Finite Group Databasebeta

← #29834983

Subgroups — #29834983

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456386C211yesmaximal, characteristic, complemented
2C4×D8×D61C211yesmaximal, characteristic, complemented
3#21448020C211yesmaximal, characteristic, complemented
4#21448003C211yesmaximal, characteristic, complemented
5#21447977C211yesmaximal, characteristic, complemented
6#21447976C211yesmaximal, characteristic, complemented
7#21447871C211yesmaximal, characteristic, complemented
8#21447746C211yesmaximal, characteristic, complemented
9#21447729C211yesmaximal, characteristic
10SD32×Dic61C211yesmaximal, characteristic, complemented
11#21447668C211yesmaximal, characteristic, complemented
12D16×Dic61C211yesmaximal, characteristic, complemented
13#21447663C211yesmaximal, characteristic, complemented
14#21447617C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840291—21no—
18C4×D4×D61—21no—
19C2×#14519720C2211yescharacteristic
20C2×#14519718C2211yescharacteristic
21#16840164C2211yescharacteristic, complemented
22#16840116—21no—
23#16840000C2211yescharacteristic
24#16839995C2211yescharacteristic
25#16839988—21no—
26Q16×Dic61C2211yescharacteristic, complemented
27SD16×Dic61—21no—
28C4×D488C2211yescharacteristic
29D8×Dic61C2211yescharacteristic
30D8×Dic61C2211yescharacteristic, complemented
31#16839844C2211yescharacteristic
32C4×#11758199—21no—
33#16839842—21no—
34#107×D61C2211yescharacteristic
35#102×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839590C2211yescharacteristic, complemented
38#16839589C2211yescharacteristic, complemented
39#16839588C2211yescharacteristic, 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
50C16×Dic61C2211yescharacteristic, 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—
60C22×C4×D61—41no—
61C2×D488C2311yescharacteristic
62C2×#11758199—21no—
63D8×D61—22no—
64D8×D61—22no—
65C2×C8×D61D411yescharacteristic
66C2×C8×D61C2311yescharacteristic
67#14519770—41no—
68#14519769—21no—
69#14519759—41no—
70#14519753—21no—
71#14519748—21no—
72Q8×Dic61—21no—
73C4×D244—21no—
74D4×Dic61—21no—
75D4×Dic61—21no—
76#14519739—21no—
77C4×C22⋊D61—41no—
78#14519737—41no—
79#14519737—41no—
80#14519733—41no—
81C4⋊C4×D61—21no—
82C22⋊C4×D61—41no—
83C42×D61D411yescharacteristic
84#14519720—22no—
85#14519718—22no—
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519710—21no—
92#14519709—21no—
93#14519701—21no—
94#14519701—21no—
95#14519699—21no—
96#14519698—21no—
97#14519698—21no—
98#14519698—21no—
99#14519697—21no—
100C4×#92418D411yescharacteristic
101C8×Dic61D411yescharacteristic
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×D61—41no—
131C2×C4×D61—84no—
132C2×C4×D61D811yescharacteristic
133C2×C4×D61D811yescharacteristic
134C2×C4×D61C2×D411yescharacteristic
135Q16×C61—21no—
136Q16×C61—21no—
137SD16×C61—42no—
138D8×C61—21no—
139D8×C61—21no—
140D8×C61—21no—
141D8×C61#4112yes—
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2411yescharacteristic
144C4⋊C4×C61—21no—
145C22⋊C4×C61—41no—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C42×C61C2×D411yescharacteristic
149C22⋊Dic61—41no—
150C22⋊Dic61—41no—
151C22⋊Dic61—41no—
152#11758199—24no—
153#11758198—41no—
154#11758198—41no—
155#11758198—41no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—41no—
160#11758196—41no—
161#11758196—41no—
162#11758196—41no—
163#11758196—41no—
164C4×Dic61—41no—
165C4×Dic61D811yescharacteristic
166C4×Dic61D811yescharacteristic
167C4×Dic61C2×D411yescharacteristic
168C2×#92418C2×D411yescharacteristic
169C2×#92418C2×D411yescharacteristic
170D488—21no—
171D488#4112yes—
172C8×D61—21no—
173C8×D61—21no—
174C8×D61—41no—
175C16×C61—21no—
176C16×C61—21no—
177#11758185—21no—
