Finite Group Databasebeta

← #29676464

Subgroups — #29676464

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458848C211yesmaximal, characteristic, complemented
2#21458740C211yesmaximal, characteristic, complemented
3#21458232C211yesmaximal, characteristic
4#21445498C211yesmaximal, characteristic, complemented
5#21445398C211yesmaximal, characteristic, complemented
6#21445327C211yesmaximal, characteristic, complemented
7#21445308C211yesmaximal, characteristic, complemented
8#21445206C211yesmaximal, characteristic, complemented
9#21445073C211yesmaximal, characteristic, complemented
10#21445068C211yesmaximal, characteristic, complemented
11#21445047C211yesmaximal, characteristic
12#21444975C211yesmaximal, characteristic, complemented
13#21444721C211yesmaximal, characteristic, complemented
14#21444656C211yesmaximal, characteristic, complemented
15C61×#3329C211yesmaximal, characteristic
16#16840490C2211yescharacteristic
17#16840482C2211yescharacteristic
18#16840168—21no—
19#16840165C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840153—21no—
23#16840142C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840103—21no—
26#16840083—21no—
27#16840062—21no—
28#16840054—21no—
29#16840054C2211yescharacteristic
30#16840044C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16840042C2211yescharacteristic
33#16839989—21no—
34#16839983—21no—
35#16839972C2211yescharacteristic
36#16839971—21no—
37#16839967C2211yescharacteristic
38#16839893C2211yescharacteristic
39#16839892C2211yescharacteristic
40#16839868—21no—
41#16839862—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#16839798—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#555C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#485—21no—
60C61×#477C2211yescharacteristic
61C61×#468—21no—
62C61×#461C2211yescharacteristic
63C61×#448—21no—
64C4×SD16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#419—21no—
67C2×Dic244—21no—
68C2×Dic244—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758200D411yescharacteristic
72C22×#92418—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519775—21no—
76#14519770—41no—
77#14519769—21no—
78#14519767—41no—
79#14519764—21no—
80#14519756—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519742—21no—
85#14519736—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—41no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708D411yescharacteristic
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519701—41no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700D411yescharacteristic
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519688—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C2×SD16×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C2×M16×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#124×C61—41no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125#121×C61—21no—
126#121×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#117×C61—21no—
133#107×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
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—42no—
148C2×Q8×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—41no—
151C2×D4×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122C2×D411yescharacteristic
159SD16×C61—41no—
160SD16×C61—81no—
161M16×C61—81no—
162C2×C8×C61—21no—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C42×C61—21no—
180C42×C61—41no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—41no—
183C22⋊Dic61—41no—
184#11758202—81no—
185#11758200—22no—
186#11758200—81no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C2×D411yescharacteristic
192#11758196—21no—
193#11758196—41no—
194#11758196—41no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—21no—
198C4×Dic61—41no—
199C4×Dic61—41no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—42no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207Dic244—42no—
208Dic244—81no—
209C23×C61—21no—
210C23×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—81no—
216Q8×C61—82no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—161no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—82no—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240Dic122—21no—
241Dic122—21no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246C8×C61—21no—
247C8×C61—21no—
248C8×C61—41no—
249#92418—21no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#92418#13712yes—
255#20261—611nomaximal, Sylow
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61D4211yescharacteristic
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61C2×#13711yescharacteristic
271C4×C61#57511yescharacteristic
272C4×C61#57611yescharacteristic
273C4×C61#57811yescharacteristic
274Dic61—41no—
275Dic61—41no—
276Dic61—81no—
277Dic61—81no—
278#3398—611no—
279#3386—611no—
280#3340—611no—
281#3329D6111yescharacteristic
282#1723—611no—
283#1704—611no—
284#1692—611no—
285#1689—611no—
286#1673—611no—
287#1650—611no—
288#1649—611no—
289#1646—611no—
290#1636—611no—
291#1592—611no—
292#1582—611no—
293C2×C61—81no—
294C2×C61—81no—
