Finite Group Databasebeta

← #30612726

Subgroups — #30612726

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462723C211yesmaximal, characteristic, complemented
2#21462666C211yesmaximal, characteristic
3#21461436C211yesmaximal, characteristic, complemented
4#21461434C211yesmaximal, characteristic, complemented
5#21461424C211yesmaximal, characteristic, complemented
6#21461422C211yesmaximal, characteristic, complemented
7#21461410C211yesmaximal, characteristic, complemented
8SD16⋊#92420C211yesmaximal, characteristic, complemented
9#21461402C211yesmaximal, characteristic, complemented
10SD16×#92420C211yesmaximal, characteristic, complemented
11#21461390C211yesmaximal, characteristic, complemented
12#21461158C211yesmaximal, characteristic
13#21461141C211yesmaximal, characteristic, complemented
14#21461036C211yesmaximal, characteristic, complemented
15#21455822C211yesmaximal, characteristic
16#16840814—21no—
17#16840813C2211yescharacteristic, complemented
18#16840812—21no—
19#16840812C2211yescharacteristic, complemented
20#16840811C2211yescharacteristic, complemented
21Q8×#92420C2211yescharacteristic, complemented
22D4×#92420—21no—
23D4×#92420—21no—
24D4×#92420C2211yescharacteristic
25D4×#92420C2211yescharacteristic
26D4×#92420C2211yescharacteristic
27D4×#92420C2211yescharacteristic
28#16840786—21no—
29#16840786—21no—
30#16840783C2211yescharacteristic
31#16840778C2211yescharacteristic
32#16840778C2211yescharacteristic
33C2×C22⋊#92420—21no—
34C2×C22⋊#92420—21no—
35C4×#11758217C2211yescharacteristic
36#16840626C2211yescharacteristic, complemented
37#16840625C2211yescharacteristic, complemented
38#16840624C2211yescharacteristic, complemented
39#16840623C2211yescharacteristic, complemented
40#16840612C2211yescharacteristic, complemented
41#16840612C2211yescharacteristic, complemented
42Q8⋊#92420C2211yescharacteristic, complemented
43Q8⋊#92420C2211yescharacteristic, complemented
44D4⋊#92420C2211yescharacteristic
45D4⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic, complemented
49#92420⋊C8C2211yescharacteristic, complemented
50#16840603C2211yescharacteristic, complemented
51C8×#92420C2211yescharacteristic, complemented
52#16840314C2211yescharacteristic
53C2×D4×Dic61C2211yescharacteristic
54#16840093C411yescharacteristic
55#16840090C411yescharacteristic
56#16840050C411yescharacteristic
57#16840050C411yescharacteristic
58#16839965C411yescharacteristic
59#16839965C411yescharacteristic
60#16839913C411yescharacteristic
61#16839853C2211yescharacteristic
62#16839853C2211yescharacteristic
63SD16×Dic61C2211yescharacteristic
64SD16×Dic61C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#450C411yescharacteristic
67C2×#11758217—21no—
68C22×#92420—21no—
69C22×#92420—21no—
70C22×#92420—21no—
71C22×#92420—21no—
72#14519844—21no—
73#14519844—21no—
74C4⋊#92420—21no—
75C4⋊#92420—21no—
76C4⋊#92420D411yescharacteristic
77C4⋊#92420D411yescharacteristic
78C4⋊#92420C2311yescharacteristic
79C4⋊#92420C2311yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C22⋊#92420—21no—
83C22⋊#92420—21no—
84C22⋊#92420—21no—
85C22⋊#92420—21no—
86C22⋊#92420—21no—
87C22⋊#92420—21no—
88C22⋊#92420D411yescharacteristic
89C22⋊#92420D411yescharacteristic
90#14519832—21no—
91#14519832—21no—
92#14519832—21no—
93#14519832—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C23×Dic61—21no—
