Finite Group Databasebeta

← #29676765

Subgroups — #29676765

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458872C211yesmaximal, characteristic
2#21458708C211yesmaximal, characteristic, complemented
3#21458622C211yesmaximal, characteristic, complemented
4#21445438C211yesmaximal, characteristic
5#21445375C211yesmaximal, characteristic, complemented
6#21445337C211yesmaximal, characteristic, complemented
7#21445289C211yesmaximal, characteristic
8#21445164C211yesmaximal, characteristic
9#21445096C211yesmaximal, characteristic, complemented
10#21445094C211yesmaximal, characteristic, complemented
11#21445005C211yesmaximal, characteristic
12#21444914C211yesmaximal, characteristic
13#21444728C211yesmaximal, characteristic, complemented
14#21444667C211yesmaximal, characteristic, complemented
15C61×#3400C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17#16840491C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840163—21no—
20#16840158C2211yescharacteristic
21#16840157—21no—
22#16840134C2211yescharacteristic
23#16840132—21no—
24#16840114C2211yescharacteristic
25#16840108C2211yescharacteristic
26#16840071—21no—
27#16840071—21no—
28#16840066—21no—
29#16840047C2211yescharacteristic
30#16840047C2211yescharacteristic
31#16840044C2211yescharacteristic
32#16839981—21no—
33#16839972—21no—
34#16839972C2211yescharacteristic
35#16839972C2211yescharacteristic
36#16839967—21no—
37#16839959C2211yescharacteristic
38#16839893C2211yescharacteristic
39#16839893C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839867—21no—
42#16839862—21no—
43Q16×Dic61—21no—
44C4×#11758202C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839507C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839492C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#559C2211yescharacteristic
57C61×#557C2211yescharacteristic
58C61×#497—21no—
59C61×#492C2211yescharacteristic
60C61×#464—21no—
61C61×#462C2211yescharacteristic
62C61×#461—21no—
63C61×#460C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#440—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202D411yescharacteristic
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73#14519777—21no—
74#14519777—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519768—21no—
78#14519768—21no—
79#14519768—41no—
80#14519768—41no—
81Q8×Dic61—21no—
82Q8×Dic61—21no—
83C4×Dic122—21no—
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86#14519711—21no—
87#14519711—21no—
88#14519710D411yescharacteristic
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—41no—
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701D411yescharacteristic
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700D411yescharacteristic
108#14519698—21no—
109#14519698—41no—
110#14519688—21no—
111#14519688—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×Q16×C61—21no—
116C2×Q16×C61D411yescharacteristic
117C2×SD16×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#125×C61—21no—
124#125×C61—41no—
125#124×C61—21no—
126#124×C61—21no—
127#123×C61—41no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61C2×D411yescharacteristic
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C22×C4×C61—41no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122C2×D411yescharacteristic
155Q16×C61—22no—
156Q16×C61—81no—
157SD16×C61—42no—
158SD16×C61—81no—
159SD16×C61—81no—
160C2×C8×C61—21no—
161C2×C8×C61C2×D411yescharacteristic
162C2×C8×C61C2×D411yescharacteristic
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—41no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—41no—
181C42×C61C2411yescharacteristic
182#11758202—41no—
183#11758202—42no—
184#11758202—81no—
185#11758200—42no—
186#11758200—81no—
187#11758200—81no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—41no—
193C4⋊Dic61—41no—
194C4⋊Dic61—41no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—41no—
199#11758196—41no—
200C4×Dic61—21no—
201C4×Dic61—41no—
202C4×Dic61—41no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—41no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—41no—
217Q8×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—81no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—161no—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238Dic122—21no—
239Dic122—21no—
240Dic122—41no—
241Dic122—41no—
242Dic122—41no—
243Dic122—41no—
244C8×C61—21no—
245C8×C61—41no—
246C8×C61#13712yes—
247#92418—21no—
248#92418—21no—
249#92418—41no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#20281—611nomaximal, Sylow
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61D4211yescharacteristic
258C4×C61—21no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—81no—
266C4×C61C2×#13711yescharacteristic
267C4×C61#57511yescharacteristic
268C4×C61#57711yescharacteristic
269C4×C61#57711yescharacteristic
270Dic61—41no—
271Dic61—41no—
272Dic61—81no—
273Dic61—81no—
274#3400—611no—
275#3400D6111yescharacteristic
276#3383—611no—
277#3374—611no—
278#1714—611no—
279#1700—611no—
280#1694—611no—
281#1686—611no—
282#1666—611no—
283#1653—611no—
284#1653—611no—
285#1640—611no—
286#1626—611no—
287#1593—611no—
