Finite Group Databasebeta

← #29835120

Subgroups — #29835120

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456455C211yesmaximal, characteristic, complemented
2#21454666C211yesmaximal, characteristic, complemented
3#21448025C211yesmaximal, characteristic, complemented
4#21448009C211yesmaximal, characteristic, complemented
5#21447967C211yesmaximal, characteristic, complemented
6#21447956C211yesmaximal, characteristic, complemented
7#21447871C211yesmaximal, characteristic, complemented
8#21447746C211yesmaximal, characteristic, complemented
9#21447729C211yesmaximal, characteristic
10#21447695C211yesmaximal, characteristic, complemented
11#21447686C211yesmaximal, characteristic, complemented
12SD32×Dic61C211yesmaximal, characteristic, complemented
13#21447663C211yesmaximal, characteristic, complemented
14#21447625C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840381—21no—
18#16840234—21no—
19C2×#14519720C2211yescharacteristic
20C2×#14519718C2211yescharacteristic
21#16840167C2211yescharacteristic, complemented
22#16840121—21no—
23#16840005—21no—
24#16840000C2211yescharacteristic
25#16839995C2211yescharacteristic
26#16839872—21no—
27#16839871—21no—
28#16839863—21no—
29Q16×Dic61C2211yescharacteristic, complemented
30C4×D488C2211yescharacteristic
31D8×Dic61C2211yescharacteristic
32D8×Dic61C2211yescharacteristic, complemented
33#16839844C2211yescharacteristic
34#16839799C2211yescharacteristic
35#16839765—21no—
36#16839716C2211yescharacteristic
37#16839590C2211yescharacteristic, complemented
38#16839590C2211yescharacteristic, complemented
39#16839587C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41C4⋊#11758185C2211yescharacteristic, complemented
42#16839566C2211yescharacteristic, complemented
43#16839564C2211yescharacteristic
44#16839564C2211yescharacteristic
45#16839563C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic, complemented
48#16839541C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839537C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C2×SD32×C61C2211yescharacteristic
53C2×D16×C61C2211yescharacteristic
54C61×#495C2211yescharacteristic, complemented
55C4×D8×C61C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic, complemented
57C61×#358C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic, complemented
59C2×D4×D61—21no—
60C2×D488C2311yescharacteristic
61C2×#11758199—21no—
62D8×D61—22no—
63D8×D61—22no—
64C2×#11758189D411yescharacteristic
65C2×C8×D61C2311yescharacteristic
66#14519777—21no—
67#14519770—41no—
68#14519763—41no—
69#14519761—41no—
70#14519754—41no—
71#14519753—21no—
72#14519748—21no—
73Q8×Dic61—21no—
74C4×D244—21no—
75D4×Dic61—21no—
76D4×Dic61—21no—
77#14519739—21no—
78C4×C22⋊D61—41no—
79#14519737—41no—
80#14519735—21no—
81#14519734D411yescharacteristic
82C2×#11758198—41no—
83C22⋊C4×D61—41no—
84#14519720—22no—
85#14519718—22no—
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88#14519711—21no—
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519708—21no—
93#14519701—21no—
94#14519701—21no—
95#14519699—21no—
96#14519698—21no—
97#14519698—21no—
98#14519698—21no—
99#14519697—21no—
100#14519688D411yescharacteristic
101#14519687D411yescharacteristic
102C8×Dic61C2311yescharacteristic
103C2×Q16×C61C2311yescharacteristic
104C2×SD16×C61—21no—
105C2×D8×C61C2311yescharacteristic
106C2×D8×C61C2311yescharacteristic
107#124×C61—21no—
108C4×D4×C61—21no—
109SD32×C61—22no—
110D16×C61—22no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C23×D61—41no—
117C2×Q8×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C22×C4×C61—41no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C22×Dic61—41no—
124C22×Dic61—41no—
125D4×D61—24no—
126D4×D61—42no—
127C2×D244—21no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61C2×D411yescharacteristic
131C2×Dic122—41no—
132Q16×C61—21no—
133Q16×C61—21no—
134SD16×C61—42no—
135D8×C61—21no—
136D8×C61—21no—
137D8×C61—21no—
138D8×C61#4112yes—
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2411yescharacteristic
141C4⋊C4×C61—21no—
