Finite Group Databasebeta

← #29664011

Subgroups — #29664011

Order 15616.

630 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456144C211yesmaximal, characteristic, complemented
2#21455975C211yesmaximal, characteristic, complemented
3#21455971C211yesmaximal, characteristic
4#21454775C211yesmaximal, characteristic, complemented
5#21445280C211yesmaximal, characteristic, complemented
6#21445273C211yesmaximal, characteristic, complemented
7#21445266C211yesmaximal, characteristic, complemented
8#21445259C211yesmaximal, characteristic, complemented
9#21445252C211yesmaximal, characteristic, complemented
10#21445245C211yesmaximal, characteristic, complemented
11#21445238C211yesmaximal, characteristic, complemented
12#21445231C211yesmaximal, characteristic, complemented
13#21444580C211yesmaximal, characteristic, complemented
14#21444522C211yesmaximal, characteristic, complemented
15C61×#1554C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17C4×#11758222C2211yescharacteristic
18#16840119—21no—
19#16840119—21no—
20#16840114C2211yescharacteristic
21#16840114C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840100—21no—
25#16840087—21no—
26#16840054C2211yescharacteristic
27#16840049C2211yescharacteristic
28#16840048C2211yescharacteristic
29#16840043C2211yescharacteristic
30#16839969C2211yescharacteristic
31#16839964C2211yescharacteristic
32#16839963C2211yescharacteristic
33#16839958C2211yescharacteristic
34#16839922—21no—
35#16839909—21no—
36#16839886C2211yescharacteristic
37#16839886C2211yescharacteristic
38#16839875—21no—
39#16839867—21no—
40#16839864—21no—
41#16839860—21no—
42#16839827C2211yescharacteristic
43#16839827C2211yescharacteristic
44#16839825—21no—
45#16839825—21no—
46#16839819C2211yescharacteristic
47#16839813—21no—
48#16839756—21no—
49#16839756—21no—
50C2×C4×#92418C2211yescharacteristic
51C4×C8×D61—21no—
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839512C2211yescharacteristic
55#16839512C2211yescharacteristic
56#16839509C2211yescharacteristic
57#16839509C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#533C2211yescharacteristic
61C61×#432C2211yescharacteristic
62C61×#431C2211yescharacteristic
63C61×#333C2211yescharacteristic
64C61×#333C2211yescharacteristic
65C61×#332C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758222D411yescharacteristic
69C2×#11758202—21no—
70C2×#11758190—21no—
71C2×#11758201—21no—
72C2×D488—21no—
73C22×#92418D411yescharacteristic
74C22×#92418D411yescharacteristic
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77#14519777C2311yescharacteristic
78#14519774C2311yescharacteristic
79#14519765—21no—
80#14519751—21no—
81#14519751—21no—
82C4×Dic122D411yescharacteristic
83C4×Dic122C2311yescharacteristic
84C4×D244—21no—
85C4×D244—21no—
86C4×D244D411yescharacteristic
87C4×D244C2311yescharacteristic
88C4×C22⋊D61—21no—
89C4×C22⋊D61—21no—
90#14519736D411yescharacteristic
91#14519734—21no—
92C42×D61—21no—
93#14519711—21no—
94#14519710—21no—
95#14519709—21no—
96#14519708—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519706—21no—
100#14519706—21no—
101#14519702—21no—
102#14519701—21no—
103#14519701—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700—21no—
108#14519700—21no—
109#14519699—21no—
110#14519699—21no—
111#14519699—21no—
112#14519697—21no—
113#14519697—21no—
114#14519697—21no—
115#14519697—21no—
116#14519697—21no—
117C22⋊#92418—21no—
118#14519689—21no—
119#14519689—21no—
120#14519688—21no—
121C4×#92418D411yescharacteristic
122C4×#92418D411yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C8×Dic61—21no—
126C8×Dic61—21no—
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—
142#97×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—22no—
152#11758222—22no—
153#11758222—42no—
