Finite Group Databasebeta

← #29836507

Subgroups — #29836507

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
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21457569C211yesmaximal, characteristic, complemented
4#21457505C211yesmaximal, characteristic
5#21448088C211yesmaximal, characteristic, complemented
6#21447935C211yesmaximal, characteristic, complemented
7#21447909C211yesmaximal, characteristic, complemented
8#21447882C211yesmaximal, characteristic, complemented
9#21447756C211yesmaximal, characteristic, complemented
10#21447754C211yesmaximal, characteristic, complemented
11#21447746C211yesmaximal, characteristic, complemented
12#21447737C211yesmaximal, characteristic, complemented
13#21447683C211yesmaximal, characteristic, complemented
14#21447643C211yesmaximal, characteristic, complemented
15C61×#2208C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18C2×Q16×D61C2211yescharacteristic, complemented
19C2×D8×D61C2211yescharacteristic
20#16840427—21no—
21#16840407—21no—
22#16840201—24no—
23C2×Dic488C2211yescharacteristic, complemented
24C2×#14519723C2211yescharacteristic, complemented
25C2×#14519723C2211yescharacteristic, complemented
26C2×#14519722C2211yescharacteristic
27C2×#14519721C2211yescharacteristic
28C2×#14519720C2211yescharacteristic
29C2×#14519720C2211yescharacteristic
30SD32×D61—24no—
31C2×#14519716C2211yescharacteristic, complemented
32C2×C16×D61C2211yescharacteristic
33#16840146C2211yescharacteristic
34#16840004C2211yescharacteristic
35#16840000C2211yescharacteristic
36#16840000C2211yescharacteristic, complemented
37#16839999C2211yescharacteristic
38#16839977—21no—
39#16839969—21no—
40#16839916—21no—
41#16839916—21no—
42#16839914—21no—
43#16839908—21no—
44C4×Dic244C2211yescharacteristic, complemented
45C8×D244C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839567C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic
50#16839563C2211yescharacteristic
51#16839541C2211yescharacteristic
52#16839541C2211yescharacteristic, complemented
53#16839538C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C2×SD32×C61C2211yescharacteristic
56C61×#494C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#358C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×Q8×D61—21no—
63C2×Q8×D61—21no—
64C2×D4×D61—21no—
65#14519796—22no—
66#14519796—41no—
67C2×Dic244C2311yescharacteristic
68C2×Dic244C2311yescharacteristic
69C2×#11758202—21no—
70C2×#11758202—21no—
71Q16×D61—22no—
72Q16×D61—22no—
73Q16×D61—41no—
74Q16×D61—41no—
75C2×#11758190—21no—
76C2×#11758200—21no—
77C2×D488C2311yescharacteristic
78C2×#11758199—21no—
79D8×D61—22no—
80D8×D61—41no—
81C2×C8×D61C2311yescharacteristic
82C2×C8×D61C2311yescharacteristic
83#14519776—41no—
84#14519765—21no—
85#14519761—41no—
86#14519757—21no—
87#14519755—41no—
88#14519755—41no—
89#14519753—21no—
90#14519753—21no—
91#14519752—21no—
92#14519751—41no—
93#14519747—41no—
94C4×Dic122—21no—
95C4×D244D411yescharacteristic
96#14519739—41no—
97C2×#11758198—41no—
98Dic488—22no—
99#14519723—22no—
100#14519723—41no—
101#14519722—22no—
102#14519721D414yescomplemented
103#14519720—22no—
104#14519720—41no—
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519702—21no—
112#14519701—21no—
113#14519701—21no—
114#14519700—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×Q16×C61—21no—
119C2×Q16×C61C2311yescharacteristic
120C2×Q16×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122C4⋊Q8×C61—21no—
123C4×D4×C61—21no—
124SD32×C61—22no—
125SD32×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C22×C4×C61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C22×Dic61—41no—
139Q8×D61—42no—
140Q8×D61—42no—
141Q8×D61—81no—
142Q8×D61—81no—
143#11758224—42no—
144#11758224—81no—
145D4×D61—42no—
146D4×D61—81no—
147C2×D244—21no—
148C2×D244—41no—
149C2×D244D811yescharacteristic
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—41no—
157C2×Dic122—41no—
158Q16×C61—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61—81no—
163D8×C61—21no—
164D8×C61C2×D412yes—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2411yescharacteristic
167C4⋊C4×C61—21no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C42×C61C2×D411yescharacteristic
172#11758202—42no—
173#11758202—42no—
174#11758200—42no—
175#11758199—42no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198D811yescharacteristic
183#11758198D811yescharacteristic
184C4⋊Dic61—21no—
185C4⋊Dic61—41no—
186C4⋊Dic61D811yescharacteristic
187#11758196—41no—
188#11758196—41no—
189#11758196—41no—
190C4×Dic61—41no—
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193Dic244—21no—
194Dic244—21no—
195Dic244—21no—
196Dic244C2×D412yes—
197D488—21no—
198D488—21no—
199#11758190—81no—
200C8×D61—21no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C22×D61—41no—
210C22×D61—82no—
211C22×D61—82no—
212C22×D61—161no—
213C22×D61C2×D811yescharacteristic
214C22×D61C2×D811yescharacteristic
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219D4×C61—22no—
220D4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—161no—
225C2×C4×C61C2×D811yescharacteristic
226C2×C4×C61C2×D811yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C22⋊D61—82no—
229C22⋊D61—82no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—82no—