102C2×#11758200C2×C411yescharacteristic
103C2×#11758200C2×C411yescharacteristic
104#14519776C2×C411yescharacteristic
105#14519775C2×C411yescharacteristic
106#14519767—21no—
107#14519756C2×C411yescharacteristic
108#14519754—21no—
109Q8×Dic61C2311yescharacteristic
110#14519742—21no—
111#14519742C2311yescharacteristic
112D4×Dic61—21no—
113D4×Dic61—21no—
114D4×Dic61C2×C411yescharacteristic
115D4×Dic61C2×C411yescharacteristic
116D4×Dic61C2311yescharacteristic
117D4×Dic61C2311yescharacteristic
118#14519733—21no—
119C2×C4⋊Dic61C2×C411yescharacteristic
120C2×C22⋊Dic61—21no—
121C2×C4×Dic61C2311yescharacteristic
122#14519709C2×C411yescharacteristic
123#14519709C2×C411yescharacteristic
124#14519708C2×C411yescharacteristic
125#14519708C2×C411yescharacteristic
126#14519706C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519702C2×C411yescharacteristic
129#14519702C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519698C2×C411yescharacteristic
134C22⋊#92418C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C22×D4×C61C2×C411yescharacteristic
138C2×SD16×C61C2×C411yescharacteristic
139C2×SD16×C61C2×C411yescharacteristic
140C22⋊Q8×C61C2×C411yescharacteristic
141#102×C61C2×C411yescharacteristic
142#102×C61C2×C411yescharacteristic
143C22⋊C8×C61C2×C411yescharacteristic
144C24×C61—21no—
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C411yescharacteristic
150C2×D4×C61C22×C411yescharacteristic
151C22×C4×C61C22×C411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—81no—
158C22×Dic61—81no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61#4111yescharacteristic
161C2×Dic122—21no—
162C2×Dic122C22×C411yescharacteristic
163#11758217—42no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—81no—
171C2×#92420—81no—
172C2×#92420—81no—
173C2×#92420—81no—
174C2×#92420C2×D411yescharacteristic
175C2×#92420C2×D411yescharacteristic
176C2×#92420C2×D411yescharacteristic
177C2×#92420C2×D411yescharacteristic
178SD16×C61—22no—
179SD16×C61—22no—
180C2×C8×C61C22×C411yescharacteristic
181C2×C8×C61C22×C411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22×C411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊Dic61—21no—
186C22⋊Dic61—21no—
187C22⋊Dic61—21no—
188C22⋊Dic61—21no—
189C22⋊Dic61—21no—
190#11758200—22no—
191#11758200—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C22×C411yescharacteristic
195C4⋊Dic61C22×C411yescharacteristic
196C4⋊Dic61C22×C411yescharacteristic
197#11758196—21no—
198#11758196—21no—
199#11758196—21no—
200#11758196—21no—
201#11758196—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61C22×C411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61#4111yescharacteristic
207C4×Dic61C2411yescharacteristic
208C2×#92418C22×C411yescharacteristic
209C2×#92418C22×C411yescharacteristic
210C23×C61—24no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—81no—
214C23×C61—81no—
215C23×C61C4×D411yescharacteristic
216Q8×C61—21no—
217Q8×C61—21no—
218D4×C61—24no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61#13112yes—
222D4×C61#13112yes—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C23×C411yescharacteristic
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—81no—
241C2×Dic61—81no—
242C2×Dic61—81no—
243C2×Dic61C4×D411yescharacteristic
244C2×Dic61C4×D411yescharacteristic
