Finite Group Databasebeta

← #29665540

Subgroups — #29665540

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456957C211yesmaximal, characteristic, complemented
2#21455981C211yesmaximal, characteristic, complemented
3#21455879C211yesmaximal, characteristic
4#21445263C211yesmaximal, characteristic, complemented
5#21445256C211yesmaximal, characteristic, complemented
6#21445235C211yesmaximal, characteristic, complemented
7#21445228C211yesmaximal, characteristic, complemented
8#21445223C211yesmaximal, characteristic, complemented
9#21445218C211yesmaximal, characteristic, complemented
10#21445201C211yesmaximal, characteristic, complemented
11#21445194C211yesmaximal, characteristic, complemented
12#21444599C211yesmaximal, characteristic, complemented
13#21444512C211yesmaximal, characteristic, complemented
14#21444477C211yesmaximal, characteristic, complemented
15C61×#2988C211yesmaximal, characteristic, complemented
16#16840319C2211yescharacteristic
17#16840312C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840160C2211yescharacteristic
20#16840109C2211yescharacteristic
21#16840109C2211yescharacteristic
22#16840103—21no—
23#16840087—21no—
24#16840087—21no—
25#16840086—21no—
26#16840063—21no—
27#16840063—21no—
28#16840057C2211yescharacteristic
29#16840057C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16839972C2211yescharacteristic
33#16839967C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839924—21no—
37#16839907—21no—
38#16839867—21no—
39#16839862—21no—
40C4×#11758201—21no—
41C4×#11758199—21no—
42#16839819C2211yescharacteristic
43#16839815C2211yescharacteristic
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C4×#41×C61C2211yescharacteristic
57C8×Q8×C61C2211yescharacteristic
58C8×Q8×C61C2211yescharacteristic
59C8×D4×C61—21no—
60C8×D4×C61—21no—
61C8×D4×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#431—21no—
64C61×#431C2211yescharacteristic
65C2×C4⋊C8×C61—21no—
66C2×C4⋊C8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519774C2311yescharacteristic
76#14519769—21no—
77#14519765—21no—
78#14519756—21no—
79#14519756—21no—
80#14519751—21no—
81#14519751—21no—
82C4×Dic122—21no—
83C4×Dic122—21no—
84C4×D244—21no—
85C4×D244—21no—
86#14519736—21no—
87#14519711—21no—
88#14519711—21no—
89#14519710—21no—
90#14519710—21no—
91#14519709—21no—
92#14519709—21no—
93#14519708—21no—
94#14519708—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
102#14519700—21no—
103#14519699—21no—
104#14519699—21no—
105#14519699—21no—
106#14519699—21no—
107#14519698—21no—
108#14519698—21no—
109#14519697—21no—
110#14519697—21no—
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519688—21no—
114C4×#92418—21no—
115C2×#41×C61D411yescharacteristic
116C22×C8×C61—21no—
117C22×C8×C61—21no—
118C22×C8×C61—21no—
119C4×Q8×C61D411yescharacteristic
120C4×Q8×C61C2311yescharacteristic
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61D411yescharacteristic
124C4×D4×C61C2311yescharacteristic
125#117×C61—21no—
126#117×C61D411yescharacteristic
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—21no—
130C4⋊C8×C61—21no—
131C4⋊C8×C61—21no—
132C4⋊C8×C61D411yescharacteristic
133C4⋊C8×C61D411yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138C22⋊C8×C61D411yescharacteristic
139C22⋊C8×C61D411yescharacteristic
140C4×C8×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144#41×C61—22no—
145#41×C61—22no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61#4111yescharacteristic
155C2×C22⋊D61—41no—
156#11758222—82no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×C4×D61—41no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—81no—
167C2×C8×C61—81no—