288#1584—611no—
289C2×C61—161no—
290C2×C61#327111yescharacteristic
291C2×C61#327811yescharacteristic
292C2×C61#328911yescharacteristic
293#559C2×D6111yescharacteristic
294#557—611no—
295#557—611no—
296#557C2×D6111yescharacteristic
297#500—611no—
298#499—1221no—
299#498—611no—
300#497—21no—
301#497—1221no—
302#492—611no—
303#492C2×D6111yescharacteristic
304#491—1221no—
305#487—611no—
306#486—611no—
307#479—1221no—
308#479—1221no—
309#479—1221no—
310#475—611no—
311#475—611no—
312#475—611no—
313#464—21no—
314#464—1221no—
315#462—611no—
316#462—611no—
317#462—1221no—
318#462C2×D6111yescharacteristic
319#461—21no—
320#461—1221no—
321#460—611no—
322#460C2×D6111yescharacteristic
323#446—611no—
324#446—611no—
325#446C2×D6111yescharacteristic
326C8×Q8—611no—
327#441—1221no—
328#440—21no—
329#440—1221no—
330C4×Q16—611no—
331C4×Q16—1221no—
332C4×Q16C2×D6111yescharacteristic
333#335—611no—
334#334—611no—
335#332—611no—
336#332—611no—
337#332—611no—
338#331—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342Q8⋊C8—611no—
343#322—611no—
344C61#2028111yescharacteristic, complemented, Sylow
345C2×Q16—21no—
346C2×Q16—611no—
347C2×Q16—1221no—
348C2×Q16—1221no—
349C2×Q16C22⋊D6111yescharacteristic
350C2×SD16—21no—
351C2×SD16—21no—
352C2×SD16—21no—
353C2×SD16—1221no—
354C2×SD16—1221no—
355C2×SD16—1221no—
356C4⋊Q8—21no—
357C4⋊Q8—611no—
358C4⋊Q8—611no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361C4⋊Q8C22×D6111yescharacteristic
362#127C22×D6111yescharacteristic
363#125—21no—
364#125—41no—
365#125—1221no—
366#125—1221no—
367#125—2441no—
368#125—2441no—
369#124—21no—
370#124—21no—
371#123—41no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—611no—
375C4×Q8—1221no—
376C4×Q8—1221no—
377C4×Q8—1221no—
378C4×Q8—1221no—
379C4×Q8C22×D6111yescharacteristic
380C4×Q8C22×D6111yescharacteristic
381C4×Q8C22×D6111yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384#107—611no—
385#107—1221no—
386#107—1221no—
387#107C22⋊D6111yescharacteristic
388#106—1221no—
389#106—1221no—
390#106—1221no—
391#106—2441no—
392C4⋊C8—21no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399C4⋊C8C22×D6111yescharacteristic
400#103—21no—
401#103—611no—
402#103—611no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103C22⋊D6111yescharacteristic
411#103C22⋊D6111yescharacteristic
412#102—21no—
413#102—1221no—
414#102—2441no—
415#97—21no—
416#97—1221no—
417#97—1221no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—1221no—
421C4×C8C22×D6111yescharacteristic
422C2×Q8—21no—
423C2×Q8—611no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8D4×D6111yescharacteristic
427C2×Q8C2×C22⋊D6111yescharacteristic
428C2×Q8C2×C22⋊D6111yescharacteristic
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C22×C4—41no—
433Q16—22no—
434Q16—81no—
435Q16—4881no—
436Q16—2441no—
437Q16—2442no—
438SD16—42no—
439SD16—81no—
440SD16—81no—
441SD16—4881no—
442SD16—4881no—
443SD16—2442no—
444C2×C8—21no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8C2×C22⋊D6111yescharacteristic
452C2×C8C2×C22⋊D6111yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—611no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4D4×D6111yescharacteristic
474C4⋊C4C2×C22⋊D6111yescharacteristic
475C4⋊C4C2×C22⋊D6111yescharacteristic
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—41no—
483C42—1221no—
484C42—2441no—
485C42—2441no—
486C42C23×D6111yescharacteristic
487C23—41no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—41no—
495Q8—41no—
496Q8—1221no—
497Q8—1221no—
498Q8—2441no—
499Q8—2441no—
500Q8—2441no—
501Q8—2441no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—81no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—161no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C2×C4#1451982211yescharacteristic
522C8—21no—
523C8—41no—
524C8—1221no—
525C8—1221no—
526C8—2441no—
527C8—2441no—
528C8—2441no—
529C8—2441no—
530C8#1451980712yes—
531C22—81no—
532C22—81no—
533C22—81no—
534C22D4×C22⋊D6111yescharacteristic
535C4—21no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—81no—
543C4—4881no—
544C4—4881no—
545C4—2441no—
546C4—2441no—
547C4C2×#1451980711yescharacteristic
548C4#1684056111yescharacteristic
549C4#1684057111yescharacteristic
550C4#1684057311yescharacteristic
551C2—161no—
552C2#2145739811yescharacteristic
553C2#2145748511yescharacteristic
554C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3400C2
socleC22×C61D42
prime radical#127×C61C23
hypercenter#3400D61
hypocenterC61#20281
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3400

Complements

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

NH ≅ G/Ntype
2. #21458708C2semidirect
3. #21458622C2semidirect
5. #21445375C2semidirect
6. #21445337C2semidirect
9. #21445096C2semidirect
10. #21445094C2semidirect
13. #21444728C2semidirect
14. #21444667C2semidirect
344. C61#20281semidirect

Maximal subgroups

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

subgroupindex
#214588722
#214587082
#214586222
#214454382
#214453752
#214453372
#214452892
#214451642
#214450962
#214450942
#214450052
#214449142
#214447282
#214446672
C61×#34002
#2028161