Finite Group Databasebeta

← #29670871

Subgroups — #29670871

Order 15616.

570 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458850C211yesmaximal, characteristic, complemented
2#21458502C211yesmaximal, characteristic, complemented
3#21458232C211yesmaximal, characteristic
4#21445451C211yesmaximal, characteristic, complemented
5#21445327C211yesmaximal, characteristic, complemented
6#21445242C211yesmaximal, characteristic, complemented
7#21445198C211yesmaximal, characteristic, complemented
8#21445182C211yesmaximal, characteristic, complemented
9#21445089C211yesmaximal, characteristic, complemented
10#21445088C211yesmaximal, characteristic, complemented
11#21445047C211yesmaximal, characteristic
12#21444976C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3340C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17#16840482C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840157—21no—
21#16840141C2211yescharacteristic
22#16840115—21no—
23#16840113C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840103—21no—
26#16840083—21no—
27#16840072—21no—
28#16840054C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16839989—21no—
32#16839982—21no—
33#16839981—21no—
34#16839972—21no—
35#16839972C2211yescharacteristic
36#16839969C2211yescharacteristic
37#16839967C2211yescharacteristic
38#16839893C2211yescharacteristic
39C8×Dic122C2211yescharacteristic
40#16839867—21no—
41#16839854—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#16839798—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#556C2211yescharacteristic
57C61×#555C2211yescharacteristic
58C61×#498C2211yescharacteristic
59C61×#485—21no—
60C61×#482—21no—
61C61×#477C2211yescharacteristic
62C61×#474C2211yescharacteristic
63C61×#465—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C61×#427—21no—
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758200D411yescharacteristic
73C22×#92418—21no—
74#14519777—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519770—41no—
78#14519768—21no—
79#14519768—41no—
80#14519764—21no—
81#14519756—21no—
82Q8×Dic61—21no—
83C4×Dic122—21no—
84C4×Dic122C2311yescharacteristic
85#14519742—21no—
86#14519736—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—41no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519702—21no—
100#14519702—21no—
101#14519701—21no—
102#14519701—41no—
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—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×SD16×C61D411yescharacteristic
116C22×C8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#126×C61—41no—
120#124×C61—21no—
121#124×C61—41no—
122C22⋊Q8×C61—21no—
123#121×C61—21no—
124#121×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61C2311yescharacteristic
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61—21no—
131#117×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—41no—
148C2×D4×C61C2×D411yescharacteristic
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122C2×D411yescharacteristic
155SD16×C61—41no—
156C2×C8×C61—21no—
157C2×C8×C61—42no—
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2×D411yescharacteristic
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—41no—
166C4⋊C4×C61C2×D411yescharacteristic
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—41no—
172C22⋊C4×C61—41no—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C42×C61—21no—
176C42×C61—41no—
177C42×C61—41no—
178C42×C61C2411yescharacteristic
179C22⋊Dic61—41no—
180C22⋊Dic61—41no—
181#11758202—42no—
182#11758202—42no—
183#11758202—81no—
184#11758202—81no—
185#11758200—22no—
186#11758200—81no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—41no—
192C4⋊Dic61—41no—
193C4⋊Dic61C2×D411yescharacteristic
194#11758196—21no—
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
207C23×C61—21no—
208C23×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—81no—
214D4×C61—21no—
215D4×C61—21no—
216D4×C61—21no—
217D4×C61—21no—
218D4×C61—161no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236Dic122—21no—
237Dic122—21no—
238Dic122—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—41no—
242C8×C61—21no—
243C8×C61—21no—
244C8×C61—41no—
245#92418—21no—
246#92418—41no—
247#92418—41no—
248#92418—41no—
249#92418—41no—
250#92418#13512yes—
251#19856—611nomaximal, Sylow
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61D4211yescharacteristic
259C4×C61—21no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61C2×#13511yescharacteristic
267C4×C61#57511yescharacteristic
268C4×C61#57711yescharacteristic
269C4×C61#57811yescharacteristic
270Dic61—41no—
271Dic61—41no—
272Dic61—81no—
273Dic61—81no—
274#3398—611no—
275#3364—611no—
276#3340—611no—
277#3340D6111yescharacteristic
278#1716—611no—
279#1692—611no—
280#1679—611no—
281#1672—611no—
282#1669—611no—
283#1652—611no—
284#1652—611no—
285#1646—611no—
286#1636—611no—
287#1582—611no—
288SD16⋊C8—611no—
289C2×C61—81no—
290C2×C61—81no—
291C2×C61#326911yescharacteristic
292C2×C61#328411yescharacteristic
293C2×C61#328911yescharacteristic
294#557—611no—
295#556—611no—
296#556C2×D6111yescharacteristic
297#555C2×D6111yescharacteristic