178#11758185—21no—
179C23×C61—41no—
180C23×C61—41no—
181C22×D61—44no—
182C22×D61—41no—
183C22×D61—82no—
184C22×D61—82no—
185C22×D61C2×D811yescharacteristic
186Q8×C61—41no—
187Q8×C61—41no—
188D4×C61—22no—
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—82no—
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—44no—
199C22⋊D61—82no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—82no—
204C2×Dic61—82no—
205C2×Dic61C2×D811yescharacteristic
206C2×Dic61C2×D811yescharacteristic
207C2×Dic61C2×D811yescharacteristic
208D244—22no—
209D244—41no—
210C4×D61—22no—
211C4×D61—21no—
212C4×D61—21no—
213C4×D61—22no—
214C4×D61—82no—
215C4×D61—82no—
216C4×D61—82no—
217C8×C61—21no—
218C8×C61C2×#4111yescharacteristic
219C8×C61C2×#4111yescharacteristic
220#92418—21no—
221#92418—21no—
222#33665—611nomaximal, Sylow
223C22×C61—42no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61C22×D811yescharacteristic
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—42no—
233C2×D61—88no—
234C2×D61—81no—
235C2×D61—81no—
236C2×D61#50812yes—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—81no—
240C4×C61—81no—
241C4×C61#52511yescharacteristic
242C4×C61#57511yescharacteristic
243Dic61—21no—
244Dic61—21no—
245Dic61—81no—
246Dic61—81no—
247Dic61—81no—
248Dic61#50912yes—
249#3168—611no—
250C2×C4×D8—611no—
251#2236—611no—
252#2233—611no—
253#2228—611no—
254#2228—611no—
255#2210—611no—
256#2210D6111yescharacteristic, complemented
257#2187—611no—
258#2184—611no—
259C4×SD32—611no—
260C4×SD32—611no—
261C4×D16—611no—
262C4×D16—611no—
263#2162—611no—
264C2×C61—82no—
265C2×C61—161no—
266C2×C61#310011yescharacteristic
267C2×C61C2×#50811yescharacteristic
268C2×C61C2×#50911yescharacteristic
269D61—22no—
270D61—82no—
271C22×D8—611no—
272#529—1221no—
273C2×C4×D4—1221no—
274C2×SD32—611no—
275C2×SD32C2×D6111yescharacteristic
276C2×D16—611no—
277C2×D16C2×D6111yescharacteristic
278#499—611no—
279#495C2×D6111yescharacteristic, complemented
280#488—1221no—
281#467—611no—
282#466—611no—
283#465—1221no—
284C4×Q16—611no—
285C4×SD16—1221no—
286C4×D8—611no—
287C4×D8—611no—
288C4×D8—611no—
289C4×D8—611no—
290C4×D8—1221no—
291C4×D8—1221no—
292C4×D8C2×D6111yescharacteristic
293C2×#107—611no—
294C2×#102—1221no—
295C2×C4×C8—611no—
296#367—611no—
297#367—611no—
298#366—611no—
299#366—611no—
300C4⋊C16—611no—
301C4⋊C16C2×D6111yescharacteristic, complemented
302#358—611no—
303#358C2×D6111yescharacteristic, complemented
304#357—611no—
305#357—611no—
306#357—611no—
307#357—611no—
308#357—611no—
309#357C2×D6111yescharacteristic, complemented
310C22⋊C16—611no—
311C22⋊C16—611no—
312C4×C16—611no—
313C4×C16—611no—
314C61#3366511yescharacteristic, complemented, Sylow
315C22×D4—1221no—
316C23×C4—2441no—
317C2×Q16C22×D6111yescharacteristic
318C2×SD16—21no—
319C2×D8—611no—
320C2×D8—1221no—
321C2×D8—1222no—
322C2×D8—1222no—
323C2×D8C22×D6111yescharacteristic
324C2×D8C22×D6111yescharacteristic
325C22×C8—611no—
326C22×C8—611no—
327#126—2441no—
328#125—1221no—
329#124—21no—
330#123—2441no—
331C22⋊Q8—1221no—
332#121—1221no—
333C4×Q8—1221no—
334C4×D4—21no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4—1221no—
338C4×D4—1221no—
339C4×D4—2441no—
340C4×D4—2441no—
341C4×D4—2441no—
342#117—2441no—
343C2×C4⋊C4—1221no—
344C2×C22⋊C4—2441no—
345C2×C42—611no—
346SD32—22no—
347SD32—1222no—
348D16—22no—
349D16—1222no—
350C2×C16—611no—
351C2×C16—611no—
352C2×C16C22×D6111yescharacteristic
353C2×C16C22×D6111yescharacteristic
354#107—611no—
355#107—611no—
356#107—611no—
357#107—1221no—
358#107C22×D6111yescharacteristic
359#106—1221no—
360#103—1221no—
361#103—1221no—
362#102—21no—
363#102—1221no—
364#102—1221no—
365#102—1221no—
366#102—1221no—
367#102—1221no—