154#11758222—42no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244C2×D411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×Dic122C2×D411yescharacteristic
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
175C22⋊C4×C61—41no—
176C42×C61—41no—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61#4111yescharacteristic
180C42×C61C2411yescharacteristic
181C22⋊Dic61—21no—
182#11758202—42no—
183#11758201—42no—
184#11758198—21no—
185#11758198—21no—
186#11758198—41no—
187#11758198—41no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61C2×D411yescharacteristic
191C4⋊Dic61C2×D411yescharacteristic
192#11758196—21no—
193#11758196—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—41no—
200C2×#92418—41no—
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205D488—42no—
206#11758190—42no—
207C8×D61—42no—
208C8×D61—42no—
209C23×C61#12111yescharacteristic
210C22×D61—21no—
211C22×D61—41no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—81no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C22⋊D61—24no—
229C22⋊D61—82no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—41no—
234D244—21no—
235D244—41no—
236D244—41no—
237D244—41no—
238D244—41no—
239D244—81no—
240D244#13512yes—
241C4×D61—24no—
242C4×D61—42no—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61—82no—
246Dic122—21no—
247Dic122—41no—
248Dic122—41no—
249Dic122#13512yes—
250C8×C61—41no—
251C8×C61—41no—
252C8×C61—41no—
253C8×C61—41no—
254#92418—21no—
255#92418—21no—
256#92418—21no—
257#92418—21no—
258#92418—41no—
259#92418—41no—
260#19278—611nomaximal, Sylow
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61C2×#12111yescharacteristic
265C2×D61—22no—
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61C2×#13511yescharacteristic
278C4×C61C2×#13511yescharacteristic
279C4×C61#57511yescharacteristic
280C4×C61#57511yescharacteristic
281Dic61—22no—
282Dic61—41no—
283Dic61—41no—
284Dic61—81no—
285#3131—611no—
286#3116—611no—
287#3115—611no—
288C8×#41—611no—
289#1684—611no—
290#1683—611no—
291#1682—611no—
292#1681—611no—
293#1680—611no—
294#1679—611no—
295#1678—611no—
296#1677—611no—
297#1568—611no—
298#1554D6111yescharacteristic, complemented
299#1553—611no—
300C2×C61—41no—
301C2×C61#304011yescharacteristic
302C2×C61#305011yescharacteristic
303C2×C61#306711yescharacteristic
304D61—42no—
305D61—161no—
306#533C2×D6111yescharacteristic
307#532—611no—
308C4×#41—611no—
309#488—1221no—
310#488—1221no—
311#487—611no—
312#487—611no—
313#486—611no—
314#486—611no—
315#485—1221no—
316#482—1221no—
317#477—611no—
318#476—611no—
319#475—611no—
320#474—611no—
321#462—611no—
322#461—611no—
323#460—611no—
324#459—611no—
325#452—1221no—
326#449—1221no—
327C8×Q8—611no—
328C8×Q8—611no—
329#442—1221no—
330#441—1221no—
331#440—1221no—
332#440—1221no—
333C8×D4—611no—
334C8×D4—611no—
335C8×D4—1221no—
336C8×D4—1221no—
337#432—611no—
338#432C2×D6111yescharacteristic
339#431—1221no—
340#431C2×D6111yescharacteristic
341#416—1221no—
342#416—1221no—
343C2×C4×C8—611no—
344C2×C4×C8—1221no—
345#335—611no—
346#335—611no—
347#334—611no—
348#334—611no—
349#333C2×D6111yescharacteristic
350#333C2×D6111yescharacteristic
351#332—611no—
352#332—611no—
353#332C2×D6111yescharacteristic
354#332C2×D6111yescharacteristic
355#331—611no—
356#331—611no—
357C61#1927811yescharacteristic, complemented, Sylow
358C2×#41—611no—
359C2×#41—1221no—
360C2×Q16—1221no—
361C2×SD16—1221no—
362C2×SD16—1221no—
363C2×D8—1221no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—1221no—
367C22×C8—1221no—
368C4⋊Q8—611no—
369C4⋊Q8C22×D6111yescharacteristic
370C4⋊Q8C22×D6111yescharacteristic
371#127—611no—
372#125—21no—
373#124—1221no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376#121—1221no—
377#121—1221no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—611no—
