Finite Group Databasebeta

← #29837039

Subgroups — #29837039

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458182C211yesmaximal, characteristic, complemented
2#21457493C211yesmaximal, characteristic
3#21448059C211yesmaximal, characteristic, complemented
4#21448007C211yesmaximal, characteristic, complemented
5#21448001C211yesmaximal, characteristic, complemented
6#21447979C211yesmaximal, characteristic, complemented
7#21447918C211yesmaximal, characteristic, complemented
8#21447857C211yesmaximal, characteristic, complemented
9#21447747C211yesmaximal, characteristic, complemented
10#21447746C211yesmaximal, characteristic, complemented
11#21447686C211yesmaximal, characteristic, complemented
12#21447642C211yesmaximal, characteristic, complemented
13#21447536C211yesmaximal, characteristic, complemented
14#357×D61C211yesmaximal, characteristic, complemented
15C61×#2220C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18#16840446—21no—
19#16840396—21no—
20C2×#14519720C2211yescharacteristic
21C2×#14519720C2211yescharacteristic, complemented
22C2×C16×D61C2211yescharacteristic, complemented
23#16840163C2211yescharacteristic, complemented
24#16840137C2211yescharacteristic
25#16840100—21no—
26#16840091—21no—
27#16840067—21no—
28#16840000C2211yescharacteristic, complemented
29#16839999C2211yescharacteristic
30#16839956—21no—
31#16839897—21no—
32#16839895—21no—
33Q16×Dic61C2211yescharacteristic, complemented
34#16839844C2211yescharacteristic
35C8×C22⋊D61C2211yescharacteristic
36#16839800C2211yescharacteristic, complemented
37#107×D61C2211yescharacteristic, complemented
38#16839589C2211yescharacteristic, complemented
39#16839587C2211yescharacteristic, complemented
40#16839580C2211yescharacteristic, complemented
41#16839564C2211yescharacteristic
42#16839564C2211yescharacteristic
43#16839564C2211yescharacteristic, complemented
44#16839564C2211yescharacteristic, complemented
45#16839563C2211yescharacteristic, complemented
46#16839563C2211yescharacteristic, complemented
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic, complemented
49C22⋊#11758185C2211yescharacteristic, complemented
50#16839541C2211yescharacteristic, complemented
51#16839537C2211yescharacteristic, complemented
52C2×SD32×C61C2211yescharacteristic, complemented
53C2×M32×C61C2211yescharacteristic, complemented
54C61×#467C2211yescharacteristic, complemented
55C61×#466C2211yescharacteristic
56C61×#366C2211yescharacteristic, complemented
57C61×#358C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic, complemented
59C22×D244—21no—
60C2×D4×D61—21no—
61C2×D4×D61—41no—
62C2×D488—22no—
63C2×D488—21no—
64C2×D488—41no—
65C2×D488C2311yescharacteristic
66C2×D488C2311yescharacteristic
67C2×#11758199—21no—
68C2×#11758199—21no—
69D8×D61—22no—
70D8×D61—41no—
71C2×C8×D61C2311yescharacteristic
72#14519773—21no—
73#14519770—41no—
74#14519768—21no—
75#14519767—41no—
76#14519759—41no—
77#14519753—21no—
78#14519752—21no—
79#14519750—41no—
80#14519747—41no—
81#14519745—41no—
82Q8×Dic61—21no—
83#14519739—21no—
84C4×C22⋊D61D411yescharacteristic
85#14519736—21no—
86#14519735—41no—
87C4⋊C4×D61—21no—
88#14519720—22no—
89#14519720—41no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92C16×D61—41no—
93#14519711C2311yescharacteristic
94#14519711C2311yescharacteristic