245C2×Dic61C22×D411yescharacteristic
246C2×Dic61C2×#4111yescharacteristic
247Dic122—21no—
248Dic122—21no—
249Dic122—81no—
250#92420—22no—
251#92420—21no—
252#92420—21no—
253#92420—21no—
254#92420—21no—
255#92420—41no—
256#92420—41no—
257#92420—41no—
258#92420—41no—
259#92420—41no—
260C8×C61—21no—
261C8×C61—21no—
262#92418—21no—
263#92418—21no—
264#17924—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—84no—
275C22×C61C2×C4×D411yescharacteristic
276C4×C61—21no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61C2×#13111yescharacteristic
280C4×C61C2×#13111yescharacteristic
281Dic61—21no—
282Dic61—21no—
283Dic61—21no—
284Dic61—41no—
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288Dic61#57711yescharacteristic
289Dic61#57711yescharacteristic
290#3101—611no—
291#2985—611no—
292#2972—611no—
293#1585—611no—
294#1585—611no—
295#1582—611no—
296#1582—611no—
297SD16⋊C8—611no—
298SD16⋊C8—611no—
299C8×SD16—611no—
300C8×SD16—611no—
301#1572—611no—
302#1490—611no—
303#1485—611no—
304#1453—611no—
305C2×C61—41no—
306C2×C61—42no—
307C2×C61—42no—
308C2×C61#292911yescharacteristic
309C2×C61#294911yescharacteristic
310C2×C61#295611yescharacteristic
311#533—611no—
312C2×C4×D4—611no—
313#483—611no—
314#483—611no—
315#476—611no—
316#476—611no—
317#461—611no—
318#461—611no—
319#450—611no—
320#450#692311yescharacteristic
321#446—611no—
322#446—611no—
323#446—1221no—
324#446—1221no—
325C8×Q8—611no—
326C8×Q8—611no—
327C4×SD16—611no—
328C4×SD16—611no—
329C4×SD16—611no—
330C4×SD16—611no—
331C8×D4—611no—
332C8×D4—611no—
333C8×D4—611no—
334C8×D4—611no—
335C8×D4—1221no—
336C8×D4—1221no—
337#433—1221no—
338#433—1221no—
339#432—611no—
340#431—611no—
341#431—611no—
342#431—611no—
343C2×C22⋊C8—1221no—
344C2×C22⋊C8—1221no—
345C4×M16—611no—
346#335—611no—
347#335—611no—
348#334—611no—
349#334—611no—
350Q8⋊C8—611no—
351Q8⋊C8—611no—
352Q8⋊C8—611no—
353Q8⋊C8—611no—
354D4⋊C8—611no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—611no—
358#322—611no—
359#322—611no—
360#322—611no—
361C82—611no—
362C61#1792411yescharacteristic, complemented, Sylow
363C22×D4C2×#692311yescharacteristic
364C23×C4—1221no—
365C2×SD16—611no—
366C2×SD16—611no—
367C2×SD16C2×#692311yescharacteristic
368C2×SD16C2×#692311yescharacteristic
369C2×M16—1221no—
370C22×C8—1221no—
371C22×C8—1221no—
372C22×C8—1221no—
373C22×C8—1221no—
374C4⋊Q8—611no—
375C4⋊Q8—611no—
376#125—1221no—
377C22⋊Q8—611no—
378C22⋊Q8—1221no—
379C22⋊Q8C2×#692311yescharacteristic
380C4×Q8—611no—
381C4×Q8—611no—
382C4×Q8—1221no—
383C4×D4—611no—
384C4×D4—611no—
385C4×D4—611no—
386C4×D4—611no—
387C4×D4—1221no—
388C4×D4—1221no—
389#117—1221no—
390C2×C4⋊C4—611no—
391C2×C22⋊C4—1221no—
392C2×C42—611no—
393#106—611no—
394#106—611no—
395#106—611no—
396#106—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408C4⋊C8—1221no—
409#103—611no—
410#103—611no—
411#103—611no—
412#103—611no—
413#102—611no—
414#102—611no—
415#102C2×#692311yescharacteristic
416#102C2×#692311yescharacteristic
417C22⋊C8—611no—
418C22⋊C8—611no—
419C22⋊C8—611no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C22⋊C8C2×#692311yescharacteristic
427#97—1221no—
428#97—1221no—
429#97—1221no—
430#97—1221no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—611no—
436C4×C8—611no—
437C4×C8—1221no—
438C4×C8—1221no—
439C4×C8—1221no—
440C24—21no—