368C4×C8—611no—
369C4×C8—611no—
370C4×C8—611no—
371C4×C8C22×D6111yescharacteristic
372C24—2441no—
373C2×Q8—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—1224no—
377C2×D4—1221no—
378C2×D4—2441no—
379C2×D4—2442no—
380C2×D4—2441no—
381C22×C4—41no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—4884no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4—2441no—
390C22×C4—2441no—
391Q16—21no—
392Q16—21no—
393SD16—42no—
394D8—21no—
395D8—21no—
396D8—21no—
397D8—612no—
398D8—1224no—
399D8—1221no—
400D8#1175822412yes—
401C2×C8—611no—
402C2×C8—611no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—2441no—
406C2×C8D4×D6111yescharacteristic
407C2×C8C23×D6111yescharacteristic
408C4⋊C4—21no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4—2441no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—2441no—
421C22⋊C4—2441no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C42—611no—
427C42—611no—
428C42—611no—
429C42—2441no—
430C42D4×D6111yescharacteristic
431C16—21no—
432C16—21no—
433C16—1221no—
434C16—1221no—
435C23—41no—
436C23—41no—
437C23—611no—
438C23—4882no—
439C23—4882no—
440C23—2444no—
441C23—2441no—
442Q8—41no—
443Q8—41no—
444D4—22no—
445D4—41no—
446D4—41no—
447D4—41no—
448D4—1222no—
449D4—4882no—
450D4—2444no—
451D4—2441no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—82no—
455C2×C4—611no—
456C2×C4—611no—
457C2×C4—611no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—4882no—
463C2×C4—4882no—
464C2×C4—4882no—
465C2×C4—4882no—
466C2×C4—4882no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4D8×D6111yescharacteristic
471C2×C4D8×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C8—21no—
474C8—1221no—
475C8—1221no—
476C8#41×D6111yescharacteristic
477C8C2×#1175822411yescharacteristic
478C22—42no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—612no—
485C22—1221no—
486C22—1221no—
487C22—4888no—
488C22—4881no—
489C22—4881no—
490C22—2442no—
491C22C2×D8×D6111yescharacteristic
492C4—21no—
493C4—21no—
494C4—81no—
495C4—81no—
496C4—612no—
497C4—1221no—
498C4—1221no—
499C4—4881no—
500C4—4881no—
501C4—4881no—
502C4#1684027211yescharacteristic
503C4#1684055811yescharacteristic
504C2—82no—
505C2—161no—
506C2—1222no—
507C2—4882no—
508C2#2145581811yescharacteristic
509C2#508×D6111yescharacteristic
510C2#509×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
prime radical#21447729C2
hypercenter#2210D61
hypocenterC61#33665
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. #21456386C2semidirect
2. C4×D8×D61C2semidirect
3. #21448020C2semidirect
4. #21448003C2semidirect
5. #21447977C2semidirect
6. #21447976C2semidirect
7. #21447871C2semidirect
8. #21447746C2semidirect
10. SD32×Dic61C2semidirect
11. #21447668C2semidirect
12. D16×Dic61C2semidirect
13. #21447663C2semidirect
14. #21447617C2semidirect
15. C61×#2210C2semidirect
21. #16840164C22semidirect
26. Q16×Dic61C22semidirect
30. D8×Dic61C22semidirect
37. #16839590C22semidirect
38. #16839589C22semidirect
39. #16839588C22semidirect
40. #16839587C22semidirect
41. C4⋊#11758185C22semidirect
42. #16839566C22semidirect
47. #16839563C22semidirect
50. C16×Dic61C22semidirect
51. C16×Dic61C22semidirect
54. C61×#495C22semidirect
56. C61×C4⋊C16C22semidirect
57. C61×#358C22semidirect
58. C61×#357C22semidirect
256. #2210D61semidirect
279. #495C2×D61semidirect
301. C4⋊C16C2×D61semidirect
303. #358C2×D61semidirect
309. #357C2×D61semidirect
314. C61#33665semidirect

Maximal subgroups

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

subgroupindex
#214563862
C4×D8×D612
#214480202
#214480032
#214479772
#214479762
#214478712
#214477462
#214477292
SD32×Dic612
#214476682
D16×Dic612
#214476632
#214476172
C61×#22102
#3366561