Finite Group Databasebeta

← #29670361

Subgroups — #29670361

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458692C211yesmaximal, characteristic, complemented
2#21458461C211yesmaximal, characteristic, complemented
3#21458135C211yesmaximal, characteristic
4#21445467C211yesmaximal, characteristic, complemented
5#21445368C211yesmaximal, characteristic, complemented
6#21445275C211yesmaximal, characteristic, complemented
7#21445246C211yesmaximal, characteristic, complemented
8#21445194C211yesmaximal, characteristic, complemented
9#21445060C211yesmaximal, characteristic, complemented
10#21445054C211yesmaximal, characteristic
11#21445024C211yesmaximal, characteristic, complemented
12#21444984C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3298C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840162—21no—
20#16840160C2211yescharacteristic
21#16840114—21no—
22#16840114C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840090—21no—
26#16840067—21no—
27#16840064—21no—
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840044—21no—
31#16840044C2211yescharacteristic
32#16840042C2211yescharacteristic
33#16840018—21no—
34#16839969C2211yescharacteristic
35#16839967—21no—
36#16839967C2211yescharacteristic
37#16839924—21no—
38Q8×#92418C2211yescharacteristic
39C8×Dic122C2211yescharacteristic
40SD16×Dic61—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839763—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56D4×Q8×C61C2211yescharacteristic
57C61×#547C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#483—21no—
60C61×#475C2211yescharacteristic
61C61×#465—21no—
62C61×#461C2211yescharacteristic
63C61×#448—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#103×C61—21no—
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758200D411yescharacteristic
70C2×#11758194—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519768—21no—
75#14519768—21no—
76#14519768—41no—
77#14519756—21no—
78#14519754—41no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82D4×Dic61—21no—
83C2×C4⋊Dic61—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—41no—
87#14519709—21no—
88#14519709—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708D411yescharacteristic
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—41no—
106#14519700D411yescharacteristic
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C22×Q8×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C22×C8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#123×C61—41no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127#121×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#106×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144#41×C61—42no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—41no—
153C2×D4×C61C2×D411yescharacteristic
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C22×C4×C61—41no—
159C22×Dic61—41no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122C2×D411yescharacteristic
163SD16×C61—22no—
164SD16×C61—81no—
165C2×C8×C61—21no—
166C2×C8×C61—81no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61C2411yescharacteristic
185C22⋊Dic61—41no—
186#11758202—81no—
187#11758200—22no—
188#11758200—42no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
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—
202#11758194—81no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210C23×C61—21no—
211Q8×C61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—82no—
217Q8×C61#13712yes—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—82no—
222D4×C61#13512yes—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61—82no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—161no—
243Dic122—21no—
244Dic122—21no—
245Dic122—41no—
246Dic122—41no—
247Dic122—81no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#19822—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61D4211yescharacteristic
265C4×C61—22no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—81no—
271C4×C61—81no—
272C4×C61C2×#13511yescharacteristic
273C4×C61C2×#13711yescharacteristic
274C4×C61#57711yescharacteristic
275C4×C61#57711yescharacteristic
276Dic61—41no—
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280#3381—611no—
281#3359—611no—
282#3331—611no—
283#3298D6111yescharacteristic
284#1718—611no—
285#1699—611no—
286#1684—611no—
287#1680—611no—
288#1671—611no—
289#1648—611no—
290#1647—611no—
291#1643—611no—
292#1637—611no—
293#1583—611no—
294SD16⋊C8—611no—
295C2×C61—42no—
296C2×C61—81no—
297C2×C61#326611yescharacteristic
298C2×C61#328211yescharacteristic
299C2×C61#328911yescharacteristic
300#557—611no—
301#554—611no—
302D4×Q8C2×D6111yescharacteristic
303#547C2×D6111yescharacteristic
304#500—611no—
305#499—1221no—
306#498—611no—
307#487—611no—
308#487—611no—
309#487—611no—
310#487—1221no—
311#487C2×D6111yescharacteristic
312#483—21no—