234C2×Dic61C2×D811yescharacteristic
235C2×Dic61C2×D811yescharacteristic
236D244—22no—
237D244—41no—
238D244—41no—
239D244—161no—
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—82no—
245C4×D61—82no—
246Dic122—22no—
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—42no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61C22×D811yescharacteristic
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—164no—
271C2×D61#51012yescomplemented
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61D4211yescharacteristic
278C4×C61#57611yescharacteristic
279Dic61—21no—
280Dic61—21no—
281Dic61—42no—
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#2208D6111yescharacteristic, complemented
293#2189—611no—
294#2188—611no—
295#2187—611no—
296#2185—611no—
297#2175—611no—
298#2168—611no—
299C2×C61—82no—
300C2×C61D4×D811yescharacteristic
301C2×C61C2×#51011yescharacteristic
302C2×C61#341611yescharacteristic
303D61—22no—
304D61—41no—
305D61—161no—
306C2×#135—611no—
307C22×Q16—611no—
308C22×Q16—611no—
309C22×D8—611no—
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#494C2×D6111yescharacteristic, complemented
327#467—611no—
328#467—611no—
329#467—611no—
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×D8C2×D6111yescharacteristic
339C8×D4—611no—
340#366—611no—
341C4⋊C16C2×D6111yescharacteristic, complemented
342#358—611no—
343#358—611no—
344#358C2×D6111yescharacteristic, complemented
345#358C2×D6111yescharacteristic, complemented
346#357—611no—
347#357—611no—
348C22⋊C16—611no—
349C22⋊C16—611no—
350#346—611no—
351C61#3383211yescharacteristic, complemented, Sylow
352C2×#41—1221no—
353C22×Q8—1221no—
354C22×Q8—1221no—
355C22×D4—1221no—
356#135—1222no—
357#135—2441no—
358C2×Q16—21no—
359C2×Q16—611no—
360C2×Q16—611no—
361C2×Q16—1221no—
362C2×Q16—1221no—
363C2×Q16—1222no—
364C2×Q16—1222no—
365C2×Q16—2441no—
366C2×Q16—2441no—
367C2×Q16C22×D6111yescharacteristic
368C2×Q16C22×D6111yescharacteristic
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8—611no—
372C2×D8—1221no—
373C2×D8—1222no—
374C2×D8—2441no—
375C2×D8C22×D6111yescharacteristic
376C22×C8—611no—
377C22×C8—611no—
378C4⋊Q8—21no—
379C4⋊Q8—2441no—
380#124—1221no—
381#123—2441no—
382C22⋊Q8—1221no—
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—21no—
392C4×D4—611no—
393C4×D4—2441no—
394C2×C22⋊C4—2441no—
395Q32—1222no—
396Q32—1222no—
397Q32—2441no—
398SD32—22no—
399SD32—22no—
400SD32—614no—
401SD32—1222no—
402SD32—1222no—
403SD32—2441no—
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—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#102—21no—
418#102—1221no—
419C22⋊C8—611no—
420C22⋊C8—611no—
421C4×C8C22×D6111yescharacteristic
422C24—2441no—
423#41—4881no—
424#41—2442no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—4881no—
430C2×Q8—4881no—
431C2×Q8—2441no—
432C2×Q8—2441no—
433C2×Q8—2442no—
434C2×Q8—2442no—
435C2×D4—21no—
436C2×D4—611no—
437C2×D4—1221no—
438C2×D4—4881no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2442no—
442C2×D4—2441no—
443C22×C4—41no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—2441no—
447C22×C4—2441no—
448C22×C4—2441no—
449Q16—21no—
450Q16—21no—
451Q16—21no—
452Q16—21no—
453Q16—81no—
454Q16—612no—
455Q16—1221no—
456Q16—1221no—
457Q16—1221no—
458Q16—2442no—
459Q16—2442no—
460SD16—4881no—
461SD16—2442no—
462D8—21no—
463D8—1221no—
464D8—1221no—
465D8—2442no—
466D8D4×D6112yes—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8D4×D6111yescharacteristic
474C2×C8C23×D6111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—611no—
479C4⋊C4—1221no—
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—2441no—
494C42D4×D6111yescharacteristic
495C16—21no—
496C16—21no—
497C16—612no—
498C16—1221no—
499C23—41no—
500C23—611no—
501C23—611no—
502C23—9761no—
503C23—4882no—
504C23—4882no—
505C23—2441no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—1222no—
511Q8—4882no—
512Q8—4882no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516D4—22no—
517D4—41no—
518D4—1222no—
519D4—9761no—
520D4—4882no—
521D4—4882no—
522D4—2441no—
523D4—2441no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—161no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—4882no—
535C2×C4—4882no—
536C2×C4—4882no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4D8×D6111yescharacteristic
541C2×C4D8×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C8—21no—
544C8—1221no—
545C8—1221no—
546C8C2×D4×D6111yescharacteristic
547C8#1451982311yescharacteristic
548C22—42no—
549C22—81no—
550C22—81no—
551C22—612no—
552C22—1221no—
553C22—1221no—
554C22—1221no—
555C22—1221no—
556C22—1221no—
557C22—9764no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22C2×D8×D6111yescharacteristic
562C4—21no—
563C4—21no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—1221no—
568C4—1221no—
569C4—4881no—
570C4—4881no—
571C4—2442no—
572C4#1684039611yescharacteristic
573C4#1684056611yescharacteristic
574C2—82no—
575C2—1222no—
576C2—9761no—
577C2—2441no—
578C2#2145749111yescharacteristic
579C2#510×D6111yescharacteristic
580C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2208C2
socleC22×C61C22×D8
prime radical#21457505C2
hypercenter#2208D61
hypocenterC61#33832
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2208

