Finite Group Databasebeta

← #29663637

Subgroups — #29663637

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457163C211yesmaximal, characteristic
2#21456962C211yesmaximal, characteristic
3#21456196C211yesmaximal, characteristic
4#21454767C211yesmaximal, characteristic
5#21445404C211yesmaximal, characteristic, complemented
6#21445373C211yesmaximal, characteristic, complemented
7#21445372C211yesmaximal, characteristic, complemented
8#21445279C211yesmaximal, characteristic, complemented
9#21445274C211yesmaximal, characteristic, complemented
10#21445250C211yesmaximal, characteristic, complemented
11#21445247C211yesmaximal, characteristic, complemented
12#21444568C211yesmaximal, characteristic
13#21444473C211yesmaximal, characteristic
14#332×D61C211yesmaximal, characteristic
15C61×#1704C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840316C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19#16840159—21no—
20#16840129—21no—
21#16840129—21no—
22#16840114C2211yescharacteristic
23#16840113C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840111—21no—
26#16840094—21no—
27#16840091—21no—
28#16840056—21no—
29#16840054C2211yescharacteristic
30#16840048C2211yescharacteristic
31#16840047C2211yescharacteristic
32#16840044C2211yescharacteristic
33#16840004—21no—
34#16840002—21no—
35#16839973C2211yescharacteristic
36#16839972C2211yescharacteristic
37#16839971—21no—
38#16839890—21no—
39#16839889—21no—
40#16839889C2211yescharacteristic
41Q8×#92418C2211yescharacteristic
42C8×Dic122C2211yescharacteristic
43#16839814C2211yescharacteristic
44#16839787—21no—
45C4⋊C8×D61C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839769—21no—
48#16839769—21no—
49#16839719—21no—
50C4×#11758189C2211yescharacteristic
51#16839515C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839509C2211yescharacteristic
56#16839509C2211yescharacteristic
57#16839508C2211yescharacteristic
58#16839507C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#477C2211yescharacteristic
62C61×#475C2211yescharacteristic
63C61×#446C2211yescharacteristic
64C61×#334C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Dic244—21no—
69C2×#11758202—21no—
70C2×#11758189D411yescharacteristic
71C2×#11758189D411yescharacteristic
72C2×C8×D61—21no—
73C2×C8×D61—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519776—21no—
78#14519776D411yescharacteristic
79#14519775—21no—
80#14519775D411yescharacteristic
81#14519768—21no—
82#14519767—21no—
83#14519757—21no—
84#14519757D411yescharacteristic
85#14519755—21no—
86#14519753—21no—
87#14519753D411yescharacteristic
88Q8×Dic61—21no—
89C4×Dic122—21no—
90C4×Dic122C2311yescharacteristic
91#14519742—21no—
92#14519735—21no—
93#14519734—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C42×D61C2311yescharacteristic
97#14519710—21no—
98#14519709—21no—
99#14519708—21no—
100#14519708—21no—
101C4⋊#92418—21no—
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105#14519706—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519702—21no—
110#14519702—21no—
111#14519702—21no—
112#14519702—21no—
113#14519701—21no—
114#14519701—21no—
115#14519701—21no—
116#14519700—21no—
117#14519700—21no—
118#14519700—21no—
119#14519700—21no—
120#14519700—21no—
121#14519689—21no—
122#14519687—21no—
123#14519687D411yescharacteristic
124#14519687D411yescharacteristic
125C4×#92418—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C4⋊Q8×C61C2311yescharacteristic
130C4⋊Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132#107×C61—21no—
133#106×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61—21no—
141#103×C61—21no—
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147Q8×D61—42no—
148Q8×D61—42no—
149C2×C4×D61—41no—
