Finite Group Databasebeta

← #29670721

Subgroups — #29670721

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458633C211yesmaximal, characteristic
2#21458513C211yesmaximal, characteristic
3#21458497C211yesmaximal, characteristic, complemented
4#21445475C211yesmaximal, characteristic
5#21445403C211yesmaximal, characteristic, complemented
6#21445263C211yesmaximal, characteristic
7#21445237C211yesmaximal, characteristic
8#21445220C211yesmaximal, characteristic, complemented
9#21445040C211yesmaximal, characteristic, complemented
10#21444976C211yesmaximal, characteristic, complemented
11#21444939C211yesmaximal, characteristic
12#21444897C211yesmaximal, characteristic
13#21444670C211yesmaximal, characteristic, complemented
14#21444636C211yesmaximal, characteristic, complemented
15Q8×Q16×C61C211yesmaximal, characteristic, complemented
16#16840497C2211yescharacteristic
17#16840491C2211yescharacteristic
18#16840136C2211yescharacteristic
19#16840124—21no—
20#16840118—21no—
21#16840114—21no—
22#16840114C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840110C2211yescharacteristic
25#16840109C2211yescharacteristic
26#16840065—21no—
27#16840054—21no—
28#16840054C2211yescharacteristic
29#16840054C2211yescharacteristic
30#16840047C2211yescharacteristic
31#16840039—21no—
32#16839986—21no—
33#16839984—21no—
34#16839972—21no—
35#16839972C2211yescharacteristic
36#16839962C2211yescharacteristic
37#16839962C2211yescharacteristic
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839860—21no—
42C4×#11758202—21no—
43C4×#11758202C2211yescharacteristic
44#16839847—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839492C2211yescharacteristic
55#16839484C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57Q82×C61C2211yescharacteristic
58C61×#500—21no—
59C61×#500C2211yescharacteristic
60C61×#477—21no—
61C61×#477—21no—
62C61×#477C2211yescharacteristic
63C61×#477C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×Q16×C61—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202D411yescharacteristic
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758201—41no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519774C2311yescharacteristic
75#14519768—21no—
76#14519768—41no—
77#14519766—21no—
78#14519765—21no—
79#14519762—41no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83C4×D244—21no—
84#14519739—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710D411yescharacteristic
88#14519708—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519701—21no—
96#14519701—41no—
97#14519701D411yescharacteristic
98#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700D411yescharacteristic
104#14519699—21no—
105#14519699—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519688—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×Q16×C61D411yescharacteristic
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—41no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131#107×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—41no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×C4×D61—41no—
155C2×Dic122—21no—
156C2×Dic122C2×D411yescharacteristic
157Q16×C61—22no—
158C2×C8×C61—21no—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2×D411yescharacteristic
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—41no—
179C42×C61—41no—
180C42×C61C2411yescharacteristic
181#11758202—22no—
182#11758202—42no—
183#11758201—42no—
184#11758201—42no—
185#11758201—82no—
186#11758198—41no—
187#11758198—41no—
188#11758198—41no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—41no—
192C4⋊Dic61—41no—
193C4⋊Dic61C2×D411yescharacteristic
194#11758196—21no—
195#11758196—41no—
196C4×Dic61—21no—
197C4×Dic61—41no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C22×D61—41no—
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—81no—
210Q8×C61—81no—
211Q8×C61—161no—
212Q8×C61—161no—
213Q8×C61#13512yes—
214Q8×C61#13712yes—
215C2×C4×C61—21no—
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×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—41no—
229D244—41no—
230D244—41no—
231D244—41no—
232D244—41no—
233D244—81no—
234C4×D61—161no—
235Dic122—21no—
236Dic122—21no—
237Dic122—41no—
238Dic122—41no—
239C8×C61—21no—
240C8×C61—21no—
241C8×C61—41no—
242#92418—21no—
243#92418—21no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#92418—41no—
248#19846—611nomaximal, Sylow
249C22×C61D4211yescharacteristic
250C2×D61—81no—
251C2×D61—81no—
252C2×D61—81no—
253C4×C61—22no—
254C4×C61—22no—
255C4×C61—21no—
256C4×C61—41no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—81no—
261C4×C61—81no—
262C4×C61C2×#13511yescharacteristic
263C4×C61C2×#13711yescharacteristic
264C4×C61#57811yescharacteristic
265C4×C61#57811yescharacteristic
266Dic61—41no—
267Dic61—41no—
268Dic61—81no—
269Q8×Q16D6111yescharacteristic, complemented
270#3375—611no—
271#3365—611no—
272#3363—611no—
273#1719—611no—
274#1704—611no—
275#1682—611no—
276#1678—611no—
277#1676—611no—
278#1645—611no—
279#1636—611no—
280#1630—611no—
281#1623—611no—
282#1584—611no—
283#1579—611no—
284C2×C61#327011yescharacteristic
285C2×C61#328511yescharacteristic
286C2×C61#328811yescharacteristic
287D61—161no—
288#559—611no—
