Finite Group Databasebeta

← #29836484

Subgroups — #29836484

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840198C211yesmaximal, characteristic, complemented
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21457563C211yesmaximal, characteristic, complemented
4D8×C22⋊D61C211yesmaximal, characteristic, complemented
5#21448084C211yesmaximal, characteristic, complemented
6#21447934C211yesmaximal, characteristic, complemented
7#21447904C211yesmaximal, characteristic, complemented
8#21447876C211yesmaximal, characteristic, complemented
9#21447858C211yesmaximal, characteristic, complemented
10#21447744C211yesmaximal, characteristic, complemented
11#21447734C211yesmaximal, characteristic, complemented
12#21447732C211yesmaximal, characteristic, complemented
13#21447680C211yesmaximal, characteristic, complemented
14#21447647C211yesmaximal, characteristic, complemented
15C61×#2187C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic, complemented
17C2×Q16×D61C2211yescharacteristic, complemented
18C2×D8×D61C2211yescharacteristic
19#16840422—21no—
20D4×C22⋊D61—21no—
21#16840198—24no—
22C2×#14519722C2211yescharacteristic, complemented
23C2×#14519721C2211yescharacteristic
24C2×#14519721C2211yescharacteristic, complemented
25C2×#14519720C2211yescharacteristic, complemented
26SD32×D61—24no—
27C2×D976C2211yescharacteristic, complemented
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic, complemented
30C2×#14519717C2211yescharacteristic, complemented
31C2×C16×D61C2211yescharacteristic, complemented
32#16840147C2211yescharacteristic, complemented
33#16840137C2211yescharacteristic
34#16839999C2211yescharacteristic, complemented
35#16839998C2211yescharacteristic
36#16839995C2211yescharacteristic
37#16839987—21no—
38#16839955—21no—
39#16839922—21no—
40#16839902—21no—
41#16839896—21no—
42#16839894—21no—
43#16839856C2211yescharacteristic, complemented
44D8×Dic61C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839566C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic, complemented
51#16839562C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839537C2211yescharacteristic, complemented
54C22×D8×C61C2211yescharacteristic
55C2×SD32×C61C2211yescharacteristic
56C2×SD32×C61C2211yescharacteristic, complemented
57C61×#467C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic
59C61×#357C2211yescharacteristic, complemented
60C61×C22⋊C16C2211yescharacteristic, complemented
61C2×#11758222—21no—
62C2×Q8×D61—21no—
63C22×C22⋊D61—41no—
64C2×D4×D61—21no—
65#14519799—22no—
66#14519799—41no—
67C2×Dic244C2311yescharacteristic
68C2×#11758202—21no—
69Q16×D61—22no—
70Q16×D61—41no—
71C2×#11758190—21no—
72C2×#11758201—21no—
73C2×D488C2311yescharacteristic
74C2×D488C2311yescharacteristic
75C2×#11758199—21no—
76C2×#11758199—21no—
77D8×D61—22no—
78D8×D61—41no—
79C2×C8×D61C2311yescharacteristic
80#14519773—21no—
81#14519766—21no—
82#14519763—41no—
83#14519755—41no—
84#14519752—21no—
85#14519751—21no—
86#14519750—41no—
87#14519749—41no—
88#14519748—21no—
89#14519746—41no—
90#14519745—41no—
91#14519742—21no—
92D4×Dic61—21no—
93#14519739—41no—
94C4×C22⋊D61D411yescharacteristic
95#14519722—22no—
96#14519721—22no—
97#14519721D414yescomplemented
98#14519720—22no—
99D976—41no—
100#14519718—22no—
101#14519718—22no—
102#14519717—22no—
103C2×#11758185C2311yescharacteristic
104C2×#11758185C2311yescharacteristic
105C16×D61—22no—
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519700—21no—
