Finite Group Databasebeta

← #30552373

Subgroups — #30552373

Order 15616.

1754 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459222C211yesmaximal, characteristic, complemented
2C2×#16840368C211yesmaximal, characteristic
3#21457075C218yesmaximal, complemented
4#21457073C218yesmaximal, complemented
5#21456368C211yesmaximal, characteristic, complemented
6#21456195C211yesmaximal, characteristic, complemented
7#21456194C211yesmaximal, characteristic, complemented
8#21455761C214yesmaximal, complemented
9#21455615C214yesmaximal, complemented
10#21454455C211yesmaximal, characteristic, complemented
11C61×#3166C211yesmaximal, characteristic, complemented
12#16840392C2214yescomplemented
13#16840391—22no—
14#16840389C2214yescomplemented
15#16840387—22no—
16#16840368—24no—
17#16840368C2214yes—
18#16840368C2214yes—
19#16840319C2211yescharacteristic
20#16840316—24no—
21#16840315C2212yes—
22C2×#14519774C2211yescharacteristic
23C2×#14519762—22no—
24C2×#14519752—22no—
25C2×#14519752C2212yes—
26C2×#14519739C2212yes—
27C2×#14519734C2211yescharacteristic
28#117×D61C2212yes—
29#16840172C2214yescomplemented
30#16840161C2214yescomplemented
31#16840128C2214yescomplemented
32#16840127C2214yescomplemented
33#16840110C2214yescomplemented
34#16840109C2214yescomplemented
35#16840089C2218yescomplemented
36#16840088C2218yescomplemented
37#16840082C2218yescomplemented
38#16840077C2218yescomplemented
39#16840065C2218yescomplemented
40#16840061C2218yescomplemented
41#16840051C2218yescomplemented
42#16840041C2218yescomplemented
43#16839824C2214yescomplemented
44#16839821C2214yescomplemented
45#16839802C2214yes—
46#16839785C2212yes—
47#16839768C2214yes—
48#16839766C2214yes—
49#16839725C2212yes—
50C4×#11758194C2211yescharacteristic
51C61×#533C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C61×#501C2214yescomplemented
54C61×#499C2214yescomplemented
55C61×#428C2214yes—
56C4×M16×C61C2211yescharacteristic
57C22×D244—22no—
58C22×D244—22no—
59C22×D244C2312yes—
60C22×C4×D61—22no—
61C22×C4×D61C2311yescharacteristic
62C2×#11758194C2312yes—
63#14519777C2312yes—
64#14519776—24no—
65#14519776C2314yes—
66#14519775—24no—
67#14519774—22no—
68#14519774C2312yes—
69#14519774C2312yes—
70#14519769C2312yes—
71#14519768C2314yes—
72#14519767—24no—
73#14519767—24no—
74#14519762—28no—
75#14519762—28no—
76#14519757—24no—
77#14519756—22no—
78#14519755—24no—
79#14519755—24no—
80#14519754—24no—
81#14519753—24no—
82#14519752—24no—
83#14519752—28no—
84#14519752—28no—
85#14519752C2318yes—
86C4×Dic122—22no—
87C4×Dic122—24no—
88#14519739—24no—
89#14519739C2318yes—
90#14519736C2312yes—
91#14519735C2314yes—
92#14519734—22no—
93#14519734D414yes—
94#14519734C2312yes—
95#14519734C2312yes—
96#14519733—22no—
97C2×#11758196D412yes—
98C4⋊C4×D61C2314yes—
99C2×#11758198—22no—
100C2×#11758198—22no—
101C2×#11758198D412yes—
102C22⋊C4×D61—22no—
103C2×C4×Dic61C2311yescharacteristic