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61#4111yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—41no—
157C2×Dic122C2×D411yescharacteristic
158C2×C8×C61—21no—
159C2×C8×C61—21no—
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×C61C2×D411yescharacteristic
169C42×C61—21no—
170C42×C61C2411yescharacteristic
171#11758202—81no—
172#11758198—21no—
173#11758198—21no—
174#11758198—41no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—41no—
181C4⋊Dic61C2×D411yescharacteristic
182#11758196—21no—
183#11758196—21no—
184#11758196—21no—
185#11758196—21no—
186#11758196—21no—
187#11758196—41no—
188#11758196—41no—
189#11758196—41no—
190C4×Dic61—21no—
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61#4111yescharacteristic
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C2×D411yescharacteristic
199C2×#92418C2×D411yescharacteristic
200Dic244—81no—
201#11758189—22no—
202#11758189—22no—
203C8×D61—42no—
204C8×D61—81no—
205C22×D61#12111yescharacteristic
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—41no—
209Q8×C61—41no—
210Q8×C61—81no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61#12111yescharacteristic
223C2×Dic61#12111yescharacteristic
224C2×Dic61#12111yescharacteristic
225C4×D61—21no—
226C4×D61—21no—
227C4×D61—41no—
228C4×D61—82no—
229C4×D61#13512yes—
230C4×D61#13712yes—
231Dic122—21no—
232Dic122—21no—
233Dic122—41no—
234Dic122—41no—
235Dic122—41no—
236Dic122—41no—
237Dic122—81no—
238Dic122—82no—
239C8×C61—21no—
240C8×C61—21no—
241C8×C61—41no—
242C8×C61—41no—
243C8×C61—41no—
244#92418—21no—
245#92418—21no—
246#92418—41no—
247#92418—41no—
248#92418—41no—
249#19243—611nomaximal, Sylow
250C22×C61C2×#12111yescharacteristic
251C2×D61—21no—
252C2×D61—21no—
253C2×D61#46412yes—
254C4×C61—21no—
255C4×C61—41no—
256C4×C61—41no—
257C4×C61—81no—
258C4×C61—81no—
259C4×C61C2×#13511yescharacteristic
260C4×C61C2×#13711yescharacteristic
261C4×C61#57711yescharacteristic
262C4×C61#57811yescharacteristic
263Dic61—21no—
264Dic61—21no—
265Dic61—41no—
266Dic61—41no—
267Dic61—81no—
268Dic61—81no—
269Dic61#46412yes—
270#3249—611no—
271#3229—611no—
272#3139—611no—
273#2962—611no—
274#1704—611no—
275#1704D6111yescharacteristic, complemented
276#1700—611no—
277#1700—611no—
278#1684—611no—
279#1684—611no—
280#1680—611no—
281#1680—611no—
282#1565—611no—
283#1542—611no—
284C2×#332—611no—
285C2×C61C2×#46411yescharacteristic
286C2×C61#306311yescharacteristic
287C2×C61#306911yescharacteristic
288D61—22no—
289#533—611no—
290#533—611no—
291C2×C4⋊Q8—611no—
292#498—1221no—
293#491—1221no—
294#491—1221no—
295#487—611no—
296#487—611no—
297#487—611no—
298#487—1221no—
299#487C2×D6111yescharacteristic
300#483—1221no—
301#483—1221no—
302#477—611no—
303#477—1221no—
304#477C2×D6111yescharacteristic
305#475—611no—
306#475—611no—
307#475—611no—
308#475C2×D6111yescharacteristic
309#467—1221no—
310#467—1221no—
311#462—611no—
312#462—611no—
313#462—1221no—
314#446—611no—
315#446—1221no—
316#446—1221no—
317#446C2×D6111yescharacteristic
318C8×Q8—611no—
319C8×Q8—611no—
320#431—611no—
321#424—1221no—
322C2×C4⋊C8—611no—
323C2×C4⋊C8—611no—
324#419—1221no—
325#419—1221no—
326#405—1221no—
327C4×M16—611no—
328#335—611no—
329#335—611no—
330#334—611no—
331#334C2×D6111yescharacteristic
332#333—611no—
333#333C2×D6111yescharacteristic
334#332—611no—
335#332—611no—
336#332—611no—
337#332—611no—
338#332—611no—
339#332C2×D6111yescharacteristic
340C61#1924311yescharacteristic, complemented, Sylow
341C22×Q8—1221no—
342C2×Q16—1221no—
343C2×Q16—1221no—
344C2×M16—611no—
345C2×M16—611no—
346C22×C8—1221no—
347C22×C8—1221no—
348C4⋊Q8—611no—
349C4⋊Q8—611no—
350C4⋊Q8—611no—
351C4⋊Q8—611no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8C22×D6111yescharacteristic
356C4⋊Q8C22×D6111yescharacteristic
