Finite Group Databasebeta

← #29670756

Subgroups — #29670756

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458555C211yesmaximal, characteristic
2#21458500C211yesmaximal, characteristic
3#21458495C211yesmaximal, characteristic, complemented
4#21445464C211yesmaximal, characteristic, complemented
5#21445371C211yesmaximal, characteristic, complemented
6#21445275C211yesmaximal, characteristic, complemented
7#21445270C211yesmaximal, characteristic
8#21445210C211yesmaximal, characteristic, complemented
9#21445095C211yesmaximal, characteristic, complemented
10#21445089C211yesmaximal, characteristic
11#21445039C211yesmaximal, characteristic, complemented
12#21444991C211yesmaximal, characteristic
13#21444670C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic
15C61×#3342C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840490C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840121—21no—
20#16840116—21no—
21#16840114C2211yescharacteristic
22#16840113C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840111—21no—
26#16840069—21no—
27#16840057C2211yescharacteristic
28#16840056—21no—
29#16840052C2211yescharacteristic
30#16840047—21no—
31#16840042C2211yescharacteristic
32#16839981—21no—
33#16839981—21no—
34#16839972C2211yescharacteristic
35#16839970—21no—
36#16839969C2211yescharacteristic
37#16839969C2211yescharacteristic
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839867—21no—
42C4×#11758202—21no—
43#16839854—21no—
44C4×#11758200C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#556C2211yescharacteristic
57C61×#556C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#485—21no—
60C61×#485—21no—
61C61×#475C2211yescharacteristic
62C61×#475C2211yescharacteristic
63C61×#467—21no—
64C4×Q16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C61×#427—21no—
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758200D411yescharacteristic
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519776—21no—
75#14519776—41no—
76#14519775—21no—
77#14519769—21no—
78#14519767—41no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519710—21no—
85#14519710—21no—
86#14519708—21no—
87#14519708—41no—
88#14519708D411yescharacteristic
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519702—21no—
96#14519702—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519700—21no—
105#14519700—41no—
106#14519700D411yescharacteristic
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110#14519688—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×Q16×C61D411yescharacteristic
116C22×C8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#126×C61—41no—
120#126×C61—41no—
121#124×C61—21no—
122#124×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130#117×C61—21no—
131#117×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142C22⋊C8×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—41no—
155C2×C8×C61—21no—
156C2×C8×C61—42no—
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61C2×D411yescharacteristic
159C4⋊C4×C61—21no—
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×C61—41no—
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C4⋊C4×C61C2×D411yescharacteristic
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—21no—
177C42×C61—41no—
178C42×C61—41no—
179C42×C61C2411yescharacteristic
180#11758202—42no—
181#11758202—42no—
182#11758202—42no—
183#11758200—22no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—41no—
189C4⋊Dic61C2×D411yescharacteristic
190#11758196—21no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—21no—
196C4×Dic61—41no—
197C4×Dic61—41no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C23×C61—21no—