168C2×C8×C61—81no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61#4111yescharacteristic
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188#11758202—42no—
189#11758201—42no—
190#11758200—42no—
191#11758199—42no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200#11758196—41no—
201#11758196—41no—
202C4×Dic61—41no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C23×C61—21no—
208C23×C61#12111yescharacteristic
209C22×D61—41no—
210Q8×C61—21no—
211Q8×C61#13512yes—
212D4×C61—24no—
213D4×C61—21no—
214D4×C61#13512yes—
215C2×C4×C61—24no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—81no—
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C22⋊D61—82no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237D244—41no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—81no—
242C4×D61—82no—
243Dic122—41no—
244Dic122—41no—
245Dic122—81no—
246Dic122—81no—
247C8×C61—21no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—41no—
252C8×C61—41no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#19419—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—22no—
261C22×C61—21no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61C2×#12111yescharacteristic
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—22no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61C2×#13511yescharacteristic
276C4×C61C2×#13511yescharacteristic
277C4×C61#57711yescharacteristic
278C4×C61#57811yescharacteristic
279Dic61—81no—
280Dic61—81no—
281Dic61—81no—
282#3228—611no—
283#3116—611no—
284#3106—611no—
285#2988D6111yescharacteristic, complemented
286#1682—611no—
287#1681—611no—
288#1678—611no—
289#1677—611no—
290#1676—611no—
291#1675—611no—
292#1672—611no—
293#1671—611no—
294#1572—611no—
295#1551—611no—
296#1542—611no—
297C2×C61—41no—
298C2×C61—42no—
299C2×C61#304611yescharacteristic
300C2×C61#305011yescharacteristic
301C2×C61#306811yescharacteristic
302D61—161no—
303#533—611no—
304#532—611no—
305C4×#41C2×D6111yescharacteristic
306#500—611no—
307#498—611no—
308#486—611no—
309#486—611no—
310#485—1221no—
311#482—1221no—
312#482—1221no—
313#482—1221no—
314#478—1221no—
315#478—1221no—
316#477—611no—
317#477—611no—
318#476—611no—
319#476—611no—
320#462—611no—
321#461—611no—
322#460—611no—
323#459—611no—
324#452—1221no—
325#449—1221no—
326C8×Q8C2×D6111yescharacteristic
327C8×Q8C2×D6111yescharacteristic
328#441—1221no—
329#440—1221no—
330C4×SD16—1221no—
331C4×D8—1221no—
332C8×D4—21no—
333C8×D4—21no—
334C8×D4C2×D6111yescharacteristic
335C8×D4C2×D6111yescharacteristic
336#432—611no—
337#431—21no—
338#431—611no—
339#431C2×D6111yescharacteristic
340C2×C4⋊C8—21no—
341C2×C4⋊C8C2×D6111yescharacteristic
342#335—611no—
343#335—611no—
344#334—611no—
345#334—611no—
346#333—611no—
347#333—611no—
348#333—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352#331—611no—
353#331—611no—
354C61#1941911yescharacteristic, complemented, Sylow
355C2×#41—1221no—
356C2×#41D24411yescharacteristic
357C2×Q16—1221no—
358C2×SD16—1221no—
359C2×SD16—1221no—
360C2×D8—1221no—
361C22×C8—21no—
362C22×C8—21no—
363C22×C8—21no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—611no—
367#127—611no—
368#125—1221no—
369#124—1221no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372#121—1221no—
373#121—1221no—
374C4×Q8—1221no—
375C4×Q8—1221no—
376C4×Q8D24411yescharacteristic
377C4×Q8C22×D6111yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—1221no—
381C4×D4—1221no—
382C4×D4D24411yescharacteristic
383C4×D4C22×D6111yescharacteristic
384#117—21no—
385#117—1221no—
386#117D24411yescharacteristic
387C2×C42—21no—
388C2×C42C22×D6111yescharacteristic
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#106—1221no—
394#106—1221no—
395#106—1221no—