95#14519710—21no—
96#14519709—21no—
97#14519708—21no—
98#14519706D411yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519697—21no—
102#14519697—21no—
103C22⋊#92418D411yescharacteristic
104#14519689D411yescharacteristic
105C8×Dic61C2311yescharacteristic
106C2×Q16×C61C2311yescharacteristic
107C2×D8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22⋊Q8×C61—21no—
110#121×C61—21no—
111SD32×C61—41no—
112M32×C61—41no—
113C2×C16×C61C2311yescharacteristic
114C2×C16×C61C2311yescharacteristic
115#107×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#103×C61—21no—
118#102×C61—21no—
119C23×D61—41no—
120C23×D61—41no—
121C2×Q8×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C2×C22⋊D61D811yescharacteristic
129D4×D61—42no—
130D4×D61—81no—
131D4×D61—84no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—42no—
135C2×D244—41no—
136C2×D244—41no—
137C2×D244—81no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61C2×D411yescharacteristic
141Q16×C61—21no—
142Q16×C61—21no—
143D8×C61—21no—
144D8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2411yescharacteristic
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C22⋊C4×C61—41no—
154C22⋊Dic61—41no—
155C22⋊Dic61D811yescharacteristic
156#11758199—42no—
157#11758199—81no—
158#11758198—41no—
159#11758198—41no—
160#11758198—41no—
161#11758198D811yescharacteristic
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—41no—
165C4⋊Dic61—41no—
166#11758196—41no—
167#11758196—41no—
168#11758196—41no—
169#11758196D811yescharacteristic
170C4×Dic61—41no—
171C4×Dic61—41no—
172C4×Dic61C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174C2×#92418C2×D411yescharacteristic
175D488—21no—
176D488—21no—
177D488—21no—
178D488—42no—
179D488#4112yes—
180C8×D61—21no—
181C8×D61—21no—
182C16×C61—21no—
183C16×C61—21no—
184#11758185—21no—
185#11758185—21no—
186C23×C61—41no—
187C23×C61C2×D811yescharacteristic
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—82no—
191C22×D61—82no—
192C22×D61—82no—
193C22×D61—82no—
194C22×D61—161no—
195C22×D61—161no—
196C22×D61C2×D811yescharacteristic
197Q8×C61—41no—
198Q8×C61—41no—
199D4×C61—41no—
200D4×C61—41no—
201D4×C61—82no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61C2×D811yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C22⋊D61—22no—
210C22⋊D61—82no—
211C22⋊D61—82no—
212C22⋊D61—161no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61C2×D811yescharacteristic
216C2×Dic61C2×D811yescharacteristic
217C2×Dic61C2×D811yescharacteristic
218D244—22no—
219D244—41no—
220D244—41no—
221D244—41no—
222D244—82no—
223D244—82no—
224C4×D61—21no—
225C4×D61—21no—
226C4×D61—82no—
227C4×D61—82no—
228C8×C61—21no—
229C8×C61C2×#4111yescharacteristic
230C8×C61H(5,𝔽2)11yescharacteristic
231#92418—21no—
232#92418—21no—
233#33877—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61C22×D811yescharacteristic
241C2×D61—21no—
242C2×D61—21no—
243C2×D61—42no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—164no—
250C2×D61—164no—
251C2×D61#50912yes—
252C4×C61—21no—
253C4×C61—81no—
254C4×C61—81no—
255C4×C61—81no—
256C4×C61#54611yescharacteristic
257C4×C61#57711yescharacteristic
258Dic61—21no—
259Dic61—21no—
260Dic61—21no—
261Dic61—81no—
262Dic61—81no—
263#3336—611no—
264D4×D8—611no—
265#2244—611no—
266#2234—611no—
