Finite Group Databasebeta

← #29676465

Subgroups — #29676465

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#21458846C211yesmaximal, characteristic, complemented
2#21458234C211yesmaximal, characteristic
3#21458113C211yesmaximal, characteristic
4#21445497C211yesmaximal, characteristic, complemented
5#21445399C211yesmaximal, characteristic, complemented
6#21445325C211yesmaximal, characteristic, complemented
7#21445306C211yesmaximal, characteristic, complemented
8#21445205C211yesmaximal, characteristic, complemented
9#21445074C211yesmaximal, characteristic, complemented
10#21445068C211yesmaximal, characteristic, complemented
11#21445046C211yesmaximal, characteristic
12#21444977C211yesmaximal, characteristic, complemented
13#21444722C211yesmaximal, characteristic, complemented
14#21444653C211yesmaximal, characteristic, complemented
15C61×#3386C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17#16840477C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840160C2211yescharacteristic
20#16840154—21no—
21#16840141C2211yescharacteristic
22#16840114C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840101—21no—
25#16840100—21no—
26#16840085—21no—
27#16840057C2211yescharacteristic
28#16840057C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840010—21no—
31#16839988—21no—
32#16839984—21no—
33#16839972C2211yescharacteristic
34#16839969—21no—
35#16839964C2211yescharacteristic
36#16839964C2211yescharacteristic
37#16839901—21no—
38#16839893C2211yescharacteristic
39#16839867—21no—
40C4×#11758201C2211yescharacteristic
41#16839840C2211yescharacteristic
42C8×D244C2211yescharacteristic
43#16839797—21no—
44#16839771—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#557C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#501—21no—
59C61×#500C2211yescharacteristic
60C61×#478—21no—
61C61×#477—21no—
62C61×#475C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#440—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758201—21no—
71C2×#11758201—21no—
72C2×#11758201D411yescharacteristic
73C2×#11758194—21no—
74C2×C8×D61—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519772—41no—
78#14519765—21no—
79#14519763—41no—
80#14519753—21no—
81#14519751—21no—
82#14519748—21no—
83C4×Dic122—21no—
84C4×D244C2311yescharacteristic
85#14519736—21no—
86#14519735—21no—
87#14519735—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—41no—
92#14519709—21no—
93#14519709—21no—
94#14519708D411yescharacteristic
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519700—41no—
108#14519698—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#125×C61—21no—
122#125×C61—41no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131#107×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#97×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×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154C2×C22⋊D61—41no—
155C2×D244C2×D411yescharacteristic
156C2×C4×D61—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159SD16×C61—22no—
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—21no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—41no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—41no—
183#11758202—42no—
184#11758202—81no—
185#11758202—81no—
186#11758201—41no—
187#11758201—81no—
188#11758201—81no—
189#11758198—21no—
190#11758198—41no—
191#11758198—41no—
192#11758198—41no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—41no—
199C4×Dic61—41no—
200C4×Dic61—41no—
201#11758194—81no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C8×D61—42no—
209C23×C61—21no—
210C22×D61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—82no—
220D4×C61—21no—
221D4×C61—21no—
222C2×C4×C61—21no—
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—82no—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C22⋊D61—161no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239D244—21no—
240D244—21no—
241C4×D61—41no—
242C4×D61—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—81no—
246C8×C61—21no—
247C8×C61—41no—
248C8×C61#13712yes—
249#92418—21no—
250#92418—21no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#20261—611nomaximal, Sylow
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61D4211yescharacteristic
260C2×D61—41no—
261C2×D61—41no—
262C2×D61—41no—
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61C2×#13711yescharacteristic
272C4×C61#57511yescharacteristic
273C4×C61#57611yescharacteristic
274C4×C61#57811yescharacteristic
275Dic61—41no—
276Dic61—81no—
277Dic61—81no—
278#3398—611no—
279#3386D6111yescharacteristic, complemented
280#3340—611no—
281#3329—611no—
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#326811yescharacteristic
295C2×C61#327611yescharacteristic
296C2×C61#328911yescharacteristic
