
173#11758198—41no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—41no—
177C4⋊Dic61—41no—
178#11758196—41no—
179#11758196—41no—
180#11758196—41no—
181#11758196—41no—
182C4×Dic61—41no—
183C2×#92418—22no—
184C2×#92418C2×D411yescharacteristic
185C2×#92418C2×D411yescharacteristic
186Dic244—21no—
187Dic244—21no—
188Dic244—21no—
189Dic244—21no—
190C8×D61—22no—
191C8×D61—21no—
192C8×D61—21no—
193C8×D61—22no—
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—22no—
203C22×D61—22no—
204C22×D61—41no—
205C22×D61C2×D811yescharacteristic
206C22×D61C2×D811yescharacteristic
207Q8×C61—41no—
208Q8×C61—41no—
209D4×C61—41no—
210D4×C61—41no—
211D4×C61—82no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61C2×D811yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C22⋊D61—82no—
220C22⋊D61—82no—
221C2×Dic61—22no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—82no—
227C2×Dic61C2×D811yescharacteristic
228C2×Dic61C2×D811yescharacteristic
229C4×D61—22no—
230C4×D61—21no—
231C4×D61—21no—
232C4×D61—22no—
233C4×D61—82no—
234C4×D61—82no—
235C4×D61#12112yes—
236C4×D61#13512yes—
237Dic122—41no—
238Dic122—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—82no—
242C8×C61—21no—
243C8×C61C22×D411yescharacteristic
244C8×C61C2×#4111yescharacteristic
245#92418—21no—
246#92418#12112yes—
247#33606—611nomaximal, Sylow
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61#51012yescomplemented
254C22×C61C22×D811yescharacteristic
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—44no—
261C2×D61#46612yes—
262C4×C61—21no—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61C2×#12111yescharacteristic
267C4×C61C2×#13511yescharacteristic
268Dic61—21no—
269Dic61—81no—
270Dic61—81no—
271Dic61—81no—
272Dic61—81no—
273Dic61#46612yes—
274C2×#510—611no—
275C22×M32—611no—
276#3050—611no—
277C2×#467—611no—
278#2220—611no—
279#2220—611no—
280#2219—611no—
281#2219—611no—
282#2217—611no—
283#2217D6111yescharacteristic, complemented
284#2216—611no—
285#2216—611no—
286#2159—611no—
287#2141—611no—
288#2139—611no—
289C2×C61—22no—
290C2×C61—161no—
291C2×C61C2×#46611yescharacteristic
292C2×C61C2×#51011yescharacteristic
293C2×C61#341411yescharacteristic
294D61—22no—
295D61—41no—
296C2×#135—611no—
297C22×Q16—611no—
298C23×C8—611no—
299#525—1221no—
300C2×C22⋊Q8—1221no—
301#510—1224no—
302C2×Q32—611no—
303C2×Q32—611no—
304C2×SD32—611no—
305C2×SD32C2×D6111yescharacteristic, complemented
306C2×M32—611no—
307C2×M32—611no—
308C2×M32—611no—
309C2×M32—611no—
310C2×M32—1224no—
311C22×C16—611no—
312C22×C16C2×D6111yescharacteristic
313#467—611no—
314#467—611no—
315#467—611no—
316#467—611no—
317#467—611no—
318#467—1221no—
319#467—1221no—
320#467C2×D6111yescharacteristic
321#466—611no—
322#466C2×D6111yescharacteristic
323#465—1221no—
324#465—1221no—
325#427—611no—
326C2×#107—611no—
327#416—1221no—
328C2×#103—1221no—
329#367—611no—
330#367C2×D6111yescharacteristic, complemented
331#366—611no—
332#366—611no—
333#358—611no—
334#358—611no—
335#358—611no—
336#358—611no—
337#358—611no—
338#358C2×D6111yescharacteristic, complemented
339#357—611no—
340#357C2×D6111yescharacteristic, complemented
341C61#3360611yescharacteristic, complemented, Sylow
342C2×#41—1221no—
343C22×Q8—1221no—
344C23×C4—611no—
345#135—1222no—
346#135—1222no—
347C2×Q16—611no—
348C2×Q16—611no—
349C2×Q16—1222no—
350C2×Q16—1222no—
351C2×Q16—1221no—
352C2×Q16C22×D6111yescharacteristic
353C2×SD16—1221no—
354C2×D8C22×D6111yescharacteristic
355C22×C8—611no—
356C22×C8—611no—
357C22×C8—611no—
358C22×C8—611no—
359C22×C8—611no—
360C22×C8—1222no—
361C22×C8—1222no—
362C22×C8C22×D6111yescharacteristic
363#123—2441no—
364C22⋊Q8—21no—
365C22⋊Q8—1221no—
366C22⋊Q8—1221no—
367C22⋊Q8—1221no—
368C22⋊Q8—1221no—
369C22⋊Q8—1221no—
370C22⋊Q8—2441no—
371C22⋊Q8—2441no—
372#121—21no—
373#121—1221no—
374#120—2441no—
375C4×D4—2441no—
376C4×D4—2441no—
377#117—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—2441no—
380C2×C22⋊C4—2441no—
381Q32—1222no—
382Q32—1222no—
383SD32—22no—
384SD32—1222no—
385M32—614no—
