Finite Group Databasebeta

← #29676704

Subgroups — #29676704

Order 15616.

556 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#21458751C211yesmaximal, characteristic, complemented
2#21458557C211yesmaximal, characteristic
3#21458245C211yesmaximal, characteristic
4#21445494C211yesmaximal, characteristic, complemented
5#21445401C211yesmaximal, characteristic, complemented
6#21445335C211yesmaximal, characteristic, complemented
7#21445310C211yesmaximal, characteristic, complemented
8#21445192C211yesmaximal, characteristic
9#21445073C211yesmaximal, characteristic
10#21445061C211yesmaximal, characteristic, complemented
11#21445038C211yesmaximal, characteristic, complemented
12#21444991C211yesmaximal, characteristic
13#21444726C211yesmaximal, characteristic, complemented
14#21444656C211yesmaximal, characteristic
15C61×#3398C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840482C2211yescharacteristic
18#16840172—21no—
19#16840165C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840120—21no—
23#16840114C2211yescharacteristic
24#16840113C2211yescharacteristic
25#16840103—21no—
26#16840100—21no—
27#16840066—21no—
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840047—21no—
31#16840047C2211yescharacteristic
32#16840046—21no—
33#16840044C2211yescharacteristic
34#16840044C2211yescharacteristic
35#16840001—21no—
36#16839982—21no—
37#16839972C2211yescharacteristic
38#16839893C2211yescharacteristic
39#16839892C2211yescharacteristic
40#16839867—21no—
41#16839866—21no—
42C4×#11758202C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#16839798—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#557C2211yescharacteristic
57C61×#556C2211yescharacteristic
58C61×#497—21no—
59C61×#494C2211yescharacteristic
60C61×#464—21no—
61C61×#462—21no—
62C61×#462C2211yescharacteristic
63C61×#461C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#441—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202D411yescharacteristic
69C22×#92418—21no—
70#14519777—21no—
71#14519777C2311yescharacteristic
72#14519777C2311yescharacteristic
73#14519776—21no—
74#14519776—41no—
75#14519770—41no—
76#14519764—21no—
77#14519756—21no—
78Q8×Dic61—21no—
79C4×Dic122—21no—
80C4×Dic122C2311yescharacteristic
81#14519742—21no—
82#14519736—21no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—41no—
87#14519710D411yescharacteristic
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519702—21no—
100#14519702—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701D411yescharacteristic
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700—41no—
108#14519700D411yescharacteristic
109#14519698—21no—
110C22⋊#92418—21no—
111#14519688—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×Q16×C61—21no—
116C2×Q16×C61—21no—
117C2×Q16×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61D411yescharacteristic
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#126×C61—41no—
124#125×C61—21no—
125#125×C61—41no—
126#124×C61—21no—
127C22⋊Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×Q8×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#117×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
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—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—41no—
153C2×Dic122C2×D411yescharacteristic
154Q16×C61—42no—
155Q16×C61—81no—
156Q16×C61—81no—
157SD16×C61—41no—
158SD16×C61—81no—
159C2×C8×C61—21no—
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D411yescharacteristic
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
175C22⋊C4×C61—21no—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—41no—
181C42×C61—41no—
182C42×C61C2411yescharacteristic
183C22⋊Dic61—41no—
184C22⋊Dic61—41no—
185#11758202—22no—
186#11758202—42no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—41no—
192C4⋊Dic61C2×D411yescharacteristic
193#11758196—21no—
194#11758196—41no—
195#11758196—41no—
196C4×Dic61—21no—
197C4×Dic61—41no—
198C4×Dic61—41no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—42no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C23×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215D4×C61—21no—
216D4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
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—41no—
233Dic122—21no—
234Dic122—21no—
235Dic122—41no—
236Dic122—41no—
237Dic122—81no—
238Dic122—161no—
239C8×C61—21no—
240C8×C61—21no—
241C8×C61—41no—
242#92418—21no—
243#92418—41no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#92418#13712yes—
248#20277—611nomaximal, Sylow
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61D4211yescharacteristic
253C4×C61—21no—
254C4×C61—41no—
255C4×C61—41no—
256C4×C61—41no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—81no—
260C4×C61—81no—
261C4×C61C2×#13711yescharacteristic
262C4×C61#57511yescharacteristic
263C4×C61#57711yescharacteristic
264C4×C61#57811yescharacteristic
265Dic61—41no—
266Dic61—41no—
267Dic61—81no—
268Dic61—81no—
269#3398D6111yescharacteristic
270#3387—611no—
271#3369—611no—
272#3342—611no—
273#1723—611no—
274#1704—611no—
275#1694—611no—
276#1690—611no—
277#1671—611no—
278#1650—611no—
279#1648—611no—