109#14519699—21no—
110#14519698—21no—
111#14519698—21no—
112#14519697—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C22×D4×C61—21no—
117C2×Q16×C61C2311yescharacteristic
118C2×D8×C61—22no—
119C2×D8×C61—21no—
120C2×D8×C61—41no—
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125SD32×C61—22no—
126SD32×C61—41no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#103×C61—21no—
131C24×C61—41no—
132C23×D61—41no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—42no—
137C2×D4×C61—41no—
138C2×D4×C61—81no—
139C22×C4×C61C2×D411yescharacteristic
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—84no—
147C2×C22⋊D61D811yescharacteristic
148C22×Dic61—41no—
149Q8×D61—42no—
150Q8×D61—81no—
151D4×D61—42no—
152D4×D61—81no—
153#11758222—42no—
154#11758222—81no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—41no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61C2×D411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—41no—
163Q16×C61—21no—
164Q16×C61—21no—
165D8×C61—21no—
166D8×C61—21no—
167D8×C61—21no—
168D8×C61—42no—
169D8×C61C2×D412yes—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C22⋊Dic61D811yescharacteristic
180#11758202—42no—
181#11758201—81no—
182#11758199—42no—
183#11758199—81no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187#11758198—41no—
188#11758198D811yescharacteristic
189C4⋊Dic61—21no—
190#11758196—41no—
191#11758196D811yescharacteristic
192C4×Dic61—41no—
193C4×Dic61C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195C2×#92418C2×D411yescharacteristic
196Dic244—21no—
197Dic244C2×D412yescomplemented
198D488—21no—
199D488—21no—
200D488—21no—
201D488—21no—
202#11758190—42no—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61—21no—
207#11758185—21no—
208#11758185C2×D412yescomplemented
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—82no—
212C23×C61—82no—
213C23×C61—161no—
214C23×C61C2×D811yescharacteristic
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—161no—
220C22×D61C2×D811yescharacteristic
221Q8×C61—41no—
222Q8×C61—41no—
223D4×C61—22no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—82no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61C2×D811yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C22⋊D61—22no—
235C22⋊D61—82no—
236C22⋊D61—82no—
237C22⋊D61—82no—
238C22⋊D61—82no—
239C22⋊D61—82no—
240C22⋊D61—161no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—82no—
244C2×Dic61C2×D811yescharacteristic
245C2×Dic61C2×D811yescharacteristic
246C2×Dic61C2×D811yescharacteristic
247D244—41no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—82no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—82no—
255C4×D61—82no—
256Dic122—22no—
257Dic122—41no—
258Dic122—82no—
259C8×C61—21no—
260C8×C61C22×D411yescharacteristic
261C8×C61H(5,𝔽2)11yescharacteristic
262#92418—21no—
263#92418—21no—
264#33831—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—42no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—164no—
275C22×C61C22×D811yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—164no—
285C2×D61#50912yescomplemented
286C4×C61—21no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61D4211yescharacteristic
290C4×C61#57611yescharacteristic
291Dic61—21no—
292Dic61—21no—
293Dic61—21no—
294Dic61—42no—
295Dic61—81no—
296C2×#509—611no—
297C22×SD32—611no—
298#3285—611no—
299D4×D8—611no—
300#2250—611no—
301#2220—611no—
302#2215—611no—
303#2210—611no—
