Finite Group Databasebeta

← #29836840

Subgroups — #29836840

Order 15616.

622 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458208C211yesmaximal, characteristic, complemented
2D8×C22⋊D61C211yesmaximal, characteristic
3#21448041C211yesmaximal, characteristic, complemented
4#21448005C211yesmaximal, characteristic, complemented
5#21448003C211yesmaximal, characteristic, complemented
6#21447964C211yesmaximal, characteristic, complemented
7#21447917C211yesmaximal, characteristic, complemented
8#21447912C211yesmaximal, characteristic, complemented
9#21447857C211yesmaximal, characteristic, complemented
10#21447746C211yesmaximal, characteristic, complemented
11SD32×Dic61C211yesmaximal, characteristic, complemented
12#21447636C211yesmaximal, characteristic, complemented
13C2×#16839563C211yesmaximal, characteristic, complemented
14#357×D61C211yesmaximal, characteristic, complemented
15C61×#2187C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840470—21no—
18D4×C22⋊D61—21no—
19C2×#14519720C2211yescharacteristic, complemented
20C22×#11758185C2211yescharacteristic, complemented
21C2×C16×D61C2211yescharacteristic, complemented
22#16840166C2211yescharacteristic, complemented
23#16840137C2211yescharacteristic
24#16840096—21no—
25#16840095—21no—
26#16840055—21no—
27#16840000C2211yescharacteristic, complemented
28#16839998C2211yescharacteristic
29#16839995C2211yescharacteristic
30#16839955—21no—
31#16839896—21no—
32#16839894—21no—
33Q16×Dic61C2211yescharacteristic, complemented
34D8×Dic61C2211yescharacteristic
35C8×C22⋊D61C2211yescharacteristic
36C2×#14519711C2211yescharacteristic, complemented
37#107×D61C2211yescharacteristic, complemented
38#16839590C2211yescharacteristic, complemented
39#16839590C2211yescharacteristic, complemented
40#16839589C2211yescharacteristic, complemented
41#16839580C2211yescharacteristic, complemented
42#16839566C2211yescharacteristic, complemented
43#16839564C2211yescharacteristic, complemented
44#16839564C2211yescharacteristic, complemented
45#16839563C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic, complemented
48#16839563C2211yescharacteristic, complemented
49#16839562C2211yescharacteristic, complemented
50#16839541C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C22×D8×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic, complemented
55C61×#467C2211yescharacteristic, complemented
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic, complemented
59C22×C22⋊D61—41no—
60C2×D4×D61—21no—
61C2×D488C2311yescharacteristic
62C2×#11758199—21no—
63C2×#11758199—21no—
64D8×D61—22no—
65D8×D61—41no—
66C2×C8×D61C2311yescharacteristic
67#14519776—21no—
68#14519773—21no—
69#14519767—41no—
70#14519757—41no—
71#14519754—41no—
72#14519753—21no—
73#14519751—21no—
74#14519749—41no—
75#14519748—21no—
76#14519746—41no—
77#14519745—41no—
78Q8×Dic61—21no—
79D4×Dic61—21no—
80C4×C22⋊D61D411yescharacteristic
81#14519737—41no—
82C2×C4⋊Dic61—21no—
83C4⋊C4×D61—21no—
84#14519720—41no—
85C2×#11758185—41no—
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88C16×D61—41no—
89#14519711—21no—
90#14519711C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519710—21no—
94#14519710—21no—
95#14519706D411yescharacteristic
96#14519701—21no—
97#14519701—21no—
98#14519699—21no—
99#14519698—21no—
100#14519698—21no—
101C22⋊#92418D411yescharacteristic