104C42×D61—22no—
105#14519710C2318yes—
106#14519708C2318yes—
107#14519706C2318yes—
108#14519699C2318yes—
109#14519699C2318yes—
110#14519697C2318yes—
111#14519697C2318yes—
112#14519689C2318yes—
113#14519688C2312yes—
114C4×#92418C2312yes—
115C2×M16×C61C2312yes—
116C4⋊Q8×C61C2312yes—
117#125×C61—22no—
118#125×C61C2312yes—
119#125×C61C2312yes—
120#125×C61C2312yes—
121C22⋊Q8×C61—22no—
122C4×Q8×C61—22no—
123#117×C61C2312yes—
124C2×C4⋊C4×C61—22no—
125C2×C4⋊C4×C61C2312yes—
126C2×C42×C61C2311yescharacteristic
127#107×C61C2318yes—
128#106×C61C2318yes—
129#97×C61C2312yes—
130C4×C8×C61C2312yes—
131C23×D61—22no—
132C23×D61C2×D411yescharacteristic
133C2×Q8×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D412yes—
139C2×D244—28no—
140C2×D244—22no—
141C2×D244—28no—
142C2×D244—24no—
143C2×D244—24no—
144C2×D244—84no—
145C2×D244—84no—
146C2×D244C2414yes—
147C2×C4×D61—24no—
148C2×C4×D61—24no—
149C2×C4×D61—28no—
150C2×C4×D61—84no—
151C2×C4×D61C2×D412yes—
152C2×C4×D61C2412yes—
153C2×Dic122—22no—
154C2×Dic122—24no—
155M16×C61—24no—
156C2×C8×C61C2414yes—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61C2414yes—
163C4⋊C4×C61C2414yes—
164C22⋊C4×C61—22no—
165C42×C61—22no—
166C42×C61C2×D412yes—
167C42×C61C2411yescharacteristic
168C42×C61C2411yescharacteristic
169C22⋊Dic61—22no—
170#11758198—24no—
171#11758198—28no—
172#11758198—28no—
173#11758198C2×D414yes—
174#11758198C2×D414yes—
175C4⋊Dic61—22no—
176C4⋊Dic61—24no—
177C4⋊Dic61C2414yes—
178#11758196—24no—
179#11758196—24no—
180#11758196—24no—
181#11758196—24no—
182#11758196C2×D414yes—
183#11758196C2×D414yes—
184C4×Dic61—22no—
185C4×Dic61—24no—
186C4×Dic61C2×D412yes—
187C4×Dic61C2412yes—
188#11758194—24no—
189C2×#92418C2414yes—
190C23×C61C22×D411yescharacteristic
191C22×D61—22no—
192C22×D61—24no—
193C22×D61—42no—
194C22×D61—44no—
195C22×D61—44no—
196C22×D61—44no—
197C22×D61C22×D412yes—
198Q8×C61—82no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—42no—
202C2×C4×C61—44no—
203C2×C4×C61C22×D412yes—
204C2×C4×C61C22×D412yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C612−1+414yes—
207C2×C4×C61C2511yescharacteristic
208C2×Dic61—22no—
209C2×Dic61—24no—
210C2×Dic61—42no—
211C2×Dic61C22×D412yes—
212C2×Dic61C22×D414yes—
213D244—28no—
214D244—44no—
215D244—84no—
216D244—84no—
217D244—88no—
218C4×D61—24no—
219C4×D61—24no—
220C4×D61—88no—
221C4×D61—84no—
222Dic122—84no—
223Dic122—82no—
224C8×C61—24no—
225#92418—24no—
226#60868—611nomaximal, Sylow
227C22×C61—22no—
228C22×C61#53612yes—
229C22×C61C23×D411yescharacteristic
230C2×D61—24no—
231C2×D61—22no—
232C2×D61—44no—
233C2×D61—44no—
234C2×D61—44no—
235C2×D61—82no—
236C2×D61—88no—
237C4×C61—22no—
238C4×C61—22no—
239C4×C61—44no—
240C4×C61—44no—
241C4×C61#53612yes—
242C4×C61C2×2−1+412yes—
243Dic61—24no—
244Dic61—22no—
245Dic61—44no—
246#3458—611no—
247C2×#540—611no—
248#3241—618no—
249#3241—618no—
250#3166—611no—
