Finite Group Databasebeta

← #29662750

Subgroups — #29662750

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456379C211yesmaximal, characteristic, complemented
2#21455884C211yesmaximal, characteristic, complemented
3#21455880C211yesmaximal, characteristic
4#21454780C211yesmaximal, characteristic, complemented
5#21445400C211yesmaximal, characteristic, complemented
6#21445393C211yesmaximal, characteristic, complemented
7#21445390C211yesmaximal, characteristic, complemented
8#21445383C211yesmaximal, characteristic, complemented
9#21445324C211yesmaximal, characteristic, complemented
10#21445317C211yesmaximal, characteristic, complemented
11#21445294C211yesmaximal, characteristic, complemented
12#21445287C211yesmaximal, characteristic, complemented
13#21444565C211yesmaximal, characteristic, complemented
14#21444479C211yesmaximal, characteristic, complemented
15C61×#1544C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17C4×#11758222C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840136C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840110C2211yescharacteristic
22#16840100—21no—
23#16840087—21no—
24#16840058C2211yescharacteristic
25#16840053C2211yescharacteristic
26#16840044C2211yescharacteristic
27#16840039C2211yescharacteristic
28#16839993—21no—
29#16839993—21no—
30#16839969C2211yescharacteristic
31#16839964C2211yescharacteristic
32#16839963C2211yescharacteristic
33#16839958C2211yescharacteristic
34#16839922—21no—
35#16839909—21no—
36#16839889C2211yescharacteristic
37#16839889C2211yescharacteristic
38C4×#11758202—21no—
39C4×#11758190—21no—
40C4×#11758201—21no—
41C4×#11758201—21no—
42C4×#11758201—21no—
43C4×D488—21no—
44#16839839C2211yescharacteristic
45#16839839C2211yescharacteristic
46#16839830—21no—
47#16839830—21no—
48#16839815C2211yescharacteristic
49#16839813—21no—
50C4×#11758189—21no—
51C2×C4×#92418C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839512C2211yescharacteristic
55#16839512C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839510C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#533C2211yescharacteristic
61C61×#431C2211yescharacteristic
62C61×#431C2211yescharacteristic
63C61×#332C2211yescharacteristic
64C61×#332C2211yescharacteristic
65C61×#332C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758222D411yescharacteristic
69C2×#11758202—21no—
70C2×#11758190—21no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×D488—21no—
75C2×#11758189—21no—
76C2×#11758189—21no—
77C22×#92418D411yescharacteristic
78C22×#92418D411yescharacteristic
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519765—21no—
82#14519751—21no—
83#14519751—21no—
84C4×Dic122D411yescharacteristic
85C4×Dic122C2311yescharacteristic
86C4×D244—21no—
87C4×D244—21no—
88C4×D244D411yescharacteristic
89C4×D244C2311yescharacteristic
90C4×C22⋊D61—21no—
91C4×C22⋊D61—21no—
92#14519736D411yescharacteristic
93#14519734—21no—
94C42×D61—21no—
95#14519711—21no—
96#14519710—21no—
97#14519709—21no—
98#14519708—21no—
99#14519708—21no—
100#14519708—21no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519702—21no—
106#14519701—21no—
107#14519701—21no—
108#14519701—21no—
109#14519701—21no—
110#14519700—21no—
111#14519699—21no—
112#14519699—21no—
113#14519699—21no—
114#14519697—21no—
115#14519697—21no—
116#14519697—21no—
117C22⋊#92418—21no—
118#14519689—21no—
119#14519689—21no—
120#14519687—21no—
121#14519687—21no—
122C4×#92418—21no—
123C4×#92418D411yescharacteristic
124C4×#92418D411yescharacteristic
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C4⋊Q8×C61C2311yescharacteristic
128C4⋊Q8×C61C2311yescharacteristic
129#125×C61—21no—
130C22⋊Q8×C61—21no—
131C22⋊Q8×C61—21no—
132C4×Q8×C61—21no—
133C4×Q8×C61—21no—
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C22⋊C8×C61—21no—
141C22⋊C8×C61—21no—
142C4×C8×C61—21no—
143C4×C8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61#4111yescharacteristic