289Q82C2×D6111yescharacteristic
290Q82C2×D6111yescharacteristic
291#557—611no—
292#500—21no—
293#500C2×D6111yescharacteristic
294#492—611no—
295#489—1221no—
296#488—1221no—
297#487—611no—
298#487—611no—
299#487—1221no—
300#486—611no—
301#486—611no—
302#478—1221no—
303#477—21no—
304#477—21no—
305#477—611no—
306#477—611no—
307#477—1221no—
308#477C2×D6111yescharacteristic
309#477C2×D6111yescharacteristic
310#475—611no—
311#474—1221no—
312#464—1221no—
313#464—1221no—
314#462—611no—
315#462—1221no—
316#460—611no—
317#460—611no—
318#446—611no—
319C8×Q8—611no—
320C8×Q8—611no—
321C8×Q8C2×D6111yescharacteristic
322#440—1221no—
323C4×Q16—21no—
324C4×Q16—611no—
325C4×Q16—1221no—
326C4×Q16C2×D6111yescharacteristic
327C4×SD16—1221no—
328#335—611no—
329#334—611no—
330#333—611no—
331#333—611no—
332#332—611no—
333#331—611no—
334Q8⋊C8—611no—
335Q8⋊C8—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338#322—611no—
339C61#1984611yescharacteristic, complemented, Sylow
340C2×Q16—611no—
341C2×Q16—1221no—
342C2×Q16C22⋊D6111yescharacteristic
343C2×SD16—1221no—
344C2×SD16—1221no—
345C2×SD16—2441no—
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—41no—
351C4⋊Q8—41no—
352C4⋊Q8—611no—
353C4⋊Q8—1221no—
354C4⋊Q8C22×D6111yescharacteristic
355C4⋊Q8C22×D6111yescharacteristic
356#127—611no—
357#125—1221no—
358#125—2441no—
359#124—1221no—
360#124—1221no—
361#123—2441no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—611no—
367C4×Q8—1221no—
368C4×Q8—1221no—
369C4×Q8C22×D6111yescharacteristic
370C4×Q8C22×D6111yescharacteristic
371C4×Q8C22×D6111yescharacteristic
372C4×D4—1221no—
373C4×D4—1221no—
374#107—21no—
375#107—21no—
376#107—21no—
377#107—21no—
378#107—611no—
379#107—1221no—
380#107—1221no—
381#107C22⋊D6111yescharacteristic
382#106—1221no—
383C4⋊C8—21no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—1221no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8C22×D6111yescharacteristic
391#103—21no—
392#103—21no—
393#103—611no—
394#103—611no—
395#103—1221no—
396#103—1221no—
397#103—1221no—
398#103—1221no—
399#103—1221no—
400#103—1221no—
401#103—2441no—
402#103C22⋊D6111yescharacteristic
403#103C22⋊D6111yescharacteristic
404#102—1221no—
405#102—1221no—
406#102—1221no—
407#102—1221no—
408#102—1221no—
409#102—1221no—
410#97—1221no—
411C4×C8—21no—
412C4×C8—611no—
413C4×C8—611no—
414C4×C8—1221no—
415C4×C8—1221no—
416C4×C8C22×D6111yescharacteristic
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—41no—
420C2×Q8—41no—
421C2×Q8—611no—
422C2×Q8—1221no—
423C2×Q8D4×D6111yescharacteristic
424C2×Q8C2×C22⋊D6111yescharacteristic
425C2×Q8C2×C22⋊D6111yescharacteristic
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1221no—
429C22×C4—2441no—
430Q16—22no—
431Q16—1222no—
432Q16—2442no—
433SD16—4882no—
434SD16—2442no—
435SD16—2442no—
436C2×C8—21no—
437C2×C8—611no—
438C2×C8—611no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8C2×C22⋊D6111yescharacteristic
444C2×C8C2×C22⋊D6111yescharacteristic
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—611no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4D4×D6111yescharacteristic
464C4⋊C4C2×C22⋊D6111yescharacteristic
465C4⋊C4C2×C22⋊D6111yescharacteristic
466C22⋊C4—2441no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—41no—
473C42—41no—
474C42—1221no—
475C42—2441no—
476C42C23×D6111yescharacteristic
477C23—2441no—
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—81no—
483Q8—81no—
484Q8—161no—
485Q8—161no—
486Q8—1221no—
487Q8—1221no—
488Q8—2441no—
489Q8—2441no—
490Q8#1451980012yes—
491Q8#1451980812yes—
492D4—4881no—
493D4—2441no—
494D4—2441no—
495D4—2441no—
496D4—2441no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—9761no—
508C2×C4—2441no—
509C2×C4C2×D4×D6111yescharacteristic
510C2×C4C22×C22⋊D6111yescharacteristic
511C2×C4#1451982211yescharacteristic
512C8—21no—
513C8—21no—
514C8—41no—
515C8—1221no—
516C8—1221no—
517C8—2441no—
518C8—2441no—
519C8—2441no—
520C8—2441no—
521C22—4881no—
522C22—4881no—
523C22—4881no—
524C22D4×C22⋊D6111yescharacteristic
525C4—22no—
526C4—21no—
527C4—22no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—81no—
533C4—81no—
534C4—4881no—
535C4—2441no—
536C4—2441no—
537C4C2×#1451980011yescharacteristic
538C4C2×#1451980811yescharacteristic
539C4#1684057511yescharacteristic
540C4#1684057711yescharacteristic
541C2—9761no—
542C2#2145739011yescharacteristic
543C2#2145756011yescharacteristic
544C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×Q16×C61C2
socleC22×C61D42
prime radical#14519774C23
hypercenterQ8×Q16D61
hypocenterC61#19846
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×Q16

Complements

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

NH ≅ G/Ntype
3. #21458497C2semidirect
5. #21445403C2semidirect
8. #21445220C2semidirect
9. #21445040C2semidirect
10. #21444976C2semidirect
13. #21444670C2semidirect
14. #21444636C2semidirect
15. Q8×Q16×C61C2semidirect
269. Q8×Q16D61semidirect
339. C61#19846semidirect

Maximal subgroups

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

subgroupindex
#214586332
#214585132
#214584972
#214454752
#214454032
#214452632
#214452372
#214452202
#214450402
#214449762
#214449392
#214448972
#214446702
#214446362
Q8×Q16×C612
#1984661