280#1645—611no—
281#1638—611no—
282#1593—611no—
283#1582—611no—
284C2×C61—81no—
285C2×C61#327111yescharacteristic
286C2×C61#327611yescharacteristic
287C2×C61#328911yescharacteristic
288Q82—611no—
289#557C2×D6111yescharacteristic
290#556—611no—
291#556C2×D6111yescharacteristic
292#501—1221no—
293#500—611no—
294#500—611no—
295#498—611no—
296#497—21no—
297#494C2×D6111yescharacteristic
298#489—1221no—
299#487—611no—
300#487—611no—
301#485—1221no—
302#485—1221no—
303#479—1221no—
304#477—611no—
305#476—611no—
306#475—611no—
307#475—611no—
308#475—611no—
309#475—1221no—
310#475—1221no—
311#467—1221no—
312#464—21no—
313#464—1221no—
314#462—21no—
315#462—611no—
316#462C2×D6111yescharacteristic
317#461C2×D6111yescharacteristic
318#446—611no—
319#446—611no—
320#446C2×D6111yescharacteristic
321#441—21no—
322#441—1221no—
323#441—1221no—
324C4×Q16—611no—
325C4×SD16C2×D6111yescharacteristic
326C8×D4—611no—
327#427—1221no—
328#334—611no—
329#334—611no—
330#333—611no—
331#332—611no—
332#332—611no—
333#332—611no—
334Q8⋊C8—611no—
335Q8⋊C8—611no—
336Q8⋊C8—611no—
337D4⋊C8—611no—
338#322—611no—
339C61#2027711yescharacteristic, complemented, Sylow
340C2×Q16—21no—
341C2×Q16—21no—
342C2×Q16—21no—
343C2×Q16—611no—
344C2×Q16—1221no—
345C2×SD16—21no—
346C2×SD16C22⋊D6111yescharacteristic
347C22×C8—1221no—
348C4⋊Q8—21no—
349C4⋊Q8—611no—
350C4⋊Q8—611no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8—2441no—
354C4⋊Q8C22×D6111yescharacteristic
355C4⋊Q8C22×D6111yescharacteristic
356#126—41no—
357#126—2441no—
358#125—21no—
359#125—41no—
360#124—21no—
361#124—1221no—
362C22⋊Q8—21no—
363C22⋊Q8—1221no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—611no—
368C4×Q8—1221no—
369C4×Q8—1221no—
370C4×Q8—1221no—
371C4×Q8C22×D6111yescharacteristic
372C4×Q8C22×D6111yescharacteristic
373C4×D4C22×D6111yescharacteristic
374#117—21no—
375#117—1221no—
376#107—611no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#107—2441no—
381#106—1221no—
382#106—1221no—
383#106—1221no—
384#106—1221no—
385#106—1221no—
386#106—1221no—
387#106C22⋊D6111yescharacteristic
388C4⋊C8—21no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—611no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8C22×D6111yescharacteristic
395#103—21no—
396#103—21no—
397#103—611no—
398#103—611no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—2441no—
407#103C22⋊D6111yescharacteristic
408#102—1221no—
409#102C22⋊D6111yescharacteristic
410C22⋊C8—1221no—
411#97—21no—
412#97—1221no—
413#97—1221no—
414C4×C8—611no—
415C4×C8—611no—
416C4×C8C22×D6111yescharacteristic
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—611no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—2441no—
423C2×Q8D4×D6111yescharacteristic
424C2×Q8C2×C22⋊D6111yescharacteristic
425C2×D4C2×C22⋊D6111yescharacteristic
426C22×C4—21no—
427Q16—42no—
428Q16—81no—
429Q16—81no—
430Q16—1222no—
431Q16—2442no—
432SD16—41no—
433SD16—81no—
434C2×C8—21no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—2442no—
442C2×C8C2×C22⋊D6111yescharacteristic
443C2×C8C2×C22⋊D6111yescharacteristic
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—611no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
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—21no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C42—21no—
472C42—21no—
473C42—41no—
474C42—41no—
475C42—1221no—
476C42—2441no—
477C42—2441no—
478C42C23×D6111yescharacteristic
479C23—21no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—41no—
485Q8—41no—
486Q8—41no—
487Q8—41no—
488Q8—1221no—
489Q8—1221no—
490Q8—9761no—
491Q8—4881no—
492Q8—2441no—
493Q8—2441no—
494D4—21no—
495D4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4C2×D4×D6111yescharacteristic
510C2×C4C22×C22⋊D6111yescharacteristic
511C2×C4#1451982211yescharacteristic
512C8—21no—
513C8—21no—
514C8—41no—
515C8—612no—
516C8—1221no—
517C8—2441no—
518C8—2441no—
519C8—2441no—
520C8—2441no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22D4×C22⋊D6111yescharacteristic
525C4—21no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—81no—
532C4—81no—
533C4—4881no—
534C4—4881no—
535C4—2441no—
536C4—2441no—
537C4#137×D6111yescharacteristic
538C4#1684056011yescharacteristic
539C4#1684057311yescharacteristic
540C4#1684057811yescharacteristic
541C2—81no—
542C2#2145740011yescharacteristic
543C2#2145745911yescharacteristic
544C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3398C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenter#3398D61
hypocenterC61#20277
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3398

Complements

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

NH ≅ G/Ntype
1. #21458751C2semidirect
4. #21445494C2semidirect
5. #21445401C2semidirect
6. #21445335C2semidirect
7. #21445310C2semidirect
10. #21445061C2semidirect
11. #21445038C2semidirect
13. #21444726C2semidirect
339. C61#20277semidirect

Maximal subgroups

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

subgroupindex
#214587512
#214585572
#214582452
#214454942
#214454012
#214453352
#214453102
#214451922
#214450732
#214450612
#214450382
#214449912
#214447262
#214446562
C61×#33982
#2027761