102#14519689D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C22×D4×C61—21no—
105C2×Q16×C61C2311yescharacteristic
106C2×D8×C61—22no—
107C2×D8×C61—21no—
108C2×D8×C61—41no—
109C2×D8×C61C2311yescharacteristic
110C2×D8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22⋊Q8×C61—21no—
113SD32×C61—22no—
114SD32×C61—41no—
115C2×C16×C61C2311yescharacteristic
116C2×C16×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#103×C61—21no—
119C24×C61—41no—
120C23×D61—41no—
121C2×Q8×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—42no—
125C2×D4×C61—41no—
126C2×D4×C61—81no—
127C22×C4×C61C2×D411yescharacteristic
128C2×C22⋊D61—41no—
129C2×C22⋊D61—41no—
130C2×C22⋊D61—41no—
131C2×C22⋊D61—41no—
132C2×C22⋊D61—41no—
133C2×C22⋊D61—84no—
134C2×C22⋊D61D811yescharacteristic
135C22×Dic61—41no—
136C22×Dic61—41no—
137D4×D61—42no—
138D4×D61—81no—
139C2×D244—21no—
140C2×C4×D61—41no—
141C2×C4×D61C2×D411yescharacteristic
142C2×Dic122—41no—
143Q16×C61—21no—
144Q16×C61—21no—
145D8×C61—21no—
146D8×C61—21no—
147D8×C61—21no—
148D8×C61—42no—
149D8×C61#4112yes—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61C2×D411yescharacteristic
153C2×C8×C61C2411yescharacteristic
154C4⋊C4×C61—21no—
155C4⋊C4×C61—41no—
156C22⋊C4×C61—41no—
157C22⋊Dic61—41no—
158C22⋊Dic61—41no—
159C22⋊Dic61D811yescharacteristic
160#11758199—42no—
161#11758199—81no—
162#11758198—41no—
163#11758198—41no—
164#11758198D811yescharacteristic
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—41no—
169C4⋊Dic61—41no—
170#11758196—41no—
171#11758196—41no—
172#11758196—41no—
173#11758196D811yescharacteristic
174C4×Dic61—41no—
175C4×Dic61C2×D411yescharacteristic
176C2×#92418C2×D411yescharacteristic
177C2×#92418C2×D411yescharacteristic
178D488—21no—
179D488—21no—
180C8×D61—21no—
181C8×D61—21no—
182C16×C61—21no—
183C16×C61—21no—
184#11758185—21no—
185#11758185—21no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—82no—
189C23×C61—82no—
190C23×C61—161no—
191C23×C61C2×D811yescharacteristic
192C22×D61—41no—
193C22×D61—82no—
194C22×D61—82no—
195C22×D61—161no—
196C22×D61C2×D811yescharacteristic
197Q8×C61—41no—
198Q8×C61—41no—
199D4×C61—22no—
200D4×C61—41no—
201D4×C61—41no—
202D4×C61—41no—
203D4×C61—82no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61C2×D811yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C22⋊D61—82no—
211C22⋊D61—82no—
212C22⋊D61—82no—
213C22⋊D61—82no—
214C22⋊D61—161no—
215C22⋊D61SD3214yescomplemented
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—82no—
220C2×Dic61—161no—
221C2×Dic61C2×D811yescharacteristic
222C2×Dic61C2×D811yescharacteristic
223C2×Dic61C2×D811yescharacteristic
224D244—41no—
225D244—41no—
226C4×D61—21no—
227C4×D61—21no—
228C4×D61—161no—
229Dic122—161no—
230C8×C61—21no—
231C8×C61C2×#4111yescharacteristic
232C8×C61H(5,𝔽2)11yescharacteristic
233#92418—21no—
234#92418—21no—
235#33862—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—42no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—164no—
245C22×C61C2×SD3212yes—
246C22×C61C22×D811yescharacteristic
247C2×D61—21no—
248C2×D61—21no—
249C2×D61—81no—
250C2×D61—81no—
251C2×D61—81no—
252C2×D61—164no—
253C2×D61C2×SD3212yes—
254C4×C61—21no—
255C4×C61—81no—
256C4×C61—81no—
257C4×C61#54611yescharacteristic
258C4×C61#57711yescharacteristic
259Dic61—21no—
260Dic61—21no—
261Dic61—81no—
262Dic61—81no—
263Dic61—81no—
264Dic61C2×SD3212yes—
265#3338—611no—
266D4×D8—611no—