386M32—1222no—
387M32—1222no—
388M32—1222no—
389C2×C16—22no—
390C2×C16—611no—
391C2×C16—611no—
392C2×C16—1222no—
393C2×C16C22×D6111yescharacteristic
394C2×C16C22×D6111yescharacteristic
395#107—611no—
396#107—611no—
397#107—1221no—
398#107C22×D6111yescharacteristic
399#107C22×D6111yescharacteristic
400#106—1221no—
401#103—21no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#102—21no—
408#102—1221no—
409C24—611no—
410#41—2442no—
411#41—2442no—
412C2×Q8—21no—
413C2×Q8—1221no—
414C2×Q8—1221no—
415C2×Q8—2441no—
416C2×Q8—2442no—
417C2×Q8—2442no—
418C2×D4—21no—
419C2×D4—41no—
420C2×D4—2441no—
421C2×D4—2441no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—2441no—
430C22×C4—2441no—
431C22×C4—2441no—
432C22×C4D4×D6111yescharacteristic
433Q16—21no—
434Q16—21no—
435Q16—1221no—
436Q16—1221no—
437Q16—1221no—
438Q16—1221no—
439Q16—2442no—
440SD16—2442no—
441D8—21no—
442D8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—611no—
446C2×C8—612no—
447C2×C8—612no—
448C2×C8—611no—
449C2×C8—1222no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8D4×D6111yescharacteristic
455C2×C8C23×D6111yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C22⋊C4—2441no—
475C22⋊C4—2441no—
476C42—2441no—
477C16—21no—
478C16—612no—
479C16—1221no—
480C16D4×D6112yes—
481C23—41no—
482C23—611no—
483C23—611no—
484C23—1222no—
485C23—1222no—
486C23—2441no—
487C23D8×D6111yescharacteristic
488Q8—41no—
489Q8—41no—
490Q8—4882no—
491Q8—2441no—
492Q8—2441no—
493Q8—2441no—
494Q8—2441no—
495D4—41no—
496D4—41no—
497D4—82no—
498D4—4882no—
499D4—4882no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—611no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—611no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—4882no—
515C2×C4—4882no—
516C2×C4—4882no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4D8×D6111yescharacteristic
522C2×C4C2×D4×D6111yescharacteristic
523C8—21no—
524C8—612no—
525C8—1221no—
526C8C2×D4×D6111yescharacteristic
527C8#41×D6111yescharacteristic
528C22—21no—
529C22—21no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—612no—
534C22—1221no—
535C22—1221no—
536C22—1221no—
537C22—1221no—
538C22—1221no—
539C22—2444no—
540C22#1684020112yescomplemented
541C22C2×D8×D6111yescharacteristic
542C4—21no—
543C4—81no—
544C4—81no—
545C4—81no—
546C4—612no—
547C4—1221no—
548C4—4881no—
549C4—4881no—
550C4—4881no—
551C4—4881no—
552C4#121×D6111yescharacteristic
553C4#135×D6111yescharacteristic
554C2—22no—
555C2—161no—
556C2—1222no—
557C2—2441no—
558C2#466×D6111yescharacteristic
559C2C2×#1684020111yescharacteristic
560C2#2145895211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2217C2
socleC22×C61C22×D8
prime radical#21455398C2
hypercenter#2217D61
hypocenterC61#33606
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. C2×#16840201C2semidirect
2. C22×#14519716C2semidirect
4. #467×D61C2semidirect
5. #21447931C2semidirect
6. #21447929C2semidirect
7. #21447928C2semidirect
8. #21447926C2semidirect
9. #21447913C2semidirect
10. #21447910C2semidirect
11. #21447908C2semidirect
12. #21447606C2semidirect
13. #21447537C2semidirect
14. #21447528C2semidirect
15. C61×#2217C2semidirect
22. C2×Dic488C22semidirect
23. C2×#14519723C22semidirect
24. C2×#14519721C22semidirect
45. #16839589C22semidirect
46. #16839588C22semidirect
47. #16839587C22semidirect
48. #16839567C22semidirect
49. #16839567C22semidirect
50. #16839566C22semidirect
51. #16839565C22semidirect
52. #16839565C22semidirect
53. #16839563C22semidirect
54. C2×SD32×C61C22semidirect
58. C61×#367C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
101. #14519716D4semidirect
253. C22×C61#510semidirect
283. #2217D61semidirect
305. C2×SD32C2×D61semidirect
330. #367C2×D61semidirect
338. #358C2×D61semidirect
340. #357C2×D61semidirect
341. C61#33606semidirect
540. 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×#145197162
#214553982
#467×D612
#214479312
#214479292
#214479282
#214479262
#214479132
#214479102
#214479082
#214476062
#214475372
#214475282
C61×#22172
#3360661