Finite Group Databasebeta

← #29837051

Subgroups — #29837051

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#21458189C211yesmaximal, characteristic, complemented
2#21457494C211yesmaximal, characteristic
3#21448060C211yesmaximal, characteristic, complemented
4#21448009C211yesmaximal, characteristic, complemented
5#21447998C211yesmaximal, characteristic, complemented
6#21447977C211yesmaximal, characteristic, complemented
7#21447929C211yesmaximal, characteristic, complemented
8#21447912C211yesmaximal, characteristic, complemented
9#21447746C211yesmaximal, characteristic, complemented
10#21447746C211yesmaximal, characteristic, complemented
11#21447681C211yesmaximal, characteristic, complemented
12#21447643C211yesmaximal, characteristic, complemented
13#21447533C211yesmaximal, characteristic, complemented
14#357×D61C211yesmaximal, characteristic, complemented
15C61×#2207C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18#16840453—21no—
19#16840397—21no—
20C2×#14519720C2211yescharacteristic
21C2×#14519716C2211yescharacteristic, complemented
22C2×C16×D61C2211yescharacteristic, complemented
23#16840164C2211yescharacteristic, complemented
24#16840101—21no—
25#16840091—21no—
26#16840069—21no—
27#16840000C2211yescharacteristic, complemented
28#16840000C2211yescharacteristic, complemented
29#16839995C2211yescharacteristic
30#16839995C2211yescharacteristic
31#16839954—21no—
32#16839894—21no—
33#16839894—21no—
34C4×D488C2211yescharacteristic
35C8×D244C2211yescharacteristic
36#16839799C2211yescharacteristic, complemented
37#107×D61C2211yescharacteristic, complemented
38#16839590C2211yescharacteristic, complemented
39#16839588C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839564C2211yescharacteristic
43#16839564C2211yescharacteristic
44#16839563C2211yescharacteristic, complemented
45#16839563C2211yescharacteristic, complemented
46#16839563C2211yescharacteristic, complemented
47#16839563C2211yescharacteristic, complemented
48#16839562C2211yescharacteristic
49#16839541C2211yescharacteristic, complemented
50#16839541C2211yescharacteristic, complemented
51#16839538C2211yescharacteristic, complemented
52C2×SD32×C61C2211yescharacteristic, complemented
53C2×SD32×C61C2211yescharacteristic, complemented
54C61×#493C2211yescharacteristic
55C4×Q16×C61C2211yescharacteristic, complemented
56C61×C4⋊C16C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic, complemented
59C2×D4×D61—21no—
60C2×D4×D61—21no—
61C2×D4×D61—41no—
62C2×D488C2311yescharacteristic
63C2×#11758199—21no—
64C2×#11758199—21no—
65D8×D61—22no—
66D8×D61—22no—
67D8×D61—41no—
68D8×D61—41no—
69C2×C8×D61C2311yescharacteristic
70C2×C8×D61C2311yescharacteristic
71#14519771—41no—
72#14519769—21no—
73#14519767—41no—
74#14519762—41no—
75#14519753—21no—
76#14519753—21no—
77#14519748—21no—
78#14519748—21no—
79#14519748—41no—
80#14519745—41no—
81#14519745—41no—
82C4×D244—21no—
83C4×D244D411yescharacteristic
84#14519735—21no—
85#14519734—41no—
86C4⋊C4×D61—21no—
87#14519720—22no—
88#14519716—41no—
89C2×#11758185C2311yescharacteristic
90C2×#11758185C2311yescharacteristic
91C16×D61—41no—
92#14519711C2311yescharacteristic
93#14519711C2311yescharacteristic
94#14519711C2311yescharacteristic
95#14519710—21no—
96#14519709—21no—
97#14519708—21no—
98C4⋊#92418D411yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519699—21no—
102#14519698—21no—
103#14519698—21no—
104#14519689D411yescharacteristic
105#14519689D411yescharacteristic
106C2×Q16×C61C2311yescharacteristic
107C2×D8×C61—21no—
108C2×D8×C61C2311yescharacteristic
109C2×D8×C61C2311yescharacteristic
110#127×C61—21no—
111C4×Q8×C61—21no—
112SD32×C61—41no—
113SD32×C61—41no—
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#103×C61—21no—
118C4×C8×C61C2311yescharacteristic
119C23×D61—41no—
120C23×D61—41no—
121C2×Q8×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—41no—
125C2×D4×C61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C2×C22⋊D61—41no—
129C2×C22⋊D61—41no—
130D4×D61—42no—
131D4×D61—42no—
132D4×D61—81no—
133D4×D61—81no—
134D4×D61—84no—
135C2×D244—21no—
136C2×D244D811yescharacteristic
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×C4×D61C2×D411yescharacteristic
140C2×C4×D61C2×D411yescharacteristic