142C22⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C42×C61C2×D411yescharacteristic
146C22⋊Dic61—41no—
147C22⋊Dic61—41no—
148C22⋊Dic61—41no—
149#11758199—24no—
150#11758198—41no—
151#11758198—41no—
152#11758198—41no—
153#11758198—41no—
154#11758198D811yescharacteristic
155#11758198D811yescharacteristic
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—41no—
160C4⋊Dic61—41no—
161#11758196—41no—
162#11758196D811yescharacteristic
163#11758196D811yescharacteristic
164C4×Dic61—41no—
165C4×Dic61—41no—
166C4×Dic61C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168C2×#92418C2×D411yescharacteristic
169D488—21no—
170D488#4112yes—
171#11758189—41no—
172C8×D61—21no—
173C8×D61—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C23×C61—41no—
180C22×D61—44no—
181C22×D61—41no—
182C22×D61—161no—
183C22×D61—161no—
184C22×D61C2×D811yescharacteristic
185Q8×C61—41no—
186Q8×C61—41no—
187D4×C61—22no—
188D4×C61—41no—
189D4×C61—41no—
190D4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—161no—
194C2×C4×C61C2×D811yescharacteristic
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C22⋊D61—44no—
198C22⋊D61—161no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—161no—
203C2×Dic61—161no—
204C2×Dic61C2×D811yescharacteristic
205C2×Dic61C2×D811yescharacteristic
206C2×Dic61C2×D811yescharacteristic
207D244—22no—
208D244—41no—
209C4×D61—21no—
210C4×D61—21no—
211C4×D61—161no—
212C4×D61—161no—
213Dic122—161no—
214C8×C61—21no—
215C8×C61C2×#4111yescharacteristic
216C8×C61C2×#4111yescharacteristic
217#92418—21no—
218#92418—21no—
219#33678—611nomaximal, Sylow
220C22×C61—42no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61C22×D811yescharacteristic
227C2×D61—21no—
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—42no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—164no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61#52511yescharacteristic
239C4×C61#57511yescharacteristic
240Dic61—21no—
241Dic61—21no—
242Dic61—21no—
243Dic61—81no—
244Dic61—81no—
245Dic61—81no—
246#3175—611no—
247#2948—611no—
248#2237—611no—
249#2234—611no—
250#2226—611no—
251#2224—611no—
252#2210—611no—
253#2210D6111yescharacteristic, complemented
254#2187—611no—
255#2184—611no—
256#2177—611no—
257#2175—611no—
258C4×SD32—611no—
259C4×D16—611no—
260#2164—611no—
261C2×C61—82no—
262C2×C61—161no—
263C2×C61#310011yescharacteristic
264C2×C61#341511yescharacteristic
265C2×C61#341611yescharacteristic
266D61—41no—
267D61—82no—
268C22×D8—611no—
269#542—1221no—
270#518—1221no—
271C2×SD32—611no—
272C2×SD32C2×D6111yescharacteristic
273C2×D16—611no—
274C2×D16C2×D6111yescharacteristic
275#500—611no—
276#495C2×D6111yescharacteristic, complemented
277#489—1221no—
278#468—1221no—
279#467—611no—
280#466—611no—
281#442—1221no—
282#442—1221no—
283#440—1221no—
284C4×Q16—611no—
285C4×D8—611no—
286C4×D8—611no—
287C4×D8—611no—
288C4×D8—611no—
289C4×D8C2×D6111yescharacteristic
290#427—611no—
291#418—1221no—
292#405—611no—
293#367—611no—
294#367—611no—
295#366—611no—
296#366—611no—
297C4⋊C16—611no—
298C4⋊C16C2×D6111yescharacteristic, complemented
299#358—611no—
300#358C2×D6111yescharacteristic, complemented
301#357—611no—
302#357—611no—
303#357—611no—
304#357—611no—
305#357—611no—
306#357C2×D6111yescharacteristic, complemented
307C22⋊C16—611no—
308C22⋊C16—611no—
309#346—611no—
310C4×C16—611no—
311C61#3367811yescharacteristic, complemented, Sylow
312C22×D4—1221no—
313C2×Q16C22×D6111yescharacteristic
314C2×SD16—21no—
315C2×D8—611no—
316C2×D8—1221no—
317C2×D8—1222no—
318C2×D8—1222no—
319C2×D8C22×D6111yescharacteristic
320C2×D8C22×D6111yescharacteristic
321C2×M16—611no—
322C22×C8—611no—
323C4⋊Q8—1221no—
324#126—2441no—
325#124—21no—
326#124—2441no—
327#123—2441no—
328C22⋊Q8—1221no—
329C22⋊Q8—2441no—
330#121—1221no—
331C4×Q8—1221no—
332C4×D4—21no—
333C4×D4—1221no—
334C4×D4—1221no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4—2441no—
338C4×D4—2441no—
339#117—611no—
340#117—1221no—
341C2×C22⋊C4—2441no—
342C2×C22⋊C4—2441no—
343SD32—22no—
344SD32—1222no—
345D16—22no—
346D16—1222no—
347C2×C16—611no—
348C2×C16—611no—
349C2×C16C22×D6111yescharacteristic