267#2240—611no—
268#2233—611no—
269#2233—611no—
270#2226—611no—
271#2217—611no—
272#2217—611no—
273#2207—611no—
274#2187—611no—
275#2187D6111yescharacteristic, complemented
276C4×SD32—611no—
277D4×C16—611no—
278C2×#357—611no—
279C2×#357—611no—
280C2×C61—22no—
281C2×C61—82no—
282C2×C61—161no—
283C2×C61#329411yescharacteristic
284C2×C61C22×SD3211yescharacteristic
285C2×C61#341511yescharacteristic
286D61—22no—
287D61—161no—
288C22×D8—611no—
289C22×D8C2×D6111yescharacteristic
290#554—1221no—
291D42—1221no—
292C2×SD32—611no—
293C2×SD32C2×D6111yescharacteristic
294C2×SD32C2×D6111yescharacteristic, complemented
295C22×C16—611no—
296C22×C16—611no—
297#500—611no—
298#493—611no—
299#484—1221no—
300#484—1221no—
301#477—1221no—
302#467—611no—
303#467C2×D6111yescharacteristic, complemented
304#466—611no—
305#466—611no—
306#459—1221no—
307#447—1221no—
308#447—1221no—
309C4×Q16—611no—
310C4×D8—611no—
311C8×D4—611no—
312C2×#107—611no—
313C2×#107—611no—
314#367—611no—
315#367—611no—
316#367—611no—
317C4⋊C16—611no—
318#358—611no—
319#357—611no—
320#357—611no—
321#357—611no—
322#357—611no—
323#357—611no—
324#357—611no—
325#357—611no—
326#357C2×D6111yescharacteristic
327#357C2×D6111yescharacteristic, complemented
328C22⋊C16—611no—
329C22⋊C16C2×D6111yescharacteristic, complemented
330C4×C16—611no—
331C61#3386211yescharacteristic, complemented, Sylow
332C22×D4—21no—
333C22×D4—1221no—
334C22×D4—2441no—
335C2×Q16C22×D6111yescharacteristic
336C2×D8—22no—
337C2×D8—21no—
338C2×D8—41no—
339C2×D8—611no—
340C2×D8—1221no—
341C2×D8—1221no—
342C2×D8—1222no—
343C2×D8—2441no—
344C2×D8C22×D6111yescharacteristic
345C2×D8C22×D6111yescharacteristic
346C22×C8—611no—
347C22×C8C22×D6111yescharacteristic
348C4⋊Q8—1221no—
349#127—1221no—
350#125—2441no—
351C22⋊Q8—21no—
352C22⋊Q8—1221no—
353C22⋊Q8—2441no—
354C22⋊Q8—2441no—
355#121—1221no—
356#121—1221no—
357#121—2441no—
358#120—2441no—
359#120—2441no—
360C4×Q8—1221no—
361C4×D4—611no—
362C4×D4—1221no—
363C4×D4—2441no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366SD32—22no—
367SD32—41no—
368SD32—2441no—
369C2×C16—611no—
370C2×C16—611no—
371C2×C16—2441no—
372C2×C16—2441no—
373C2×C16C22×D6111yescharacteristic
374C2×C16C22×D6111yescharacteristic
375#107—611no—
376#107—611no—
377#107—611no—
378#107—1221no—
379#107—1221no—
380#107—1221no—
381#107C22×D6111yescharacteristic
382C4⋊C8—611no—
383#103—21no—
384#103—1221no—
385#103—1221no—
386#102—1221no—
387#102—1221no—
388#102—1221no—
389C22⋊C8—611no—
390C22⋊C8—611no—
391C4×C8—611no—
392C24—41no—
393C24—2441no—
394C2×Q8—21no—
395C2×Q8—2441no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—81no—
401C2×D4—611no—
402C2×D4—1221no—
403C2×D4—4881no—
404C2×D4—4884no—
405C2×D4—2441no—
406C2×D4—2441no—
407C2×D4—2441no—
408C2×D4—2442no—
409C2×D4—2441no—
410C2×D4—2441no—
411C22×C4—611no—
412C22×C4—2441no—
413C22×C4—2441no—
414C22×C4—2441no—
415C22×C4D4×D6111yescharacteristic
416Q16—21no—
417Q16—21no—
418D8—21no—
419D8—21no—
420D8—21no—
421D8—42no—
422D8—1221no—
423D8—1221no—
424D8—4881no—
425D8—2442no—
426D8#1175822412yes—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—611no—
430C2×C8—611no—
431C2×C8—1221no—
432C2×C8—1221no—
433C2×C8D4×D6111yescharacteristic
434C2×C8C23×D6111yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—41no—
437C4⋊C4—611no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—1221no—
441C4⋊C4—2441no—