297D61—81no—
298#557C2×D6111yescharacteristic
299#556—611no—
300#555—611no—
301D4×Q8C2×D6111yescharacteristic
302#501—21no—
303#500—611no—
304#500C2×D6111yescharacteristic
305#498—611no—
306#497—1221no—
307#494—611no—
308#487—611no—
309#487—611no—
310#485—1221no—
311#485—1221no—
312#482—1221no—
313#478—21no—
314#477—21no—
315#477—611no—
316#477—611no—
317#475C2×D6111yescharacteristic
318#474—611no—
319#474C2×D6111yescharacteristic
320#468—1221no—
321#465—1221no—
322#464—1221no—
323#462—611no—
324#462—1221no—
325#461—611no—
326#461—611no—
327#448—1221no—
328#446—611no—
329#446C2×D6111yescharacteristic
330#441—1221no—
331#440—21no—
332C4×SD16—611no—
333C4×SD16C2×D6111yescharacteristic
334#436—611no—
335C8×D4—611no—
336#427—1221no—
337#419—1221no—
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—611no—
355C2×SD16—1221no—
356C2×SD16—1221no—
357C2×SD16C22⋊D6111yescharacteristic
358C2×M16—1221no—
359C22×C8—1221no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—611no—
363C4⋊Q8—611no—
364C4⋊Q8C22×D6111yescharacteristic
365C4⋊Q8C22×D6111yescharacteristic
366#126—2441no—
367#125—21no—
368#125—41no—
369#124—1221no—
370#124—2441no—
371C22⋊Q8—21no—
372C22⋊Q8—41no—
373C22⋊Q8—1221no—
374#121—1221no—
375#121—1221no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8—1221no—
379C4×Q8C22×D6111yescharacteristic
380C4×Q8C22×D6111yescharacteristic
381C4×D4—21no—
382C4×D4—611no—
383C4×D4C22×D6111yescharacteristic
384#117—1221no—
385#117—1221no—
386#117—1221no—
387#107—21no—
388#107—21no—
389#107—21no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#107—2441no—
394#106—21no—
395#106—611no—
396#106—1221no—
397#106—1221no—
398#106C22⋊D6111yescharacteristic
399C4⋊C8—21no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405#103—21no—
406#103—611no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—2441no—
415#103C22⋊D6111yescharacteristic
416#102—21no—
417#102—611no—
418#102—1221no—
419#102C22⋊D6111yescharacteristic
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422#97—21no—
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—1221no—
433C2×Q8—1221no—
434C2×Q8D4×D6111yescharacteristic
435C2×Q8C2×C22⋊D6111yescharacteristic
436C2×D4—611no—
437C2×D4—2441no—
438C2×D4C2×C22⋊D6111yescharacteristic
439C22×C4—21no—
440C22×C4—41no—
441C22×C4—1221no—
442Q16—4881no—
443Q16—4881no—
444Q16—2442no—
445SD16—22no—
446SD16—4881no—
447SD16—4881no—
448SD16—2441no—
449M16—4881no—
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—21no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—611no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—2441no—
476C4⋊C4D4×D6111yescharacteristic
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—41no—
481C22⋊C4—1221no—
482C22⋊C4—2441no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C42—21no—
487C42—21no—
488C42—41no—
489C42—2441no—
490C42—2441no—
491C42C23×D6111yescharacteristic
492C23—21no—
493C23—1221no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—82no—
503Q8—4881no—
504Q8—2441no—
505Q8—2441no—
506D4—21no—
507D4—21no—
508D4—1221no—
509D4—1221no—
510D4—9761no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—82no—
521C2×C4—1221no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4#1451982211yescharacteristic
529C8—21no—
530C8—41no—
531C8—1221no—
532C8—1221no—
533C8—2441no—
534C8—2441no—
535C8—2441no—
536C8—2441no—
537C8#1451980812yes—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—2441no—
542C22—2441no—
543C22—2441no—
544C22D4×C22⋊D6111yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—81no—
552C4—81no—
553C4—4881no—
554C4—4881no—
555C4—2441no—
556C4C2×#1451980811yescharacteristic
557C4#1684055811yescharacteristic
558C4#1684056611yescharacteristic
559C4#1684057711yescharacteristic
560C2—81no—
561C2—4881no—
562C2#2145735311yescharacteristic
563C2#2145746511yescharacteristic
564C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3386C2
socleC22×C61D42
prime radical#16840477C22
hypercenter#3386D61
hypocenterC61#20261
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3386

Complements

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

NH ≅ G/Ntype
1. #21458846C2semidirect
4. #21445497C2semidirect
5. #21445399C2semidirect
6. #21445325C2semidirect
7. #21445306C2semidirect
8. #21445205C2semidirect
9. #21445074C2semidirect
10. #21445068C2semidirect
12. #21444977C2semidirect
13. #21444722C2semidirect
14. #21444653C2semidirect
15. C61×#3386C2semidirect
279. #3386D61semidirect
349. C61#20261semidirect

Maximal subgroups

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

subgroupindex
#214588462
#214582342
#214581132
#214454972
#214453992
#214453252
#214453062
#214452052
#214450742
#214450682
#214450462
#214449772
#214447222
#214446532
C61×#33862
#2026161