251#3166D6111yescharacteristic, complemented
252#3138—611no—
253#3138—611no—
254#3094—614no—
255#3077—614no—
256#2921—611no—
257C2×C61—22no—
258C2×C61C2×#53611yescharacteristic
259C2×C61#358512yes—
260D61—42no—
261D61—84no—
262#544—614no—
263#544—614no—
264#544—1222no—
265#543—1222no—
266#540—614no—
267#540—614no—
268#540—1224no—
269#533—612no—
270#533—611no—
271#533—1224no—
272#533C2×D6111yescharacteristic
273C2×#127—611no—
274C2×#125C2×D6111yescharacteristic
275C2×#123—1222no—
276C2×#121—612no—
277C2×#121—1222no—
278C2×C4×D4—612no—
279C2×#117—611no—
280C2×#117—612no—
281#501—614no—
282#501C2×D6114yescomplemented
283#499—614no—
284#499C2×D6114yescomplemented
285#490—614no—
286#490—614no—
287#486—614no—
288#486—614no—
289#482—618no—
290#482—618no—
291#481—618no—
292#480—618no—
293#478—618no—
294#478—618no—
295#476—618no—
296#474—618no—
297#433—614no—
298#433—614no—
299#428—614no—
300#428C2×D6114yes—
301#423—612no—
302#418—614no—
303#418—614no—
304#407—612no—
305C4×M16—611no—
306C4×M16C2×D6111yescharacteristic
307C61#6086811yescharacteristic, complemented, Sylow
308C22×D4—612no—
309C22×D4—1222no—
310C22×D4—1222no—
311C23×C4—611no—
312C23×C4—1222no—
313C2×M16—612no—
314C2×M16C22×D6112yes—
315C4⋊Q8—614no—
316C4⋊Q8—612no—
317C4⋊Q8—1224no—
318C4⋊Q8—1224no—
319C4⋊Q8C22×D6112yes—
320#127—612no—
321#127—612no—
322#127—1222no—
323#125—22no—
324#125—614no—
325#125—612no—
326#125—1224no—
327#125—1224no—
328#125C22×D6112yes—
329#125C22×D6112yes—
330#125C22×D6112yes—
331#123—1228no—
332#123—1228no—
333C22⋊Q8—22no—
334C22⋊Q8—1224no—
335C22⋊Q8—1224no—
336C22⋊Q8—1224no—
337C22⋊Q8—1224no—
338C22⋊Q8—1224no—
339C22⋊Q8—1222no—
340#121—618no—
341#121—1228no—
342#121—1228no—
343#121—1224no—
344C4×Q8—22no—
345C4×Q8—1224no—
346C4×Q8—1222no—
347C4×D4—618no—
348C4×D4—1224no—
349#117—614no—
350#117—612no—
351#117—612no—
352#117—612no—
353#117—614no—
354#117—1222no—
355#117—1222no—
356#117C22×D6112yes—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—614no—
360C2×C4⋊C4C22×D6112yes—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1222no—
365C2×C42—611no—
366C2×C42—1222no—
367C2×C42C22×D6111yescharacteristic
368#107—618no—
369#107C22×D6118yes—
370#106—618no—
371#106C22×D6118yes—
372C4⋊C8—618no—
373#102—618no—
374#102—618no—
375#102—618no—
376#102—618no—
377C22⋊C8—618no—
378#97—612no—
379#97C22×D6112yes—
380C4×C8—612no—
381C4×C8C22×D6112yes—
382C24—611no—
383C24—1222no—
384C2×Q8—22no—
385C2×Q8—1224no—
386C2×Q8—1222no—
387C2×D4—614no—
388C2×D4—1228no—
389C2×D4—1224no—
390C2×D4—1224no—
391C2×D4—1222no—
392C2×D4—1228no—
393C2×D4—4884no—
394C2×D4—4884no—
395C22×C4—22no—
396C22×C4—611no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—1228no—
401C22×C4—1224no—
402C22×C4—1224no—
403C22×C4—4884no—
404C22×C4D4×D6112yes—
405C22×C4C23×D6111yescharacteristic
406M16—24no—
407M16—1224no—
408C2×C8—614no—