149C2×C22⋊D61—21no—
150C2×C22⋊D61—41no—
151#11758222—41no—
152#11758222—41no—
153#11758222—82no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C2×D411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—41no—
162C2×Dic122—21no—
163C2×Dic122C2×D411yescharacteristic
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C42×C61—41no—
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61#4111yescharacteristic
179C42×C61C2411yescharacteristic
180C22⋊Dic61—21no—
181#11758202—42no—
182#11758201—42no—
183#11758201—42no—
184#11758201—42no—
185#11758198—21no—
186#11758198—21no—
187#11758198—41no—
188#11758198—41no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C2×D411yescharacteristic
192C4⋊Dic61C2×D411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C2×#92418—22no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—41no—
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206D488—42no—
207#11758190—42no—
208#11758189—42no—
209#11758189—42no—
210C23×C61#12111yescharacteristic
211C22×D61—21no—
212C22×D61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61#12111yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C22⋊D61—42no—
231C22⋊D61—82no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—41no—
236D244—21no—
237D244—21no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—41no—
242D244—81no—
243C4×D61—42no—
244C4×D61—41no—
245C4×D61—41no—
246C4×D61—42no—
247C4×D61—82no—
248Dic122—21no—
249Dic122—21no—
250Dic122—81no—
251C8×C61—41no—
252C8×C61—41no—
253C8×C61—41no—
254C8×C61—41no—
255#92418—21no—
256#92418—21no—
257#92418—21no—
258#92418—41no—
259#92418—41no—
260#92418#13512yes—
261#19159—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61C2×#12111yescharacteristic
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61#47112yes—
278C4×C61C2×#13511yescharacteristic
279C4×C61#57511yescharacteristic
280C4×C61#57511yescharacteristic
281C4×C61#57511yescharacteristic
282Dic61—41no—
283Dic61—41no—
284Dic61—41no—
285Dic61—81no—
286#3167—611no—
287#3107—611no—
288#3106—611no—
289#2965—611no—
290#1704—611no—
291#1703—611no—
292#1702—611no—
293#1701—611no—
294#1692—611no—
295#1691—611no—
296#1686—611no—
297#1685—611no—
298#1564—611no—
299#1544D6111yescharacteristic, complemented
300#1543—611no—
301C2×C61—41no—
302C2×C61#303511yescharacteristic
303C2×C61#305411yescharacteristic
304C2×C61C2×#47111yescharacteristic
305D61—81no—
306D61—161no—
307#533C2×D6111yescharacteristic
308#532—611no—
309C4×#41—611no—
310#498—611no—
311#492—611no—
312#487—611no—
313#486—611no—
314#485—1221no—
315#482—1221no—
316#477—611no—
317#476—611no—
318#475—611no—
319#474—611no—
320#465—1221no—
321#465—1221no—
322#462—611no—
323#461—611no—
324#460—611no—
325#459—611no—
326#452—1221no—
327#449—1221no—
328#446—611no—
329#446—611no—
330C4×Q16—1221no—
331C4×SD16—1221no—
332C4×SD16—1221no—
333C4×SD16—1221no—
334C4×SD16—1221no—
335C4×D8—1221no—
336#436—611no—
337#436—611no—
338#435—1221no—
339#435—1221no—
340#431—611no—
341#431—1221no—
342#431C2×D6111yescharacteristic
343#431C2×D6111yescharacteristic
344C4×M16—1221no—
345C2×C4×C8—611no—
346#335—611no—
347#335—611no—
348#334—611no—
349#334—611no—
350#333—611no—
351#333—611no—
352#332C2×D6111yescharacteristic
353#332C2×D6111yescharacteristic
354#332C2×D6111yescharacteristic
355#332C2×D6111yescharacteristic
356#331—611no—
357#331—611no—
358C61#1915911yescharacteristic, complemented, Sylow
359C2×#41—611no—
360C2×#41—1221no—
361C2×Q16—1221no—
362C2×SD16—1221no—
363C2×SD16—1221no—
364C2×SD16—1221no—
365C2×SD16—1221no—
366C2×D8—1221no—
367C2×M16—1221no—
368C2×M16—1221no—
369C22×C8—611no—
370C22×C8—611no—
371C4⋊Q8—611no—
372C4⋊Q8C22×D6111yescharacteristic
373C4⋊Q8C22×D6111yescharacteristic
374#127—611no—
375#125—21no—
376#124—1221no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379#121—1221no—
380#121—1221no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8—611no—
