Finite Group Databasebeta

← #29670659

Subgroups — #29670659

Order 15616.

576 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#21458515C211yesmaximal, characteristic, complemented
2#21458477C211yesmaximal, characteristic
3#21458207C211yesmaximal, characteristic, complemented
4#21445473C211yesmaximal, characteristic, complemented
5#21445359C211yesmaximal, characteristic, complemented
6#21445274C211yesmaximal, characteristic, complemented
7#21445258C211yesmaximal, characteristic
8#21445221C211yesmaximal, characteristic, complemented
9#21445024C211yesmaximal, characteristic, complemented
10#21444977C211yesmaximal, characteristic, complemented
11#21444939C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic
13#21444667C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3372C211yesmaximal, characteristic, complemented
16#16840497C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840136C2211yescharacteristic
20#16840119—21no—
21#16840118—21no—
22#16840114C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840109C2211yescharacteristic
25#16840097—21no—
26#16840095—21no—
27#16840061—21no—
28#16840059—21no—
29#16840057C2211yescharacteristic
30#16840057C2211yescharacteristic
31#16840054C2211yescharacteristic
32#16840044C2211yescharacteristic
33#16840001—21no—
34#16839984—21no—
35#16839979—21no—
36#16839962C2211yescharacteristic
37#16839957C2211yescharacteristic
38Q8×#92418C2211yescharacteristic
39C8×Dic122C2211yescharacteristic
40C4×#11758202C2211yescharacteristic
41C4×#11758201—21no—
42#16839842—21no—
43D4×#92418C2211yescharacteristic
44C2×#14519710—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#557C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#491—21no—
59C61×#487C2211yescharacteristic
60C61×#479—21no—
61C61×#477C2211yescharacteristic
62C61×#461—21no—
63C61×#461C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#440—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758202D411yescharacteristic
68C2×#11758201—21no—
69C2×#11758201—41no—
70C2×#11758199—21no—
71C22×#92418—21no—
72#14519777C2311yescharacteristic
73#14519774C2311yescharacteristic
74#14519768—21no—
75#14519766—21no—
76#14519765—21no—
77#14519762—41no—
78#14519756—21no—
79#14519754—41no—
80C4×Dic122C2311yescharacteristic
81C4×D244—21no—
82D4×Dic61—21no—
83#14519739—21no—
84C2×C4⋊Dic61—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701D411yescharacteristic
100#14519700—21no—
101#14519700—21no—
102#14519700—41no—
103#14519700D411yescharacteristic
104#14519699—21no—
105#14519699—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110C22⋊#92418—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×SD16×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61C2311yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120#125×C61—21no—
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
131C2×C4⋊C4×C61—21no—
132#107×C61—21no—
133#106×C61D411yescharacteristic
134C4⋊C8×C61—21no—
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
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—41no—
151C22×Dic61—41no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×C4×D61—41no—
156C2×Dic122—21no—
157C2×Dic122C2×D411yescharacteristic
158SD16×C61—22no—
159SD16×C61—42no—
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×C61—41no—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—41no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—41no—
182C42×C61C2411yescharacteristic
183C22⋊Dic61—41no—
184#11758202—41no—
185#11758201—42no—
186#11758201—82no—
187#11758199—42no—
188#11758198—41no—