304#2207—611no—
305#2187D6111yescharacteristic, complemented
306#2186—611no—
307#2185—611no—
308#2184—611no—
309#2175—611no—
310#2168—611no—
311C2×C61—41no—
312C2×C61—82no—
313C2×C61—161no—
314C2×C61D4×D811yescharacteristic
315C2×C61C2×#50911yescharacteristic
316C2×C61#341611yescharacteristic
317D61—22no—
318D61—161no—
319D61—161no—
320C2×#135—611no—
321C22×Q16—611no—
322C22×D8—611no—
323C22×D8C2×D6111yescharacteristic
324#548—1221no—
325D42—1221no—
326#509—1224no—
327C2×SD32—611no—
328C2×SD32—611no—
329C2×SD32—611no—
330C2×SD32—611no—
331C2×SD32—1224no—
332C2×SD32C2×D6111yescharacteristic
333C2×SD32C2×D6111yescharacteristic, complemented
334C2×D16—611no—
335C2×D16—611no—
336C2×D16—611no—
337C2×M32—611no—
338C22×C16—611no—
339#495—611no—
340#493—611no—
341#467C2×D6111yescharacteristic, complemented
342#466—611no—
343#466—611no—
344#466—611no—
345#464—1221no—
346#459—1221no—
347#452—1221no—
348#448—1221no—
349#447—1221no—
350#447—1221no—
351C4×Q16—611no—
352C4×D8—611no—
353C8×D4—611no—
354#366—611no—
355C4⋊C16—611no—
356#358—611no—
357#358—611no—
358#357—611no—
359#357—611no—
360#357C2×D6111yescharacteristic
361#357C2×D6111yescharacteristic, complemented
362C22⋊C16—611no—
363C22⋊C16C2×D6111yescharacteristic, complemented
364#346—611no—
365C61#3383111yescharacteristic, complemented, Sylow
366C2×#41—1221no—
367C22×Q8—1221no—
368C22×D4—21no—
369C22×D4—1221no—
370C22×D4—2441no—
371#135—1222no—
372#135—2441no—
373C2×Q16—611no—
374C2×Q16—1221no—
375C2×Q16—1222no—
376C2×Q16—2441no—
377C2×Q16C22×D6111yescharacteristic
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×D8—22no—
381C2×D8—21no—
382C2×D8—41no—
383C2×D8—611no—
384C2×D8—611no—
385C2×D8—1221no—
386C2×D8—1221no—
387C2×D8—1222no—
388C2×D8—2441no—
389C2×D8C22×D6111yescharacteristic
390C2×D8C22×D6111yescharacteristic
391C22×C8—611no—
392C22×C8C22×D6111yescharacteristic
393#127—1221no—
394#124—1221no—
395#124—2441no—
396C22⋊Q8—21no—
397C22⋊Q8—2441no—
398#121—1221no—
399#121—1221no—
400#121—1221no—
401#121—2441no—
402#121—2441no—
403#120—2441no—
404#120—2441no—
405C4×Q8—1221no—
406C4×D4—611no—
407C4×D4—1221no—
408C4×D4—2441no—
409SD32—22no—
410SD32—41no—
411SD32—614no—
412SD32—1222no—
413SD32—1222no—
414SD32—1222no—
415D16—1222no—
416D16—1222no—
417D16—2441no—
418M32—1222no—
419C2×C16—611no—
420C2×C16—611no—
421C2×C16—1222no—
422C2×C16C22×D6111yescharacteristic
423C2×C16C22×D6111yescharacteristic
424#107—611no—
425#107C22×D6111yescharacteristic
426C4⋊C8—611no—
427#103—21no—
428#103—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433C22⋊C8—611no—
434C22⋊C8—611no—
435C4×C8—611no—
436C24—41no—
437C24—2441no—
438#41—4881no—
439#41—2442no—
440C2×Q8—21no—
441C2×Q8—1221no—
442C2×Q8—4881no—
443C2×Q8—2441no—
444C2×Q8—2442no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—41no—
449C2×D4—81no—
450C2×D4—611no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—4881no—
454C2×D4—4884no—
455C2×D4—2441no—
456C2×D4—2441no—
457C2×D4—2441no—
458C2×D4—2441no—
459C2×D4—2442no—
460C2×D4—2441no—
461C2×D4—2441no—
462C2×D4—2441no—
463C22×C4—611no—
464C22×C4—2441no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4D4×D6111yescharacteristic
468Q16—21no—
469Q16—21no—
470Q16—612no—
471Q16—1221no—
472Q16—2442no—
473SD16—4881no—
474SD16—2442no—
475D8—21no—
476D8—21no—
477D8—21no—
478D8—42no—
479D8—1221no—
480D8—1221no—
481D8—1221no—
482D8—1221no—
483D8—4881no—
484D8—2442no—