298#498—611no—
299#498—611no—
300#498C2×D6111yescharacteristic
301#497—1221no—
302#494—611no—
303#488—1221no—
304#487—611no—
305#487—611no—
306#485—21no—
307#485—1221no—
308#482—21no—
309#482—1221no—
310#479—1221no—
311#477—611no—
312#477C2×D6111yescharacteristic
313#474—611no—
314#474—611no—
315#474C2×D6111yescharacteristic
316#465—21no—
317#465—1221no—
318#464—1221no—
319#464—1221no—
320#462—611no—
321#462—611no—
322#462—1221no—
323#461—611no—
324#446—611no—
325C8×Q8—611no—
326#441—1221no—
327C4×Q16—1221no—
328C4×SD16—611no—
329C4×SD16C2×D6111yescharacteristic
330C8×D4—611no—
331C8×D4C2×D6111yescharacteristic
332#427—21no—
333#427—1221no—
334#335—611no—
335#334—611no—
336#333—611no—
337#332—611no—
338#332—611no—
339#332—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342D4⋊C8—611no—
343D4⋊C8—611no—
344#322—611no—
345C61#1985611yescharacteristic, complemented, Sylow
346C2×Q16—1221no—
347C2×Q16—1221no—
348C2×Q16—1221no—
349C2×Q16—1221no—
350C2×SD16—611no—
351C2×SD16—1221no—
352C2×SD16C22⋊D6111yescharacteristic
353C22×C8—21no—
354C22×C8—1221no—
355C4⋊Q8—611no—
356C4⋊Q8—611no—
357C4⋊Q8—1221no—
358C4⋊Q8C22×D6111yescharacteristic
359C4⋊Q8C22×D6111yescharacteristic
360#126—41no—
361#126—2441no—
362#125—1221no—
363#125—2441no—
364#124—21no—
365#124—41no—
366#124—1221no—
367C22⋊Q8—21no—
368C22⋊Q8—1221no—
369#121—21no—
370#121—21no—
371C4×Q8—21no—
372C4×Q8—611no—
373C4×Q8—1221no—
374C4×Q8—1221no—
375C4×Q8—1221no—
376C4×Q8C22×D6111yescharacteristic
377C4×D4C22×D6111yescharacteristic
378C4×D4C22×D6111yescharacteristic
379#117—21no—
380#117—21no—
381#117—21no—
382#117—1221no—
383#107—21no—
384#107—21no—
385#107—1221no—
386#107—1221no—
387#107—1221no—
388#106—21no—
389#106—21no—
390#106—611no—
391#106—1221no—
392#106—1221no—
393#106—2441no—
394#106C22⋊D6111yescharacteristic
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8C22×D6111yescharacteristic
401#103—21no—
402#103—611no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—2441no—
410#103C22⋊D6111yescharacteristic
411#102—21no—
412#102—611no—
413#102—1221no—
414#102—1221no—
415#102C22⋊D6111yescharacteristic
416C22⋊C8—21no—
417C22⋊C8—1221no—
418#97—1221no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8—1221no—
422C4×C8C22×D6111yescharacteristic
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—611no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8C2×C22⋊D6111yescharacteristic
429C2×D4—41no—
430C2×D4D4×D6111yescharacteristic
431C2×D4C2×C22⋊D6111yescharacteristic
432C22×C4—21no—
433C22×C4—21no—
434Q16—4881no—
435Q16—4881no—
436Q16—2442no—
437Q16—2442no—
438SD16—41no—
439SD16—1222no—
440SD16—4881no—
441C2×C8—21no—
442C2×C8—42no—
443C2×C8—611no—
444C2×C8—611no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—2442no—
450C2×C8C2×C22⋊D6111yescharacteristic
451C2×C8C2×C22⋊D6111yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—611no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4D4×D6111yescharacteristic
469C4⋊C4C2×C22⋊D6111yescharacteristic
470C4⋊C4C2×C22⋊D6111yescharacteristic
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—2441no—
478C22⋊C4—2441no—
479C42—21no—
480C42—41no—
481C42—41no—
482C42—1221no—
483C42—2441no—
484C42—2441no—
485C42C23×D6111yescharacteristic
486C23—21no—
487C23—21no—
488Q8—21no—
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493Q8—1221no—
494Q8—1221no—
495Q8—2441no—
496Q8—2441no—
497Q8—2441no—
498Q8—2441no—
499D4—21no—
500D4—21no—
501D4—21no—
502D4—21no—
503D4—161no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4C2×D4×D6111yescharacteristic
519C2×C4C22×C22⋊D6111yescharacteristic
520C2×C4#1451982211yescharacteristic
521C8—21no—
522C8—21no—
523C8—41no—
524C8—612no—
525C8—1221no—
526C8—2441no—
527C8—2441no—
528C8—2441no—
529C8—2441no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22D4×C22⋊D6111yescharacteristic
537C4—21no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—81no—
543C4—81no—
544C4—4881no—
545C4—4881no—
546C4—2441no—
547C4—2441no—
548C4#135×D6111yescharacteristic
549C4#1684056011yescharacteristic
550C4#1684056911yescharacteristic
551C4#1684057711yescharacteristic
552C2—81no—
553C2—81no—
554C2#2145737011yescharacteristic
555C2#2145754311yescharacteristic
556C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3340C2
socleC22×C61D42
prime radicalC61×#555C22
hypercenter#3340D61
hypocenterC61#19856
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3340

Complements

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

NH ≅ G/Ntype
1. #21458850C2semidirect
2. #21458502C2semidirect
4. #21445451C2semidirect
5. #21445327C2semidirect
6. #21445242C2semidirect
7. #21445198C2semidirect
8. #21445182C2semidirect
9. #21445089C2semidirect
10. #21445088C2semidirect
12. #21444976C2semidirect
13. #21444656C2semidirect
14. #21444631C2semidirect
345. C61#19856semidirect

Maximal subgroups

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

subgroupindex
#214588502
#214585022
#214582322
#214454512
#214453272
#214452422
#214451982
#214451822
#214450892
#214450882
#214450472
#214449762
#214446562
#214446312
C61×#33402
#1985661