357#125—1221no—
358#125—1221no—
359C22⋊Q8—611no—
360C22⋊Q8—611no—
361C22⋊Q8—1221no—
362C22⋊Q8—1221no—
363C22⋊Q8—1221no—
364C4×Q8—611no—
365C4×Q8—1221no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8C22×D6111yescharacteristic
369#117—1221no—
370#117—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C42—611no—
374#107—21no—
375#107—1221no—
376#106—21no—
377#106—1221no—
378#106—1221no—
379#106—1221no—
380C4⋊C8—21no—
381C4⋊C8—611no—
382C4⋊C8—611no—
383C4⋊C8—611no—
384C4⋊C8—1221no—
385C4⋊C8—1221no—
386C4⋊C8—1221no—
387C4⋊C8—1221no—
388C4⋊C8—1221no—
389C4⋊C8C22×D6111yescharacteristic
390C4⋊C8C22×D6111yescharacteristic
391C4⋊C8C22×D6111yescharacteristic
392#103—21no—
393#103—21no—
394#103—21no—
395#103—21no—
396#103—1221no—
397#103—1221no—
398#103—1221no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408C22⋊C8—1221no—
409#97—21no—
410#97—611no—
411#97—611no—
412#97—1221no—
413C4×C8—611no—
414C4×C8—1221no—
415C4×C8—1221no—
416C4×C8—1221no—
417C4×C8C22×D6111yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—611no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—2441no—
425C2×Q8—2442no—
426C2×Q8—2442no—
427C2×Q8D4×D6111yescharacteristic
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—2441no—
432Q16—4881no—
433Q16—4881no—
434M16—1222no—
435M16—1222no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—611no—
440C2×C8—611no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—4881no—
445C2×C8—2442no—
446C2×C8D4×D6111yescharacteristic
447C2×C8D4×D6111yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—611no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
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
469C22⋊C4—1221no—
470C22⋊C4—1221no—
471C22⋊C4—2441no—
472C42—21no—
473C42—611no—
474C42—611no—
475C42—611no—
476C42—1221no—
477C42—2441no—
478C42C23×D6111yescharacteristic
479C23—611no—
480Q8—21no—
481Q8—21no—
482Q8—41no—
483Q8—41no—
484Q8—81no—
485Q8—1221no—
486Q8—1221no—
487Q8—4881no—
488Q8—4882no—
489Q8—2441no—
490Q8—2441no—
491Q8—2441no—
492Q8—2441no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—611no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—611no—
501C2×C4—611no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—4882no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4#41×D6111yescharacteristic
513C8—21no—
514C8—21no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—1221no—
519C8—1221no—
520C8—2441no—
521C8—2441no—
522C8—2441no—
523C22—612no—
524C22—1221no—
525C22—1221no—
526C22#121×D6111yescharacteristic
527C4—21no—
528C4—41no—
529C4—41no—
530C4—81no—
531C4—81no—
532C4—612no—
533C4—1221no—
534C4—1221no—
535C4—4881no—
536C4—4881no—
537C4—2441no—
538C4—2441no—
539C4#135×D6111yescharacteristic
540C4#137×D6111yescharacteristic
541C4#1684057111yescharacteristic
542C4#1684057811yescharacteristic
543C2—1222no—
544C2#464×D6111yescharacteristic
545C2#2145550711yescharacteristic
546C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1704C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1704D61
hypocenterC61#19243
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1704

Complements

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

NH ≅ G/Ntype
5. #21445404C2semidirect
6. #21445373C2semidirect
7. #21445372C2semidirect
8. #21445279C2semidirect
9. #21445274C2semidirect
10. #21445250C2semidirect
11. #21445247C2semidirect
15. C61×#1704C2semidirect
275. #1704D61semidirect
340. C61#19243semidirect

Maximal subgroups

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

subgroupindex
#214571632
#214569622
#214561962
#214547672
#214454042
#214453732
#214453722
#214452792
#214452742
#214452502
#214452472
#214445682
#214444732
#332×D612
C61×#17042
#1924361