295C2×C61#326811yescharacteristic
296C2×C61#327611yescharacteristic
297C2×C61#328911yescharacteristic
298#557—611no—
299#556—611no—
300#555C2×D6111yescharacteristic
301D4×Q8C2×D6111yescharacteristic
302#501—1221no—
303#500—611no—
304#500—611no—
305#498—611no—
306#497—1221no—
307#494—611no—
308#487—611no—
309#487C2×D6111yescharacteristic
310#485—21no—
311#485—1221no—
312#482—1221no—
313#478—1221no—
314#477—611no—
315#477—1221no—
316#477C2×D6111yescharacteristic
317#475—611no—
318#474—611no—
319#474—611no—
320#468—21no—
321#465—1221no—
322#464—1221no—
323#462—611no—
324#462—1221no—
325#461—611no—
326#461C2×D6111yescharacteristic
327#448—21no—
328#446—611no—
329#446—611no—
330#441—1221no—
331#440—1221no—
332C4×SD16—611no—
333C4×SD16C2×D6111yescharacteristic
334#436C2×D6111yescharacteristic
335C8×D4—611no—
336#427—1221no—
337#419—21no—
338#335—611no—
339#334—611no—
340#333—611no—
341#332—611no—
342#332—611no—
343#332—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346D4⋊C8—611no—
347D4⋊C8—611no—
348#322—611no—
349C61#2026111yescharacteristic, complemented, Sylow
350C22×Q8—21no—
351C2×Q16—1221no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×SD16—21no—
355C2×SD16—611no—
356C2×SD16—1221no—
357C2×SD16C22⋊D6111yescharacteristic
358C2×M16—21no—
359C22×C8—1221no—
360C4⋊Q8—21no—
361C4⋊Q8—611no—
362C4⋊Q8—611no—
363C4⋊Q8—1221no—
364C4⋊Q8C22×D6111yescharacteristic
365C4⋊Q8C22×D6111yescharacteristic
366#126—2441no—
367#125—1221no—
368#125—2441no—
369#124—41no—
370#124—1221no—
371C22⋊Q8—21no—
372C22⋊Q8—41no—
373C22⋊Q8—1221no—
374#121—21no—
375#121—21no—
376C4×Q8—611no—
377C4×Q8—1221no—
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8C22×D6111yescharacteristic
381C4×D4—21no—
382C4×D4C22×D6111yescharacteristic
383C4×D4C22×D6111yescharacteristic
384#117—21no—
385#117—21no—
386#117—1221no—
387#107—21no—
388#107—1221no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#107—2441no—
394#106—611no—
395#106—1221no—
396#106—1221no—
397#106—1221no—
398#106C22⋊D6111yescharacteristic
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103—611no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—2441no—
415#103C22⋊D6111yescharacteristic
416#102—611no—
417#102—1221no—
418#102—1221no—
419#102C22⋊D6111yescharacteristic
420C22⋊C8—21no—
421C22⋊C8—1221no—
422#97—1221no—
423#97—1221no—
424C4×C8—611no—
425C4×C8—611no—
426C4×C8C22×D6111yescharacteristic
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—41no—
430C2×Q8—42no—
431C2×Q8—81no—
432C2×Q8—611no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8C2×C22⋊D6111yescharacteristic
436C2×D4—41no—
437C2×D4D4×D6111yescharacteristic
438C2×D4C2×C22⋊D6111yescharacteristic
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—41no—
442Q16—4881no—
443Q16—4881no—
444Q16—2442no—
445SD16—41no—
446SD16—81no—
447SD16—1222no—
448SD16—4881no—
449M16—81no—
450C2×C8—21no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—2442no—
458C2×C8C2×C22⋊D6111yescharacteristic
459C2×C8C2×C22⋊D6111yescharacteristic
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—41no—
466C4⋊C4—611no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4D4×D6111yescharacteristic
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C42—21no—
487C42—41no—
488C42—1221no—
489C42—2441no—
490C42—2441no—
491C42C23×D6111yescharacteristic
492C23—21no—
493C23—21no—
494Q8—21no—
495Q8—21no—
496Q8—41no—
497Q8—41no—
498Q8—81no—
499Q8—82no—
500Q8—1221no—
501Q8—1221no—
502Q8—2441no—
503Q8—2441no—
504Q8—2441no—
505Q8—2441no—
506D4—21no—
507D4—21no—
508D4—21no—
509D4—21no—
510D4—161no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4#1451982211yescharacteristic
529C8—21no—
530C8—21no—
531C8—41no—
532C8—612no—
533C8—1221no—
534C8—2441no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22D4×C22⋊D6111yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—81no—
551C4—81no—
552C4—4881no—
553C4—4881no—
554C4—2441no—
555C4—2441no—
556C4#137×D6111yescharacteristic
557C4#1684056011yescharacteristic
558C4#1684056511yescharacteristic
559C4#1684057811yescharacteristic
560C2—81no—
561C2—81no—
562C2#2145735511yescharacteristic
563C2#2145745911yescharacteristic
564C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3329C2
socleC22×C61D42
prime radicalC61×#555C22
hypercenter#3329D61
hypocenterC61#20261
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3329

Complements

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

NH ≅ G/Ntype
1. #21458848C2semidirect
2. #21458740C2semidirect
4. #21445498C2semidirect
5. #21445398C2semidirect
6. #21445327C2semidirect
7. #21445308C2semidirect
8. #21445206C2semidirect
9. #21445073C2semidirect
10. #21445068C2semidirect
12. #21444975C2semidirect
13. #21444721C2semidirect
14. #21444656C2semidirect
349. C61#20261semidirect

Maximal subgroups

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

subgroupindex
#214588482
#214587402
#214582322
#214454982
#214453982
#214453272
#214453082
#214452062
#214450732
#214450682
#214450472
#214449752
#214447212
#214446562
C61×#33292
#2026161