267#2232—611no—
268#2228—611no—
269#2220D6111yescharacteristic, complemented
270#2217—611no—
271#2207—611no—
272#2187—611no—
273#2187—611no—
274#2175—611no—
275#2168—611no—
276#2140—611no—
277C2×#357—611no—
278C2×C61—41no—
279C2×C61—161no—
280C2×C61#329411yescharacteristic
281C2×C61C2×#50911yescharacteristic
282C2×C61#341611yescharacteristic
283D61—22no—
284D61—82no—
285D61—161no—
286C22×D8—611no—
287C22×D8—611no—
288#552—1221no—
289D42—1221no—
290C2×SD32—611no—
291C2×SD32—611no—
292C2×SD32C2×D6111yescharacteristic, complemented
293C2×M32C2×D6111yescharacteristic, complemented
294C22×C16—611no—
295#499—611no—
296#493—611no—
297#485—1221no—
298#483—1221no—
299#479—1221no—
300#467—611no—
301#467C2×D6111yescharacteristic, complemented
302#466—611no—
303#466C2×D6111yescharacteristic
304#459—1221no—
305#447—1221no—
306#447—1221no—
307C4×Q16—611no—
308C4×D8—611no—
309C8×D4—611no—
310#427—611no—
311C2×#107—611no—
312#367—611no—
313#366—611no—
314#366C2×D6111yescharacteristic, complemented
315C4⋊C16—611no—
316#358C2×D6111yescharacteristic, complemented
317#357—611no—
318#357—611no—
319#357—611no—
320#357—611no—
321#357—611no—
322#357—611no—
323#357—611no—
324#357—611no—
325#357C2×D6111yescharacteristic, complemented
326C22⋊C16—611no—
327C22⋊C16—611no—
328#346—611no—
329C61#3387711yescharacteristic, complemented, Sylow
330C22×D4—1221no—
331C22×D4—1221no—
332C22×D4—2441no—
333C2×Q16C22×D6111yescharacteristic
334C2×D8—611no—
335C2×D8—611no—
336C2×D8—1222no—
337C2×D8—1221no—
338C2×D8—1221no—
339C2×D8—1222no—
340C2×D8—1221no—
341C2×D8—2441no—
342C2×D8—2441no—
343C2×D8C22×D6111yescharacteristic
344C22×C8—611no—
345C22×C8C22×D6111yescharacteristic
346#127—1221no—
347#126—2441no—
348#125—1221no—
349#125—2441no—
350#123—2441no—
351C22⋊Q8—21no—
352C22⋊Q8—1221no—
353#121—21no—
354#121—1221no—
355#121—2441no—
356#120—2441no—
357#120—2441no—
358C4×Q8—1221no—
359C4×D4—611no—
360C4×D4—1221no—
361#117—1221no—
362#117—2441no—
363C2×C4⋊C4—1221no—
364SD32—41no—
365SD32—1222no—
366SD32—2441no—
367M32—41no—
368C2×C16—611no—
369C2×C16—611no—
370C2×C16—2441no—
371C2×C16C22×D6111yescharacteristic
372C2×C16C22×D6111yescharacteristic
373#107—611no—
374#107—611no—
375#107—1221no—
376#107C22×D6111yescharacteristic
377#107C22×D6111yescharacteristic
378#106—1221no—
379#106—1221no—
380C4⋊C8—611no—
381#103—21no—
382#103—1221no—
383#103—1221no—
384#102—21no—
385#102—1221no—
386#102—1221no—
387C22⋊C8—611no—
388C22⋊C8—611no—
389C4×C8—611no—
390C24—2441no—
391C24—2441no—
392C2×Q8—21no—
393C2×D4—21no—
394C2×D4—41no—
395C2×D4—611no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—4881no—
399C2×D4—4881no—
400C2×D4—4884no—
401C2×D4—2442no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4—2441no—
405C2×D4—2441no—
406C2×D4—2442no—
407C2×D4—2441no—
408C22×C4—611no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4D4×D6111yescharacteristic
412Q16—21no—
413Q16—21no—
414D8—21no—
415D8—21no—
416D8—612no—
417D8—1221no—
418D8—1221no—
419D8—1221no—
420D8—4881no—
421D8—2442no—
422D8—2442no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—611no—
426C2×C8—611no—
427C2×C8—1221no—
428C2×C8—1221no—
429C2×C8D4×D6111yescharacteristic
430C2×C8C23×D6111yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—611no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C4⋊C4—2441no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—611no—