485D8D4×D6112yes—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8D4×D6111yescharacteristic
493C2×C8C23×D6111yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—611no—
497C4⋊C4—1221no—
498C4⋊C4—2441no—
499C22⋊C4—41no—
500C22⋊C4—611no—
501C22⋊C4—611no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C42—611no—
509C42—2441no—
510C16—21no—
511C16—21no—
512C16—612no—
513C16—1221no—
514C23—41no—
515C23—41no—
516C23—82no—
517C23—82no—
518C23—161no—
519C23—611no—
520C23—9761no—
521C23—4882no—
522C23—4882no—
523C23—2441no—
524C23—2441no—
525C23D8×D6111yescharacteristic
526Q8—41no—
527Q8—41no—
528Q8—1222no—
529Q8—4882no—
530Q8—2441no—
531D4—22no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—82no—
536D4—1222no—
537D4—9761no—
538D4—4882no—
539D4—4882no—
540D4—4882no—
541D4—4882no—
542D4—4882no—
543D4—4882no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—4882no—
558C2×C4—4882no—
559C2×C4—4882no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4D8×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C8—21no—
565C8—1221no—
566C8—1221no—
567C8C2×D4×D6111yescharacteristic
568C8#1451982211yescharacteristic
569C22—21no—
570C22—21no—
571C22—21no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—164no—
579C22—612no—
580C22—1221no—
581C22—1221no—
582C22—9764no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—4881no—
589C22C2×D8×D6111yescharacteristic
590C4—21no—
591C4—81no—
592C4—81no—
593C4—1221no—
594C4—1221no—
595C4—1221no—
596C4—4881no—
597C4—2442no—
598C4#1684039411yescharacteristic
599C4#1684056611yescharacteristic
600C2—41no—
601C2—82no—
602C2—161no—
603C2—1222no—
604C2—9761no—
605C2—9761no—
606C2#2145748911yescharacteristic
607C2#509×D6111yescharacteristic
608C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2187C2
socleC22×C61C22×D8
hypercenter#2187D61
hypocenterC61#33831
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. C2×#16840198C2semidirect
2. C2×SD32×D61C2semidirect
3. #21457563C2semidirect
4. D8×C22⋊D61C2semidirect
5. #21448084C2semidirect
6. #21447934C2semidirect
7. #21447904C2semidirect
8. #21447876C2semidirect
9. #21447858C2semidirect
10. #21447744C2semidirect
11. #21447734C2semidirect
12. #21447732C2semidirect
13. #21447680C2semidirect
14. #21447647C2semidirect
15. C61×#2187C2semidirect
16. C2×#14519799C22semidirect
17. C2×Q16×D61C22semidirect
22. C2×#14519722C22semidirect
24. C2×#14519721C22semidirect
25. C2×#14519720C22semidirect
27. C2×D976C22semidirect
29. C2×#14519718C22semidirect
30. C2×#14519717C22semidirect
31. C2×C16×D61C22semidirect
32. #16840147C22semidirect
34. #16839999C22semidirect
43. #16839856C22semidirect
46. #16839587C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839566C22semidirect
50. #16839564C22semidirect
51. #16839562C22semidirect
52. #16839541C22semidirect
53. #16839537C22semidirect
56. C2×SD32×C61C22semidirect
57. C61×#467C22semidirect
59. C61×#357C22semidirect
60. C61×C22⋊C16C22semidirect
97. #14519721D4semidirect
197. Dic244C2×D4semidirect
208. #11758185C2×D4semidirect
285. C2×D61#509semidirect
305. #2187D61semidirect
333. C2×SD32C2×D61semidirect
341. #467C2×D61semidirect
361. #357C2×D61semidirect
363. C22⋊C16C2×D61semidirect
365. C61#33831semidirect

Maximal subgroups

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

subgroupindex
C2×#168401982
C2×SD32×D612
#214575632
D8×C22⋊D612
#214480842
#214479342
#214479042
#214478762
#214478582
#214477442
#214477342
#214477322
#214476802
#214476472
C61×#21872
#3383161