441C2×Q8—611no—
442C2×Q8—1221no—
443C2×Q8C22×#692311yescharacteristic
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4C22×#692311yescharacteristic
448C2×D4C22×#692311yescharacteristic
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—4881no—
457C22×C4—4881no—
458C22×C4C22×#692311yescharacteristic
459SD16—22no—
460SD16—22no—
461SD16—1222no—
462SD16—1222no—
463M16—2442no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—4881no—
477C2×C8—4881no—
478C2×C8—4881no—
479C2×C8—4881no—
480C2×C8C22×#692311yescharacteristic
481C2×C8C22×#692311yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—611no—
484C4⋊C4—611no—
485C4⋊C4—611no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4C22×#692311yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—1221no—
496C22⋊C4—1221no—
497C22⋊C4—1221no—
498C22⋊C4—1221no—
499C22⋊C4—1221no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—1221no—
505C42—1221no—
506C23—24no—
507C23—21no—
508C23—21no—
509C23—81no—
510C23—81no—
511C23D4×#692311yescharacteristic
512Q8—21no—
513Q8—21no—
514Q8—1221no—
515Q8—1221no—
516Q8—4881no—
517D4—24no—
518D4—21no—
519D4—21no—
520D4#1451986412yes—
521D4#1451986412yes—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—4881no—
537C2×C4—4881no—
538C2×C4—4881no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4D4×#692311yescharacteristic
545C2×C4C23×#692311yescharacteristic
546C8—21no—
547C8—21no—
548C8—1221no—
549C8—1221no—
550C8—1221no—
551C8—1221no—
552C8—1222no—
553C8—1221no—
554C8—1221no—
555C8—2441no—
556C8—2441no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—21no—
564C22—21no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—84no—
570C22C2×D4×#692311yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—611no—
575C4—611no—
576C4—1221no—
577C4—1221no—
578C4—1221no—
579C4—2441no—
580C4—2441no—
581C4—2441no—
582C4—2441no—
583C4C2×#1451986411yescharacteristic
584C4C2×#1451986411yescharacteristic
585C2—41no—
586C2—42no—
587C2—42no—
588C2#2146253411yescharacteristic
589C2#2146257911yescharacteristic
590C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#450C4
socleC22×C61C2×C4×D4
prime radicalC22×D4×C61C2×C4
cosocleC4×Dic61C24
hypercenter#450#6923
hypocenterC61#17924
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #21462723C2semidirect
3. #21461436C2semidirect
4. #21461434C2semidirect
5. #21461424C2semidirect
6. #21461422C2semidirect
7. #21461410C2semidirect
8. SD16⋊#92420C2semidirect
9. #21461402C2semidirect
10. SD16×#92420C2semidirect
11. #21461390C2semidirect
13. #21461141C2semidirect
14. #21461036C2semidirect
17. #16840813C22semidirect
19. #16840812C22semidirect
20. #16840811C22semidirect
21. Q8×#92420C22semidirect
36. #16840626C22semidirect
37. #16840625C22semidirect
38. #16840624C22semidirect
39. #16840623C22semidirect
40. #16840612C22semidirect
41. #16840612C22semidirect
42. Q8⋊#92420C22semidirect
43. Q8⋊#92420C22semidirect
48. #92420⋊C8C22semidirect
49. #92420⋊C8C22semidirect
50. #16840603C22semidirect
51. C8×#92420C22semidirect
362. C61#17924semidirect

Maximal subgroups

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

subgroupindex
#214627232
#214626662
#214614362
#214614342
#214614242
#214614222
#214614102
SD16⋊#924202
#214614022
SD16×#924202
#214613902
#214611582
#214611412
#214610362
#214558222
#1792461