Complements

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

NH ≅ G/Ntype
1. C2×#16840201C2semidirect
2. C2×SD32×D61C2semidirect
3. #21457569C2semidirect
5. #21448088C2semidirect
6. #21447935C2semidirect
7. #21447909C2semidirect
8. #21447882C2semidirect
9. #21447756C2semidirect
10. #21447754C2semidirect
11. #21447746C2semidirect
12. #21447737C2semidirect
13. #21447683C2semidirect
14. #21447643C2semidirect
15. C61×#2208C2semidirect
18. C2×Q16×D61C22semidirect
23. C2×Dic488C22semidirect
24. C2×#14519723C22semidirect
25. C2×#14519723C22semidirect
31. C2×#14519716C22semidirect
36. #16840000C22semidirect
44. C4×Dic244C22semidirect
46. #16839587C22semidirect
47. #16839567C22semidirect
48. #16839565C22semidirect
52. #16839541C22semidirect
53. #16839538C22semidirect
56. C61×#494C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
60. C61×#358C22semidirect
102. #14519721D4semidirect
271. C2×D61#510semidirect
292. #2208D61semidirect
326. #494C2×D61semidirect
341. C4⋊C16C2×D61semidirect
344. #358C2×D61semidirect
345. #358C2×D61semidirect
351. C61#33832semidirect

Maximal subgroups

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

subgroupindex
C2×#168402012
C2×SD32×D612
#214575692
#214575052
#214480882
#214479352
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