313#483—1221no—
314#479—1221no—
315#478—1221no—
316#477—611no—
317#476—611no—
318#475—611no—
319#475—1221no—
320#475C2×D6111yescharacteristic
321#474—611no—
322#470—1221no—
323#465—21no—
324#462—611no—
325#461—611no—
326#461—1221no—
327#461C2×D6111yescharacteristic
328#452—1221no—
329#448—21no—
330C8×Q8—611no—
331C8×Q8—611no—
332C4×SD16—611no—
333C4×SD16—1221no—
334C4×SD16—1221no—
335C4×SD16C2×D6111yescharacteristic
336#436—611no—
337C8×D4C2×D6111yescharacteristic
338#417—1221no—
339C2×#103—21no—
340#335—611no—
341#334—611no—
342#333—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348D4⋊C8—611no—
349D4⋊C8—611no—
350#322—611no—
351C61#1982211yescharacteristic, complemented, Sylow
352C2×#41—21no—
353C22×Q8—21no—
354C2×Q16—1221no—
355C2×SD16—21no—
356C2×SD16—611no—
357C2×SD16—1221no—
358C2×SD16C22⋊D6111yescharacteristic
359C2×M16—1221no—
360C22×C8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—611no—
363C4⋊Q8—611no—
364C4⋊Q8—1221no—
365C4⋊Q8C22×D6111yescharacteristic
366C4⋊Q8C22×D6111yescharacteristic
367#125—1221no—
368#125—1221no—
369#125—2441no—
370#123—41no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—41no—
374C22⋊Q8—1221no—
375C22⋊Q8—2441no—
376#121—21no—
377C4×Q8—611no—
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8C22×D6111yescharacteristic
381C4×D4—21no—
382C4×D4—1221no—
383C4×D4C22×D6111yescharacteristic
384C4×D4C22×D6111yescharacteristic
385C2×C4⋊C4—21no—
386C2×C4⋊C4—1221no—
387#107—1221no—
388#107—1221no—
389#107—2441no—
390#106—21no—
391#106—611no—
392#106—1221no—
393#106—1221no—
394#106—1221no—
395#106—1221no—
396#106C22⋊D6111yescharacteristic
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8C22×D6111yescharacteristic
403#103—21no—
404#103—21no—
405#103—21no—
406#103—21no—
407#103—611no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—2441no—
417#103C22⋊D6111yescharacteristic
418#102—611no—
419#102—1221no—
420#102—1221no—
421#102C22⋊D6111yescharacteristic
422C22⋊C8—21no—
423C22⋊C8—1221no—
424C4×C8—611no—
425C4×C8—611no—
426C4×C8—1221no—
427C4×C8—1221no—
428C4×C8C22×D6111yescharacteristic
429#41—42no—
430#41—81no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—41no—
434C2×Q8—42no—
435C2×Q8—81no—
436C2×Q8—611no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8C2×C22⋊D6111yescharacteristic
440C2×D4—41no—
441C2×D4D4×D6111yescharacteristic
442C2×D4C2×C22⋊D6111yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—2441no—
448Q16—4881no—
449SD16—22no—
450SD16—81no—
451SD16—1222no—
452SD16—2442no—
453M16—4881no—
454C2×C8—21no—
455C2×C8—81no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8C2×C22⋊D6111yescharacteristic
463C2×C8C2×C22⋊D6111yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—611no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4D4×D6111yescharacteristic
483C4⋊C4C2×C22⋊D6111yescharacteristic
484C4⋊C4C2×C22⋊D6111yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—2441no—
490C42—21no—
491C42—1221no—
492C42—2441no—
493C42C23×D6111yescharacteristic
494C23—21no—
495C23—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—82no—
502Q8—1221no—
503Q8—1221no—
504Q8—4881no—
505Q8—2441no—
506Q8—2441no—
507Q8#1451980712yes—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—82no—
512D4#1451979812yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—9761no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4#1451982211yescharacteristic
533C8—21no—
534C8—21no—
535C8—41no—
536C8—1221no—
537C8—1221no—
538C8—2441no—
539C8—2441no—
540C8—2441no—
541C8—2441no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22D4×C22⋊D6111yescharacteristic
549C4—22no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—4881no—
557C4—4881no—
558C4—2441no—
559C4—2441no—
560C4C2×#1451979811yescharacteristic
561C4C2×#1451980711yescharacteristic
562C4#1684056811yescharacteristic
563C4#1684057311yescharacteristic
564C2—42no—
565C2—81no—
566C2#2145732511yescharacteristic
567C2#2145752611yescharacteristic
568C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3298C2
socleC22×C61D42
prime radicalC61×#547C22
hypercenter#3298D61
hypocenterC61#19822
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3298

Complements

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

NH ≅ G/Ntype
1. #21458692C2semidirect
2. #21458461C2semidirect
4. #21445467C2semidirect
5. #21445368C2semidirect
6. #21445275C2semidirect
7. #21445246C2semidirect
8. #21445194C2semidirect
9. #21445060C2semidirect
11. #21445024C2semidirect
12. #21444984C2semidirect
13. #21444663C2semidirect
14. #21444631C2semidirect
351. C61#19822semidirect

Maximal subgroups

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

subgroupindex
#214586922
#214584612
#214581352
#214454672
#214453682
#214452752
#214452462
#214451942
#214450602
#214450542
#214450242
#214449842
#214446632
#214446312
C61×#32982
#1982261