381C4×Q8—611no—
382C4×D4—611no—
383C4×D4—611no—
384C4×D4—1221no—
385C4×D4—1221no—
386C4×D4—1221no—
387C4×D4—1221no—
388#117—21no—
389#117—611no—
390#117—1221no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—1221no—
394#107—1221no—
395#106—1221no—
396#106—1221no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—1221no—
400C4⋊C8—1221no—
401C4⋊C8C22×D6111yescharacteristic
402C4⋊C8C22×D6111yescharacteristic
403C4⋊C8C22×D6111yescharacteristic
404C4⋊C8C22×D6111yescharacteristic
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#102—1221no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421C22⋊C8—21no—
422C22⋊C8—21no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426#97—21no—
427#97—1221no—
428C4×C8—21no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—1221no—
434C4×C8—1221no—
435#41—1222no—
436#41—1222no—
437#41—2442no—
438#41—2442no—
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—
459D8—2442no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—2442no—
471C2×C8—2442no—
472C2×C8—2441no—
473C2×C8—2441no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—611no—
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C22⋊C4—41no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C42—41no—
493C42—1221no—
494C42—1221no—
495C42—1221no—
496C42#1175822211yescharacteristic
497C42D4×D6111yescharacteristic
498C42D4×D6111yescharacteristic
499C42C23×D6111yescharacteristic
500C23—1221no—
501C23—2441no—
502C23#1451975011yescharacteristic
503Q8—41no—
504Q8—41no—
505Q8—81no—
506Q8—612no—
507Q8—1221no—
508Q8—2441no—
509Q8—2441no—
510D4—612no—
511D4—1224no—
512D4—1221no—
513D4—4881no—
514D4—4882no—
515D4—2441no—
516D4—2441no—
517D4—2441no—
518D4—2441no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—1224no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—4882no—
531C2×C4—2442no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4#1451975011yescharacteristic
536C2×C4#1451975011yescharacteristic
537C2×C4#1451975011yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4C2×D4×D6111yescharacteristic
540C2×C4C2×#1175822211yescharacteristic
541C8—41no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—1221no—
546C8—1221no—
547C8—1221no—
548C8—1221no—
549C8—2441no—
550C8—2441no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—1222no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—2441no—
559C22—2441no—
560C22C2×#1451975011yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—81no—
566C4—81no—
567C4—1222no—
568C4—4881no—
569C4—2441no—
570C4—2441no—
571C4#135×D6111yescharacteristic
572C4#135×D6111yescharacteristic
573C4#1684056111yescharacteristic
574C4#1684056111yescharacteristic
575C2—41no—
576C2—9761no—
577C2—2442no—
578C2#2145531611yescharacteristic
579C2#2145539911yescharacteristic
580C2#2145553811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1554C2
socleC22×C61C2×#121
prime radical#21455971C2
hypercenter#1554D61
hypocenterC61#19278
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1554

Complements

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

NH ≅ G/Ntype
1. #21456144C2semidirect
2. #21455975C2semidirect
4. #21454775C2semidirect
5. #21445280C2semidirect
6. #21445273C2semidirect
7. #21445266C2semidirect
8. #21445259C2semidirect
9. #21445252C2semidirect
10. #21445245C2semidirect
11. #21445238C2semidirect
12. #21445231C2semidirect
13. #21444580C2semidirect
14. #21444522C2semidirect
15. C61×#1554C2semidirect
298. #1554D61semidirect
357. C61#19278semidirect

Maximal subgroups

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

subgroupindex
#214561442
#214559752
#214559712
#214547752
#214452802
#214452732
#214452662
#214452592
#214452522
#214452452
#214452382
#214452312
#214445802
#214445222
C61×#15542
#1927861