350C2×C16C22×D6111yescharacteristic
351#107—611no—
352#107—611no—
353#107—611no—
354#107—1221no—
355#107C22×D6111yescharacteristic
356#106—1221no—
357#103—1221no—
358#103—1221no—
359#102—21no—
360#102—1221no—
361#102—1221no—
362#102—1221no—
363#102—1221no—
364#102—1221no—
365#97—611no—
366#97—611no—
367C4×C8—611no—
368C4×C8C22×D6111yescharacteristic
369C24—2441no—
370C2×Q8—21no—
371C2×Q8—2441no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—1224no—
375C2×D4—1221no—
376C2×D4—2441no—
377C2×D4—2442no—
378C2×D4—2441no—
379C22×C4—41no—
380C22×C4—611no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4—2441no—
385Q16—21no—
386Q16—21no—
387SD16—42no—
388D8—21no—
389D8—21no—
390D8—21no—
391D8—612no—
392D8—1224no—
393D8—1221no—
394D8#1175822412yes—
395M16—2441no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8D4×D6111yescharacteristic
401C2×C8C23×D6111yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—611no—
404C4⋊C4—611no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—2441no—
409C4⋊C4—2441no—
410C4⋊C4—2441no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—611no—
415C22⋊C4—611no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C22⋊C4—2441no—
419C22⋊C4—2441no—
420C22⋊C4—2441no—
421C22⋊C4—2441no—
422C22⋊C4—2441no—
423C42—611no—
424C42—2441no—
425C42—2441no—
426C42D4×D6111yescharacteristic
427C16—21no—
428C16—21no—
429C16—1221no—
430C16—1221no—
431C23—41no—
432C23—41no—
433C23—611no—
434C23—9761no—
435C23—9761no—
436C23—2444no—
437C23—2441no—
438Q8—41no—
439Q8—41no—
440Q8—9761no—
441D4—22no—
442D4—41no—
443D4—41no—
444D4—41no—
445D4—1222no—
446D4—9761no—
447D4—2444no—
448D4—2441no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—161no—
452C2×C4—611no—
453C2×C4—611no—
454C2×C4—611no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—9761no—
458C2×C4—9761no—
459C2×C4—9761no—
460C2×C4—9761no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4D8×D6111yescharacteristic
465C2×C4D8×D6111yescharacteristic
466C2×C4C2×D4×D6111yescharacteristic
467C8—21no—
468C8—1221no—
469C8—1221no—
470C8#41×D6111yescharacteristic
471C8C2×#1175822411yescharacteristic
472C22—42no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—1221no—
479C22—1221no—
480C22—1221no—
481C22—9764no—
482C22—4881no—
483C22—4881no—
484C22—2442no—
485C22C2×D8×D6111yescharacteristic
486C4—21no—
487C4—21no—
488C4—81no—
489C4—81no—
490C4—1221no—
491C4—1221no—
492C4—1221no—
493C4—4881no—
494C4—4881no—
495C4—4881no—
496C4#1684027211yescharacteristic
497C4#1684055811yescharacteristic
498C2—82no—
499C2—161no—
500C2—4882no—
501C2—2441no—
502C2#2145581811yescharacteristic
503C2#2145895911yescharacteristic
504C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
prime radical#21447729C2
hypercenter#2210D61
hypocenterC61#33678
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. #21456455C2semidirect
2. #21454666C2semidirect
3. #21448025C2semidirect
4. #21448009C2semidirect
5. #21447967C2semidirect
6. #21447956C2semidirect
7. #21447871C2semidirect
8. #21447746C2semidirect
10. #21447695C2semidirect
11. #21447686C2semidirect
12. SD32×Dic61C2semidirect
13. #21447663C2semidirect
14. #21447625C2semidirect
15. C61×#2210C2semidirect
21. #16840167C22semidirect
29. Q16×Dic61C22semidirect
32. D8×Dic61C22semidirect
37. #16839590C22semidirect
38. #16839590C22semidirect
39. #16839587C22semidirect
40. #16839587C22semidirect
41. C4⋊#11758185C22semidirect
42. #16839566C22semidirect
47. #16839563C22semidirect
50. #16839537C22semidirect
51. C16×Dic61C22semidirect
54. C61×#495C22semidirect
56. C61×C4⋊C16C22semidirect
57. C61×#358C22semidirect
58. C61×#357C22semidirect
253. #2210D61semidirect
276. #495C2×D61semidirect
298. C4⋊C16C2×D61semidirect
300. #358C2×D61semidirect
306. #357C2×D61semidirect
311. C61#33678semidirect

Maximal subgroups

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

subgroupindex
#214564552
#214546662
#214480252
#214480092
#214479672
#214479562
#214478712
#214477462
#214477292
#214476952
#214476862
SD32×Dic612
#214476632
#214476252
C61×#22102
#3367861