Finite Group Databasebeta

← #29670636

Subgroups — #29670636

Order 15616.

600 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458522C211yesmaximal, characteristic
2#21458464C211yesmaximal, characteristic, complemented
3#21458213C211yesmaximal, characteristic, complemented
4#21445471C211yesmaximal, characteristic, complemented
5#21445399C211yesmaximal, characteristic, complemented
6#21445253C211yesmaximal, characteristic, complemented
7#21445233C211yesmaximal, characteristic, complemented
8#21445218C211yesmaximal, characteristic, complemented
9#21445025C211yesmaximal, characteristic, complemented
10#21444991C211yesmaximal, characteristic, complemented
11#21444946C211yesmaximal, characteristic, complemented
12#21444890C211yesmaximal, characteristic, complemented
13#21444670C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15C61×#3304C211yesmaximal, characteristic, complemented
16#16840472C2211yescharacteristic
17#16840472C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840134C2211yescharacteristic
20#16840126—21no—
21#16840116—21no—
22#16840114C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840096—21no—
26#16840096—21no—
27#16840069—21no—
28#16840062—21no—
29#16840057C2211yescharacteristic
30#16840057C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840052C2211yescharacteristic
33#16840000—21no—
34#16839974—21no—
35#16839964C2211yescharacteristic
36#16839954—21no—
37#16839954C2211yescharacteristic
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839872—21no—
41C4×#11758201—21no—
42C4×#11758199C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#107×D61—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#556C2211yescharacteristic
57C61×#550C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#475C2211yescharacteristic
61C61×#465—21no—
62C61×#460C2211yescharacteristic
63C61×#452—21no—
64C4×Q16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C61×#416—21no—
67C2×#11758200—41no—
68C2×#11758201—21no—
69C2×#11758199—21no—
70C2×#11758199D411yescharacteristic
71C2×C8×D61—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519769—21no—
75#14519769—21no—
76#14519755—41no—
77#14519755—41no—
78#14519753—21no—
79#14519753—21no—
80C4×D244—21no—
81C4×D244—21no—
82C4×D244C2311yescharacteristic
83C4⋊C4×D61—21no—
84C4⋊C4×D61—21no—
85#14519711—21no—
86#14519711—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710D411yescharacteristic
90#14519708—21no—
91#14519708—21no—
92C4⋊#92418—21no—
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519700—41no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698D411yescharacteristic
109#14519697D411yescharacteristic
110#14519689—21no—
111#14519688—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×Q16×C61D411yescharacteristic
117C2×SD16×C61—21no—
118C22×C8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120#127×C61—21no—
121#127×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61D411yescharacteristic
138#103×C61D411yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144#41×C61—42no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—41no—
157C2×D244C2×D411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163Q16×C61—41no—
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—41no—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—41no—
184C42×C61C2411yescharacteristic
185#11758201—42no—
186#11758200—82no—
187#11758199—22no—
188#11758199—42no—
189#11758198—21no—
190#11758198—41no—
191#11758198—41no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—41no—
198#11758196—41no—
199#11758196—41no—
200#11758196—41no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C8×D61—42no—
208C23×C61—21no—
209C23×C61—41no—
210C22×D61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—81no—
224D4×C61—82no—
225D4×C61—161no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—82no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241D244—21no—
242D244#13512yes—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61—161no—
246C4×D61—161no—
247Dic122—81no—
248Dic122—81no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—41no—
252#92418—21no—
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#19840—611nomaximal, Sylow
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61D4211yescharacteristic
266C2×D61—22no—
267C2×D61—41no—
268C2×D61—41no—
269C4×C61—21no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—81no—
276C4×C61C2×#13511yescharacteristic
277C4×C61#57511yescharacteristic
278C4×C61#57711yescharacteristic
279C4×C61#57811yescharacteristic
280Dic61—41no—
281Dic61—81no—
282Dic61—81no—
283#3366—611no—
284#3360—611no—
285#3338—611no—
286#3304D6111yescharacteristic, complemented
287#1719—611no—
288#1704—611no—
289#1681—611no—
290#1678—611no—
291#1675—611no—
292#1643—611no—
293#1638—611no—
294#1631—611no—
295#1622—611no—
296#1584—611no—
297#1580—611no—
298C2×C61—81no—
299C2×C61—161no—
300C2×C61#326711yescharacteristic
301C2×C61#328511yescharacteristic
302C2×C61#328811yescharacteristic
303D61—42no—
304#556C2×D6111yescharacteristic
305#554—611no—
306#554—611no—
307#550C2×D6111yescharacteristic
308#500—611no—
309#492—611no—
310#490—1221no—
311#488—1221no—