189#11758198—41no—
190#11758198—41no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—41no—
194C4⋊Dic61C2×D411yescharacteristic
195#11758196—21no—
196#11758196—41no—
197C4×Dic61—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—42no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C23×C61—21no—
206C22×D61—41no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—81no—
214D4×C61—21no—
215D4×C61#13512yes—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—161no—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—41no—
232C2×Dic61—161no—
233D244—41no—
234D244—41no—
235D244—41no—
236D244—41no—
237D244—81no—
238C4×D61—161no—
239Dic122—21no—
240Dic122—21no—
241Dic122—81no—
242C8×C61—21no—
243C8×C61—21no—
244C8×C61—41no—
245#92418—21no—
246#92418—21no—
247#92418—41no—
248#92418—41no—
249#92418—41no—
250#92418—41no—
251#19842—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61D4211yescharacteristic
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C4×C61—21no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—81no—
266C4×C61—81no—
267C4×C61C2×#13511yescharacteristic
268C4×C61#57511yescharacteristic
269C4×C61#57811yescharacteristic
270C4×C61#57811yescharacteristic
271Dic61—41no—
272Dic61—41no—
273Dic61—81no—
274#3372D6111yescharacteristic, complemented
275#3365—611no—
276#3361—611no—
277#3338—611no—
278#1719—611no—
279#1698—611no—
280#1684—611no—
281#1681—611no—
282#1676—611no—
283#1643—611no—
284#1636—611no—
285#1630—611no—
286#1622—611no—
287#1584—611no—
288SD16⋊C8—611no—
289C2×C61—42no—
290C2×C61#326911yescharacteristic
291C2×C61#328511yescharacteristic
292C2×C61#328811yescharacteristic
293D61—161no—
294#559—611no—
295#557C2×D6111yescharacteristic
296#554—611no—
297#554C2×D6111yescharacteristic
298#500—611no—
299#492—611no—
300#491—21no—
301#488—1221no—
302#488—1221no—
303#487—611no—
304#487C2×D6111yescharacteristic
305#486—611no—
306#486—611no—
307#484—1221no—
308#484—1221no—
309#479—21no—
310#478—1221no—
311#478—1221no—
312#477—611no—
313#477—611no—
314#477—611no—
315#477C2×D6111yescharacteristic
316#475—611no—
317#467—1221no—
318#464—1221no—
319#463—1221no—
320#461—21no—
321#461C2×D6111yescharacteristic
322#460—611no—
323#459—611no—
324#446C2×D6111yescharacteristic
325C8×Q8—611no—
326C8×Q8—611no—
327#440—21no—
328C4×Q16—611no—
329C4×SD16—1221no—
330C4×SD16C2×D6111yescharacteristic
331C4×D8—1221no—
332C8×D4—611no—
333C2×#107—1221no—
334#335—611no—
335#335—611no—
336#333—611no—
337#333—611no—
338#332—611no—
339#331—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342Q8⋊C8—611no—
343D4⋊C8—611no—
344#322—611no—
345C61#1984211yescharacteristic, complemented, Sylow
346C2×Q16—611no—
347C2×SD16—21no—
348C2×SD16—1221no—
349C2×SD16—2441no—
350C2×SD16C22⋊D6111yescharacteristic
351C2×D8—1221no—
352C22×C8—1221no—
353C4⋊Q8—21no—
354C4⋊Q8—611no—
355C4⋊Q8C22×D6111yescharacteristic
356C4⋊Q8C22×D6111yescharacteristic
357#127—611no—
358#125—21no—
359#125—21no—
360#125—41no—
361#125—1221no—
362#124—1221no—
363#124—1221no—
364#123—2441no—
365C22⋊Q8—21no—
366C22⋊Q8—41no—
367C22⋊Q8—1221no—
368C22⋊Q8—2441no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—611no—
372C4×Q8C22×D6111yescharacteristic
373C4×Q8C22×D6111yescharacteristic
374C4×D4—21no—
375C4×D4—1221no—
376C4×D4—1221no—
377C4×D4—1221no—
378C4×D4C22×D6111yescharacteristic
379C2×C4⋊C4—21no—
380C2×C4⋊C4—1221no—
381#107—21no—
382#107—611no—
383#107—1221no—
384#107—1221no—
385#107—1221no—
386#107—1221no—
387#107—1221no—
388#106—1221no—
389#106C22⋊D6111yescharacteristic
390C4⋊C8—21no—
391C4⋊C8—611no—
392C4⋊C8—611no—