141Q16×C61—21no—
142Q16×C61—21no—
143D8×C61—21no—
144D8×C61—21no—
145D8×C61—21no—
146D8×C61—21no—
147D8×C61—81no—
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2411yescharacteristic
150C4⋊C4×C61—21no—
151C4⋊C4×C61—41no—
152C42×C61—41no—
153C42×C61C2×D411yescharacteristic
154C22⋊Dic61—41no—
155C22⋊Dic61—41no—
156#11758199—42no—
157#11758199—42no—
158#11758198—41no—
159#11758198—41no—
160#11758198—41no—
161#11758198—41no—
162#11758198D811yescharacteristic
163#11758198D811yescharacteristic
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61D811yescharacteristic
168#11758196—41no—
169#11758196—41no—
170#11758196—41no—
171#11758196—41no—
172#11758196—41no—
173C4×Dic61—41no—
174C2×#92418C2×D411yescharacteristic
175C2×#92418C2×D411yescharacteristic
176D488—21no—
177D488#4112yes—
178C8×D61—21no—
179C8×D61—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—
188C22×D61—41no—
189C22×D61—82no—
190C22×D61—82no—
191C22×D61—82no—
192C22×D61—82no—
193C22×D61—161no—
194C22×D61—161no—
195C22×D61C2×D811yescharacteristic
196C22×D61C2×D811yescharacteristic
197Q8×C61—41no—
198Q8×C61—41no—
199D4×C61—41no—
200D4×C61—41no—
201D4×C61—41no—
202D4×C61—41no—
203D4×C61—82no—
204D4×C61—161no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61C2×D811yescharacteristic
208C2×C4×C61C2×D811yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C22⋊D61—82no—
211C22⋊D61—82no—
212C22⋊D61—82no—
213C22⋊D61—82no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61C2×D811yescharacteristic
218C2×Dic61C2×D811yescharacteristic
219D244—22no—
220D244—22no—
221D244—41no—
222C4×D61—21no—
223C4×D61—21no—
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—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61C22×D811yescharacteristic
241C2×D61—21no—
242C2×D61—21no—
243C2×D61—21no—
244C2×D61—21no—
245C2×D61—21no—
246C2×D61—42no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—164no—
250C2×D61—164no—
251C2×D61#50912yes—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—81no—
255C4×C61—81no—
256C4×C61#54611yescharacteristic
257C4×C61#57711yescharacteristic
258Dic61—21no—
259Dic61—21no—
260Dic61—81no—
261Dic61—81no—
262Dic61—81no—
263#3336—611no—
264D4×D8—611no—
265#2244—611no—
266#2234—611no—
267#2232—611no—
268#2228—611no—
269#2220—611no—
270#2217—611no—
271#2207D6111yescharacteristic, complemented
272#2187—611no—
273#2187—611no—
274#2175—611no—
275#2168—611no—
276#2140—611no—
277C2×#357—611no—
278C2×C61—161no—
279C2×C61—161no—
280C2×C61#329411yescharacteristic
281C2×C61C2×#50911yescharacteristic
282C2×C61#341611yescharacteristic
283D61—22no—
284D61—41no—
285D61—82no—
286C22×D8—611no—
287C22×D8—611no—
288#552—1221no—
289D42—1221no—
290C2×SD32—611no—
291C2×SD32C2×D6111yescharacteristic, complemented
292C2×SD32C2×D6111yescharacteristic, complemented
293C2×M32—611no—
294C22×C16—611no—
295#499—611no—
296#493C2×D6111yescharacteristic
297#485—1221no—
298#483—1221no—
299#479—1221no—
300#467—611no—
301#467—611no—
302#466—611no—
303#466—611no—
304#459—1221no—
305#447—1221no—
306#447—1221no—
307C4×Q16C2×D6111yescharacteristic, complemented
308C4×D8—611no—
309C8×D4—611no—
310#427—611no—
311C2×#107—611no—
312#367—611no—
313#366—611no—
314#366—611no—
315C4⋊C16C2×D6111yescharacteristic, complemented
316#358—611no—
317#357—611no—
318#357—611no—
319#357—611no—
320#357—611no—
321#357—611no—
322#357—611no—
323#357—611no—
324#357C2×D6111yescharacteristic, complemented
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—21no—
335C2×D8—611no—
336C2×D8—1221no—
337C2×D8—1221no—
338C2×D8—1222no—
339C2×D8—1222no—
340C2×D8—2441no—
341C2×D8—2441no—
342C2×D8C22×D6111yescharacteristic
343C2×D8C22×D6111yescharacteristic
344C22×C8—611no—
345C22×C8—611no—
346#127—21no—
347#126—2441no—
348#125—1221no—
349#125—2441no—
350#123—2441no—
351C22⋊Q8—1221no—
352C22⋊Q8—1221no—
353#121—1221no—
354#121—1221no—
355#121—2441no—
356#120—2441no—
357#120—2441no—
358C4×Q8—21no—
359C4×D4—611no—
360C4×D4—1221no—
361#117—1221no—
362#117—2441no—
363C2×C4⋊C4—1221no—
364SD32—41no—
365SD32—41no—
366SD32—1222no—
367M32—2441no—
368C2×C16—611no—
369C2×C16—611no—
370C2×C16—2441no—
371C2×C16C22×D6111yescharacteristic
372C2×C16C22×D6111yescharacteristic
373#107—611no—