442C4⋊C4—2441no—
443C4⋊C4—2441no—
444C4⋊C4—2441no—
445C4⋊C4—2441no—
446C22⋊C4—41no—
447C22⋊C4—611no—
448C22⋊C4—611no—
449C22⋊C4—2441no—
450C22⋊C4—2441no—
451C22⋊C4—2441no—
452C22⋊C4—2441no—
453C42—611no—
454C42—2441no—
455C16—21no—
456C16—21no—
457C16—1221no—
458C16—1221no—
459C23—41no—
460C23—41no—
461C23—82no—
462C23—82no—
463C23—161no—
464C23—611no—
465C23—9761no—
466C23—4882no—
467C23—4882no—
468C23—2441no—
469C23D8×D6111yescharacteristic
470Q8—41no—
471Q8—41no—
472Q8—9761no—
473D4—22no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—82no—
478D4—614no—
479D4—9761no—
480D4—4882no—
481D4—4882no—
482D4—4882no—
483D4—4882no—
484D4—2441no—
485D4—2441no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—611no—
491C2×C4—611no—
492C2×C4—611no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—9761no—
496C2×C4—9761no—
497C2×C4—4882no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4D8×D6111yescharacteristic
502C2×C4C2×D4×D6111yescharacteristic
503C8—21no—
504C8—1221no—
505C8—1221no—
506C8C2×#1175822411yescharacteristic
507C8#1451982211yescharacteristic
508C22—21no—
509C22—21no—
510C22—42no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—164no—
517C22—612no—
518C22—1221no—
519C22—1221no—
520C22—9764no—
521C22—4881no—
522C22—4881no—
523C22—4881no—
524C22SD32×D6112yes—
525C22C2×D8×D6111yescharacteristic
526C4—21no—
527C4—81no—
528C4—81no—
529C4—612no—
530C4—1221no—
531C4—1221no—
532C4—4881no—
533C4—4881no—
534C4—4881no—
535C4#1684039911yescharacteristic
536C4#1684057211yescharacteristic
537C2—22no—
538C2—82no—
539C2—161no—
540C2—1222no—
541C2—9761no—
542C2#2145762511yescharacteristic
543C2C2×SD32×D6111yescharacteristic
544C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2187C2
socleC22×C61C22×D8
prime radicalD8×C22⋊D61C2
hypercenter#2187D61
hypocenterC61#33862
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. #21458208C2semidirect
3. #21448041C2semidirect
4. #21448005C2semidirect
5. #21448003C2semidirect
6. #21447964C2semidirect
7. #21447917C2semidirect
8. #21447912C2semidirect
9. #21447857C2semidirect
10. #21447746C2semidirect
11. SD32×Dic61C2semidirect
12. #21447636C2semidirect
13. C2×#16839563C2semidirect
14. #357×D61C2semidirect
15. C61×#2187C2semidirect
19. C2×#14519720C22semidirect
20. C22×#11758185C22semidirect
21. C2×C16×D61C22semidirect
22. #16840166C22semidirect
27. #16840000C22semidirect
33. Q16×Dic61C22semidirect
36. C2×#14519711C22semidirect
37. #107×D61C22semidirect
38. #16839590C22semidirect
39. #16839590C22semidirect
40. #16839589C22semidirect
41. #16839580C22semidirect
42. #16839566C22semidirect
43. #16839564C22semidirect
44. #16839564C22semidirect
47. #16839563C22semidirect
48. #16839563C22semidirect
49. #16839562C22semidirect
50. #16839541C22semidirect
51. C16×Dic61C22semidirect
54. C2×SD32×C61C22semidirect
55. C61×#467C22semidirect
57. C61×#357C22semidirect
58. C61×C22⋊C16C22semidirect
215. C22⋊D61SD32semidirect
275. #2187D61semidirect
294. C2×SD32C2×D61semidirect
303. #467C2×D61semidirect
327. #357C2×D61semidirect
329. C22⋊C16C2×D61semidirect
331. C61#33862semidirect

Maximal subgroups

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

subgroupindex
#214582082
D8×C22⋊D612
#214480412
#214480052
#214480032
#214479642
#214479172
#214479122
#214478572
#214477462
SD32×Dic612
#214476362
C2×#168395632
#357×D612
C61×#21872
#3386261