445C22⋊C4—611no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C42—611no—
451C42—2441no—
452C42—2441no—
453C16—21no—
454C16—21no—
455C16—1221no—
456C16—1221no—
457C23—41no—
458C23—611no—
459C23—9761no—
460C23—9761no—
461C23—4882no—
462C23—4882no—
463C23—4882no—
464C23—4882no—
465C23—2441no—
466C23—2441no—
467C23D8×D6111yescharacteristic
468Q8—41no—
469Q8—41no—
470D4—41no—
471D4—41no—
472D4—82no—
473D4—1222no—
474D4—1222no—
475D4—9761no—
476D4—4882no—
477D4—4882no—
478D4—4882no—
479D4—4882no—
480D4—2441no—
481D4—2441no—
482D4—2441no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—611no—
489C2×C4—611no—
490C2×C4—611no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—4882no—
494C2×C4—4882no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4D8×D6111yescharacteristic
498C2×C4C2×D4×D6111yescharacteristic
499C8—21no—
500C8—1221no—
501C8—1221no—
502C8#41×D6111yescharacteristic
503C8#1451982211yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—612no—
511C22—1221no—
512C22—1221no—
513C22—9764no—
514C22—9764no—
515C22—4881no—
516C22—4881no—
517C22—4881no—
518C22—4881no—
519C22—4881no—
520C22—2442no—
521C22C2×D8×D6111yescharacteristic
522C4—21no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—1221no—
527C4—1221no—
528C4—1221no—
529C4—4881no—
530C4—4881no—
531C4#1684040111yescharacteristic
532C4#1684056911yescharacteristic
533C2—41no—
534C2—161no—
535C2—1222no—
536C2—9761no—
537C2—4882no—
538C2#2145762711yescharacteristic
539C2#509×D6111yescharacteristic
540C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2220C2
socleC22×C61C22×D8
prime radical#21457493C2
hypercenter#2220D61
hypocenterC61#33877
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. #21458182C2semidirect
3. #21448059C2semidirect
4. #21448007C2semidirect
5. #21448001C2semidirect
6. #21447979C2semidirect
7. #21447918C2semidirect
8. #21447857C2semidirect
9. #21447747C2semidirect
10. #21447746C2semidirect
11. #21447686C2semidirect
12. #21447642C2semidirect
13. #21447536C2semidirect
14. #357×D61C2semidirect
15. C61×#2220C2semidirect
21. C2×#14519720C22semidirect
22. C2×C16×D61C22semidirect
23. #16840163C22semidirect
28. #16840000C22semidirect
33. Q16×Dic61C22semidirect
36. #16839800C22semidirect
37. #107×D61C22semidirect
38. #16839589C22semidirect
39. #16839587C22semidirect
40. #16839580C22semidirect
43. #16839564C22semidirect
44. #16839564C22semidirect
45. #16839563C22semidirect
46. #16839563C22semidirect
48. #16839562C22semidirect
49. C22⋊#11758185C22semidirect
50. #16839541C22semidirect
51. #16839537C22semidirect
52. C2×SD32×C61C22semidirect
53. C2×M32×C61C22semidirect
54. C61×#467C22semidirect
56. C61×#366C22semidirect
57. C61×#358C22semidirect
58. C61×#357C22semidirect
269. #2220D61semidirect
292. C2×SD32C2×D61semidirect
293. C2×M32C2×D61semidirect
301. #467C2×D61semidirect
314. #366C2×D61semidirect
316. #358C2×D61semidirect
325. #357C2×D61semidirect
329. C61#33877semidirect

Maximal subgroups

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

subgroupindex
#214581822
#214574932
#214480592
#214480072
#214480012
#214479792
#214479182
#214478572
#214477472
#214477462
#214476862
#214476422
#214475362
#357×D612
C61×#22202
#3387761