374#107—611no—
375#107—611no—
376#107—1221no—
377#107C22×D6111yescharacteristic
378#106—1221no—
379#106—1221no—
380C4⋊C8—611no—
381#103—21no—
382#103—1221no—
383#103—1221no—
384#102—1221no—
385#102—1221no—
386#102—1221no—
387C22⋊C8—611no—
388C22⋊C8—611no—
389C4×C8C22×D6111yescharacteristic
390C24—2441no—
391C24—2441no—
392C2×Q8—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—611no—
398C2×D4—1221no—
399C2×D4—4881no—
400C2×D4—4881no—
401C2×D4—4884no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4—2442no—
405C2×D4—2442no—
406C2×D4—2441no—
407C2×D4—2441no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—2441no—
411C22×C4—2441no—
412Q16—21no—
413Q16—21no—
414D8—21no—
415D8—21no—
416D8—21no—
417D8—21no—
418D8—81no—
419D8—612no—
420D8—1221no—
421D8—2442no—
422D8—2442no—
423C2×C8—611no—
424C2×C8—611no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1221no—
429C2×C8D4×D6111yescharacteristic
430C2×C8C23×D6111yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—41no—
433C4⋊C4—611no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C4⋊C4—2441no—
442C22⋊C4—611no—
443C22⋊C4—611no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C42—41no—
451C42—2441no—
452C42D4×D6111yescharacteristic
453C16—21no—
454C16—21no—
455C16—1221no—
456C16—1221no—
457C23—41no—
458C23—41no—
459C23—611no—
460C23—611no—
461C23—9761no—
462C23—9761no—
463C23—4882no—
464C23—4882no—
465C23—4882no—
466C23—4882no—
467C23—2441no—
468Q8—41no—
469Q8—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—82no—
475D4—161no—
476D4—1222no—
477D4—1222no—
478D4—4882no—
479D4—4882no—
480D4—4882no—
481D4—4882no—
482D4—2441no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—611no—
486C2×C4—611no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—4882no—
492C2×C4—4882no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4D8×D6111yescharacteristic
497C2×C4D8×D6111yescharacteristic
498C2×C4C2×D4×D6111yescharacteristic
499C8—21no—
500C8—1221no—
501C8—1221no—
502C8#41×D6111yescharacteristic
503C8#1451982211yescharacteristic
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—612no—
511C22—1221no—
512C22—1221no—
513C22—1221no—
514C22—1221no—
515C22—1221no—
516C22—9764no—
517C22—9764no—
518C22—4881no—
519C22—4881no—
520C22—2442no—
521C22C2×D8×D6111yescharacteristic
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—1221no—
527C4—1221no—
528C4—4881no—
529C4—4881no—
530C4—4881no—
531C4#1684040911yescharacteristic
532C4#1684057211yescharacteristic
533C2—161no—
534C2—161no—
535C2—1222no—
536C2—4882no—
537C2—2441no—
538C2#2145763511yescharacteristic
539C2#509×D6111yescharacteristic
540C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2207C2
socleC22×C61C22×D8
prime radical#21457494C2
hypercenter#2207D61
hypocenterC61#33877
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #21458189C2semidirect
3. #21448060C2semidirect
4. #21448009C2semidirect
5. #21447998C2semidirect
6. #21447977C2semidirect
7. #21447929C2semidirect
8. #21447912C2semidirect
9. #21447746C2semidirect
10. #21447746C2semidirect
11. #21447681C2semidirect
12. #21447643C2semidirect
13. #21447533C2semidirect
14. #357×D61C2semidirect
15. C61×#2207C2semidirect
21. C2×#14519716C22semidirect
22. C2×C16×D61C22semidirect
23. #16840164C22semidirect
27. #16840000C22semidirect
28. #16840000C22semidirect
36. #16839799C22semidirect
37. #107×D61C22semidirect
38. #16839590C22semidirect
39. #16839588C22semidirect
40. #16839587C22semidirect
41. #16839566C22semidirect
44. #16839563C22semidirect
45. #16839563C22semidirect
46. #16839563C22semidirect
47. #16839563C22semidirect
49. #16839541C22semidirect
50. #16839541C22semidirect
51. #16839538C22semidirect
52. C2×SD32×C61C22semidirect
53. C2×SD32×C61C22semidirect
55. C4×Q16×C61C22semidirect
56. C61×C4⋊C16C22semidirect
57. C61×#357C22semidirect
58. C61×#357C22semidirect
271. #2207D61semidirect
291. C2×SD32C2×D61semidirect
292. C2×SD32C2×D61semidirect
307. C4×Q16C2×D61semidirect
315. C4⋊C16C2×D61semidirect
324. #357C2×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
#214581892
#214574942
#214480602
#214480092
#214479982
#214479772
#214479292
#214479122
#214477462
#214476812
#214476432
#214475332
#357×D612
C61×#22072
#3387761