312#487—611no—
313#487—611no—
314#486—611no—
315#486C2×D6111yescharacteristic
316#484—1221no—
317#484—1221no—
318#482—21no—
319#479—1221no—
320#478—1221no—
321#477—611no—
322#477—611no—
323#476—611no—
324#476—611no—
325#475C2×D6111yescharacteristic
326#467—1221no—
327#465—21no—
328#463—1221no—
329#461—611no—
330#460C2×D6111yescharacteristic
331#459—611no—
332#459—1221no—
333#452—21no—
334#446—611no—
335C8×Q8—611no—
336#442—1221no—
337C4×Q16C2×D6111yescharacteristic
338C4×SD16—1221no—
339C4×D8—611no—
340C8×D4—611no—
341C8×D4C2×D6111yescharacteristic
342C2×#107—1221no—
343#416—21no—
344#335—611no—
345#334—611no—
346#333—611no—
347#333—611no—
348#332—611no—
349#331—611no—
350Q8⋊C8—611no—
351Q8⋊C8—611no—
352D4⋊C8—611no—
353D4⋊C8—611no—
354#322—611no—
355C61#1984011yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C2×Q16C22⋊D6111yescharacteristic
358C2×SD16—21no—
359C2×SD16—1221no—
360C2×SD16—2441no—
361C2×D8—611no—
362C2×D8—1221no—
363C22×C8—21no—
364C22×C8—1221no—
365C4⋊Q8—611no—
366C4⋊Q8—611no—
367C4⋊Q8C22×D6111yescharacteristic
368#127—21no—
369#127C22×D6111yescharacteristic
370#126—41no—
371#125—1221no—
372#125—1221no—
373#124—21no—
374C22⋊Q8—21no—
375C22⋊Q8—1221no—
376C22⋊Q8—1221no—
377C22⋊Q8—2441no—
378C22⋊Q8—2441no—
379#121—21no—
380#121—41no—
381C4×Q8—21no—
382C4×Q8C22×D6111yescharacteristic
383C4×Q8C22×D6111yescharacteristic
384C4×D4—21no—
385C4×D4—611no—
386C4×D4—1221no—
387C4×D4—1221no—
388C4×D4C22×D6111yescharacteristic
389#117—21no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392#107—611no—
393#107—1221no—
394#107—1221no—
395#107—1221no—
396#107—1221no—
397#107C22⋊D6111yescharacteristic
398#106—21no—
399#106—1221no—
400#106—1221no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8C22×D6111yescharacteristic
407#103—21no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—2441no—
414#103C22⋊D6111yescharacteristic
415#103C22⋊D6111yescharacteristic
416#102—21no—
417#102—21no—
418#102—21no—
419#102—611no—
420#102—611no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426C22⋊C8—21no—
427C22⋊C8—1221no—
428#97—1221no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—1221no—
432C4×C8C22×D6111yescharacteristic
433#41—42no—
434#41—81no—
435C2×Q8—21no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8C2×C22⋊D6111yescharacteristic
439C2×Q8C2×C22⋊D6111yescharacteristic
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—611no—
446C2×D4D4×D6111yescharacteristic
447C22×C4—21no—
448C22×C4—41no—
449C22×C4—1221no—
450C22×C4—2441no—
451C22×C4—2441no—
452Q16—41no—
453SD16—81no—
454SD16—4882no—
455SD16—2442no—
456D8—1222no—
457D8—2442no—
458C2×C8—21no—
459C2×C8—81no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—2442no—
467C2×C8C2×C22⋊D6111yescharacteristic
468C2×C8C2×C22⋊D6111yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—611no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4D4×D6111yescharacteristic
484C4⋊C4C2×C22⋊D6111yescharacteristic
485C4⋊C4C2×C22⋊D6111yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—1221no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C42—21no—
494C42—21no—
495C42—41no—
496C42C23×D6111yescharacteristic
497C23—21no—
498C23—41no—
499C23—1221no—
500Q8—21no—
501Q8—21no—
502Q8—21no—
503Q8—21no—
504Q8—41no—
505Q8—41no—
506Q8—4881no—
507Q8—4881no—
508D4—21no—
509D4—21no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—81no—
515D4—82no—
516D4—161no—
517D4—612no—
518D4—1221no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—82no—
528C2×C4—1221no—
529C2×C4—9761no—
530C2×C4—9761no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C22×C22⋊D6111yescharacteristic
537C2×C4#1451982211yescharacteristic
538C8—21no—
539C8—21no—
540C8—41no—
541C8—1221no—
542C8—1221no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—1222no—
554C22—2441no—
555C22—2441no—
556C22D4×C22⋊D6111yescharacteristic
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—81no—
564C4—4881no—
565C4—4881no—
566C4—2441no—
567C4#135×D6111yescharacteristic
568C4#1684056011yescharacteristic
569C4#1684057311yescharacteristic
570C4#1684057611yescharacteristic
571C2—81no—
572C2—161no—
573C2—2442no—
574C2#2145734011yescharacteristic
575C2#2145755411yescharacteristic
576C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3304C2
socleC22×C61D42
prime radical#21458522C2
hypercenter#3304D61
hypocenterC61#19840
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3304

Complements

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

NH ≅ G/Ntype
2. #21458464C2semidirect
3. #21458213C2semidirect
4. #21445471C2semidirect
5. #21445399C2semidirect
6. #21445253C2semidirect
7. #21445233C2semidirect
8. #21445218C2semidirect
9. #21445025C2semidirect
10. #21444991C2semidirect
11. #21444946C2semidirect
12. #21444890C2semidirect
13. #21444670C2semidirect
14. #21444641C2semidirect
15. C61×#3304C2semidirect
286. #3304D61semidirect
355. C61#19840semidirect

Maximal subgroups

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

subgroupindex
#214585222
#214584642
#214582132
#214454712
#214453992
#214452532
#214452332
#214452182
#214450252
#214449912
#214449462
#214448902
#214446702
#214446412
C61×#33042
#1984061