396#106—1221no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8C22⋊D6111yescharacteristic
405C4⋊C8C22⋊D6111yescharacteristic
406C4⋊C8C22×D6111yescharacteristic
407C4⋊C8C22×D6111yescharacteristic
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#102—1221no—
413#102—1221no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420C22⋊C8—21no—
421C22⋊C8—21no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C22⋊C8C22⋊D6111yescharacteristic
425C22⋊C8C22⋊D6111yescharacteristic
426#97—1221no—
427C4×C8—21no—
428C4×C8—21no—
429C4×C8—1221no—
430C4×C8C22×D6111yescharacteristic
431C4×C8C22×D6111yescharacteristic
432#41—22no—
433#41—22no—
434#41—4882no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8C2×D24411yescharacteristic
439C2×D4—21no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—2441no—
444C2×D4C2×D24411yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—2441no—
449C22×C4C2×D24411yescharacteristic
450C22×C4#1175822211yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452Q16—2442no—
453SD16—2442no—
454SD16—2442no—
455D8—2442no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—81no—
459C2×C8—81no—
460C2×C8—81no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8C2×C22⋊D6111yescharacteristic
466C2×C8C2×C22⋊D6111yescharacteristic
467C2×C8C2×C22⋊D6111yescharacteristic
468C2×C8C2×C22⋊D6111yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4C2×D24411yescharacteristic
480C4⋊C4C2×D24411yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—2441no—
491C42C2×D24411yescharacteristic
492C42#1175822211yescharacteristic
493C42C2×C22⋊D6111yescharacteristic
494C42C23×D6111yescharacteristic
495C23—21no—
496C23—2441no—
497C23#1451975111yescharacteristic
498Q8—21no—
499Q8—4881no—
500Q8—4881no—
501Q8—2441no—
502Q8—2441no—
503Q8#1451979912yes—
504D4—24no—
505D4—21no—
506D4—4881no—
507D4—4882no—
508D4—2441no—
509D4—2441no—
510D4—2441no—
511D4—2441no—
512D4#1451979912yes—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—81no—
525C2×C4—4882no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4#1451975111yescharacteristic
530C2×C4#1451975111yescharacteristic
531C2×C4#1451975111yescharacteristic
532C2×C4C22×C22⋊D6111yescharacteristic
533C2×C4C22×D24411yescharacteristic
534C2×C4C2×#1175822211yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—21no—
549C22—41no—
550C22—41no—
551C22—4881no—
552C22—4881no—
553C22—4881no—
554C22C2×#1451975111yescharacteristic
555C4—21no—
556C4—21no—
557C4—22no—
558C4—21no—
559C4—21no—
560C4—41no—
561C4—41no—
562C4—4881no—
563C4—4881no—
564C4—4881no—
565C4C2×#1451979911yescharacteristic
566C4C2×#1451979911yescharacteristic
567C4#1684057311yescharacteristic
568C4#1684057711yescharacteristic
569C2—41no—
570C2—42no—
571C2—9761no—
572C2#2145536911yescharacteristic
573C2#2145540011yescharacteristic
574C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2988C2
socleC22×C61C2×#121
prime radical#21455879C2
hypercenter#2988D61
hypocenterC61#19419
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #21456957C2semidirect
2. #21455981C2semidirect
4. #21445263C2semidirect
5. #21445256C2semidirect
6. #21445235C2semidirect
7. #21445228C2semidirect
8. #21445223C2semidirect
9. #21445218C2semidirect
10. #21445201C2semidirect
11. #21445194C2semidirect
12. #21444599C2semidirect
13. #21444512C2semidirect
14. #21444477C2semidirect
15. C61×#2988C2semidirect
285. #2988D61semidirect
354. C61#19419semidirect

Maximal subgroups

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

subgroupindex
#214569572
#214559812
#214558792
#214452632
#214452562
#214452352
#214452282
#214452232
#214452182
#214452012
#214451942
#214445992
#214445122
#214444772
C61×#29882
#1941961