384C4×Q8—611no—
385C4×D4—611no—
386C4×D4—611no—
387C4×D4—1221no—
388C4×D4—1221no—
389C4×D4—1221no—
390C4×D4—1221no—
391#117—21no—
392#117—611no—
393#117—1221no—
394C2×C42—1221no—
395C2×C42C22×D6111yescharacteristic
396#107—1221no—
397#107—1221no—
398#106—1221no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8C22×D6111yescharacteristic
407C4⋊C8C22×D6111yescharacteristic
408C4⋊C8C22×D6111yescharacteristic
409C4⋊C8C22×D6111yescharacteristic
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427#97—1221no—
428#97—1221no—
429C4×C8—21no—
430C4×C8—21no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—1221no—
436#41—4882no—
437#41—2441no—
438#41—2441no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—611no—
442C2×Q8—1221no—
443C2×D4—611no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—2441no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—2441no—
453C22×C4#1175822211yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4D4×D6111yescharacteristic
456Q16—2442no—
457SD16—2442no—
458SD16—2442no—
459SD16—2442no—
460SD16—2442no—
461D8—2442no—
462M16—2442no—
463M16—2442no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—1222no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—2441no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—611no—
483C4⋊C4—611no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C22⋊C4—41no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C22⋊C4—1221no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C42—41no—
495C42—1221no—
496C42—1221no—
497C42—1221no—
498C42#1175822211yescharacteristic
499C42D4×D6111yescharacteristic
500C42D4×D6111yescharacteristic
501C42C23×D6111yescharacteristic
502C23—1221no—
503C23—2441no—
504C23#1451975011yescharacteristic
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—1221no—
510Q8—1221no—
511Q8—4881no—
512D4—1221no—
513D4—1221no—
514D4—4881no—
515D4—4882no—
516D4—2442no—
517D4—2441no—
518D4—2441no—
519D4—2441no—
520D4—2441no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—4882no—
532C2×C4—2442no—
533C2×C4—2441no—
534C2×C4—2442no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4#1451975011yescharacteristic
538C2×C4#1451975011yescharacteristic
539C2×C4#1451975011yescharacteristic
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×#1175822211yescharacteristic
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—612no—
548C8—1221no—
549C8—1221no—
550C8—1221no—
551C8—2441no—
552C8—2441no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22—2441no—
560C22—2441no—
561C22—2441no—
562C22C2×#1451975011yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—81no—
567C4—81no—
568C4—4881no—
569C4—2441no—
570C4—2441no—
571C4—2441no—
572C4#1684002812yes—
573C4#135×D6111yescharacteristic
574C4#1684055911yescharacteristic
575C4#1684055911yescharacteristic
576C4#1684055911yescharacteristic
577C2—41no—
578C2—9761no—
579C2—4881no—
580C2#2145527211yescharacteristic
581C2#2145542711yescharacteristic
582C2C2×#1684002811yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1544C2
socleC22×C61C2×#121
prime radical#21455880C2
hypercenter#1544D61
hypocenterC61#19159
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
1. #21456379C2semidirect
2. #21455884C2semidirect
4. #21454780C2semidirect
5. #21445400C2semidirect
6. #21445393C2semidirect
7. #21445390C2semidirect
8. #21445383C2semidirect
9. #21445324C2semidirect
10. #21445317C2semidirect
11. #21445294C2semidirect
12. #21445287C2semidirect
13. #21444565C2semidirect
14. #21444479C2semidirect
15. C61×#1544C2semidirect
299. #1544D61semidirect
358. C61#19159semidirect

Maximal subgroups

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

subgroupindex
#214563792
#214558842
#214558802
#214547802
#214454002
#214453932
#214453902
#214453832
#214453242
#214453172
#214452942
#214452872
#214445652
#214444792
C61×#15442
#1915961