393C4⋊C8—611no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8C22×D6111yescharacteristic
397#103—21no—
398#103—21no—
399#103—21no—
400#103—21no—
401#103—611no—
402#103—611no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—2441no—
408#103C22⋊D6111yescharacteristic
409#102—1221no—
410#102—1221no—
411#102—1221no—
412#102—1221no—
413#102—1221no—
414#102—1221no—
415#102C22⋊D6111yescharacteristic
416C22⋊C8—1221no—
417#97—21no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—1221no—
421C4×C8—1221no—
422C4×C8C22×D6111yescharacteristic
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—611no—
426C2×Q8—1221no—
427C2×Q8D4×D6111yescharacteristic
428C2×Q8C2×C22⋊D6111yescharacteristic
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4C2×C22⋊D6111yescharacteristic
433C22×C4—21no—
434C22×C4—41no—
435C22×C4—2441no—
436C22×C4—2441no—
437Q16—2441no—
438SD16—22no—
439SD16—42no—
440SD16—4882no—
441SD16—2442no—
442D8—2442no—
443C2×C8—21no—
444C2×C8—611no—
445C2×C8—611no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—2442no—
451C2×C8C2×C22⋊D6111yescharacteristic
452C2×C8C2×C22⋊D6111yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—611no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C4⋊C4D4×D6111yescharacteristic
471C4⋊C4C2×C22⋊D6111yescharacteristic
472C4⋊C4C2×C22⋊D6111yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—41no—
475C22⋊C4—2441no—
476C22⋊C4—2441no—
477C22⋊C4—2441no—
478C22⋊C4—2441no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—1221no—
483C42C23×D6111yescharacteristic
484C23—21no—
485C23—2441no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493Q8—1221no—
494Q8—1221no—
495Q8—4881no—
496D4—21no—
497D4—4881no—
498D4—2441no—
499D4—2441no—
500D4—2441no—
501D4—2441no—
502D4#1451979712yes—
503C2×C4—21no—
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—161no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—9761no—
516C2×C4—9761no—
517C2×C4—2441no—
518C2×C4C2×D4×D6111yescharacteristic
519C2×C4C22×C22⋊D6111yescharacteristic
520C2×C4#1451982211yescharacteristic
521C8—21no—
522C8—21no—
523C8—41no—
524C8—1221no—
525C8—1221no—
526C8—2441no—
527C8—2441no—
528C8—2441no—
529C8—2441no—
530C22—22no—
531C22—41no—
532C22—41no—
533C22—4881no—
534C22—4881no—
535C22—4881no—
536C22D4×C22⋊D6111yescharacteristic
537C4—21no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—81no—
544C4—81no—
545C4—4881no—
546C4—2441no—
547C4—2441no—
548C4C2×#1451979711yescharacteristic
549C4#1684056111yescharacteristic
550C4#1684057711yescharacteristic
551C4#1684057811yescharacteristic
552C2—42no—
553C2—9761no—
554C2#2145736811yescharacteristic
555C2#2145755611yescharacteristic
556C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3372C2
socleC22×C61D42
prime radical#16839957C22
hypercenter#3372D61
hypocenterC61#19842
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3372

Complements

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

NH ≅ G/Ntype
1. #21458515C2semidirect
3. #21458207C2semidirect
4. #21445473C2semidirect
5. #21445359C2semidirect
6. #21445274C2semidirect
8. #21445221C2semidirect
9. #21445024C2semidirect
10. #21444977C2semidirect
11. #21444939C2semidirect
13. #21444667C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3372C2semidirect
274. #3372D61semidirect
345. C61#19842semidirect

Maximal subgroups

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

subgroupindex
#214585152
#214584772
#214582072
#214454732
#214453592
#214452742
#214452582
#214452212
#214450242
#214449772
#214449392
#214448912
#214446672
SD16⋊#924182
C61×#33722
#1984261