409C2×C8C23×D6114yes—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—614no—
416C4⋊C4—614no—
417C4⋊C4—614no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1224no—
421C4⋊C4—1224no—
422C4⋊C4—1224no—
423C4⋊C4—1222no—
424C4⋊C4C23×D6114yes—
425C4⋊C4C23×D6114yes—
426C22⋊C4—22no—
427C22⋊C4—614no—
428C22⋊C4—614no—
429C22⋊C4—1228no—
430C22⋊C4—1228no—
431C22⋊C4—1224no—
432C22⋊C4—1222no—
433C42—22no—
434C42—612no—
435C42—612no—
436C42—1224no—
437C42—1222no—
438C42D4×D6112yes—
439C42C23×D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—612no—
442C23—1222no—
443C23—1224no—
444C23—2442no—
445C23—2444no—
446C23—2444no—
447C23—2444no—
448C23C2×D4×D6111yescharacteristic
449Q8—82no—
450Q8—4882no—
451Q8—4884no—
452D4—1228no—
453D4—4884no—
454D4—4884no—
455D4—4888no—
456D4—2444no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—614no—
462C2×C4—612no—
463C2×C4—1222no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—1224no—
467C2×C4—4888no—
468C2×C4—4884no—
469C2×C4—2442no—
470C2×C4C2×D4×D6112yes—
471C2×C4C2×D4×D6112yes—
472C2×C4C2×D4×D6111yescharacteristic
473C2×C4#1451982514yes—
474C2×C4C24×D6111yescharacteristic
475C8—24no—
476C8—1224no—
477C22—22no—
478C22—1224no—
479C22—1222no—
480C22—4888no—
481C22—4882no—
482C22—2444no—
483C22—2444no—
484C22—2444no—
485C22#1684033712yes—
486C22C22×D4×D6111yescharacteristic
487C4—22no—
488C4—22no—
489C4—44no—
490C4—44no—
491C4—1224no—
492C4—1222no—
493C4—2444no—
494C4#1684033712yes—
495C4C2×#1451982512yes—
496C2—22no—
497C2—4884no—
498C2—2442no—
499C2C2×#1684033711yescharacteristic
500C2#2146059612yes—
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3166C2
socleC22×C61C23×D4
prime radicalC2×#16840368C2
hypercenter#3166D61
hypocenterC61#60868
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
1. #21459222C2semidirect
3. #21457075C2semidirect
4. #21457073C2semidirect
5. #21456368C2semidirect
6. #21456195C2semidirect
7. #21456194C2semidirect
8. #21455761C2semidirect
9. #21455615C2semidirect
10. #21454455C2semidirect
11. C61×#3166C2semidirect
12. #16840392C22semidirect
14. #16840389C22semidirect
29. #16840172C22semidirect
30. #16840161C22semidirect
31. #16840128C22semidirect
32. #16840127C22semidirect
33. #16840110C22semidirect
34. #16840109C22semidirect
35. #16840089C22semidirect
36. #16840088C22semidirect
37. #16840082C22semidirect
38. #16840077C22semidirect
39. #16840065C22semidirect
40. #16840061C22semidirect
41. #16840051C22semidirect
42. #16840041C22semidirect
43. #16839824C22semidirect
44. #16839821C22semidirect
53. C61×#501C22semidirect
54. C61×#499C22semidirect
251. #3166D61semidirect
282. #501C2×D61semidirect
284. #499C2×D61semidirect
307. C61#60868semidirect

Maximal subgroups

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

subgroupindex
#214592222
C2×#168403682
#214570752
#214570732
#214563682
#214561952
#214561942
#214557612
#214556152
#214544552
C61×#31662
#6086861