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—41no—
210Q8×C61—41no—
211Q8×C61—41no—
212Q8×C61—41no—
213D4×C61—21no—
214D4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231Dic122—21no—
232Dic122—41no—
233Dic122—41no—
234Dic122—81no—
235Dic122—161no—
236Dic122#13512yes—
237C8×C61—21no—
238C8×C61—21no—
239C8×C61—41no—
240#92418—21no—
241#92418—21no—
242#92418—41no—
243#92418—41no—
244#92418—41no—
245#92418—41no—
246#19848—611nomaximal, Sylow
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61D4211yescharacteristic
251C4×C61—21no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—41no—
256C4×C61—41no—
257C4×C61—81no—
258C4×C61—81no—
259C4×C61C2×#13511yescharacteristic
260C4×C61#57511yescharacteristic
261C4×C61#57811yescharacteristic
262C4×C61#57811yescharacteristic
263Dic61—22no—
264Dic61—41no—
265Dic61—81no—
266Dic61—81no—
267#3369—611no—
268#3364—611no—
269#3363—611no—
270#3342D6111yescharacteristic
271#1718—611no—
272#1700—611no—
273#1684—611no—
274#1683—611no—
275#1674—611no—
276#1653—611no—
277#1652—611no—
278#1645—611no—
279#1638—611no—
280#1584—611no—
281SD16⋊C8—611no—
282C2×C61—81no—
283C2×C61#326911yescharacteristic
284C2×C61#328511yescharacteristic
285C2×C61#328911yescharacteristic
286Q82—611no—
287#557—611no—
288#556C2×D6111yescharacteristic
289#556C2×D6111yescharacteristic
290#500C2×D6111yescharacteristic
291#498—611no—
292#489—1221no—
293#488—1221no—
294#487—611no—
295#487—611no—
296#487—611no—
297#487—611no—
298#487—1221no—
299#485—21no—
300#485—21no—
301#479—1221no—
302#477—611no—
303#477—1221no—
304#476—611no—
305#475—1221no—
306#475C2×D6111yescharacteristic
307#475C2×D6111yescharacteristic
308#474—611no—
309#467—21no—
310#464—1221no—
311#464—1221no—
312#462—611no—
313#462—611no—
314#462—611no—
315#462—1221no—
316#446—611no—
317C8×Q8—611no—
318C8×Q8—611no—
319#441—1221no—
320C4×Q16—1221no—
321C4×Q16—1221no—
322C4×Q16C2×D6111yescharacteristic
323C4×SD16—611no—
324C8×D4C2×D6111yescharacteristic
325#427—21no—
326#335—611no—
327#335—611no—
328#333—611no—
329#332—611no—
330#332—611no—
331#332—611no—
332Q8⋊C8—611no—
333Q8⋊C8—611no—
334Q8⋊C8—611no—
335D4⋊C8—611no—
336#322—611no—
337C61#1984811yescharacteristic, complemented, Sylow
338C2×Q16—1221no—
339C2×Q16—1221no—
340C2×Q16—1221no—
341C2×Q16C22⋊D6111yescharacteristic
342C2×SD16—611no—
343C22×C8—21no—
344C4⋊Q8—611no—
345C4⋊Q8—611no—
346C4⋊Q8—1221no—
347C4⋊Q8—1221no—
348C4⋊Q8—1221no—
349C4⋊Q8—2441no—
350C4⋊Q8C22×D6111yescharacteristic
351C4⋊Q8C22×D6111yescharacteristic
352#126—41no—
353#126—41no—
354#125—1221no—
355#125—2441no—
356#124—21no—
357#124—21no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—611no—
363C4×Q8—1221no—
364C4×Q8—1221no—
365C4×Q8—1221no—
366C4×Q8—1221no—
367C4×Q8C22×D6111yescharacteristic
368C4×Q8C22×D6111yescharacteristic
369C4×D4C22×D6111yescharacteristic
370#117—21no—
371#117—21no—
372#107—21no—
373#107—21no—
374#107—1221no—
375#107—1221no—
376#107C22⋊D6111yescharacteristic
377#106—21no—
378#106—21no—
379#106—611no—
380#106—1221no—
381#106—2441no—
382C4⋊C8—611no—
383C4⋊C8—611no—
384C4⋊C8—611no—
385C4⋊C8—1221no—
386C4⋊C8—1221no—
387C4⋊C8—1221no—
388C4⋊C8C22×D6111yescharacteristic
389#103—21no—
390#103—21no—
391#103—611no—
392#103—1221no—
393#103—1221no—
394#103—1221no—
395#103—1221no—
396#103—1221no—
397#103—1221no—
398#103—1221no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—2441no—
403#103C22⋊D6111yescharacteristic
404#103C22⋊D6111yescharacteristic
405#102—611no—
406#102—1221no—
407#102—1221no—
408C22⋊C8—21no—
409#97—1221no—
410C4×C8—611no—
411C4×C8—611no—
412C4×C8—1221no—
413C4×C8—1221no—
414C4×C8C22×D6111yescharacteristic
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—611no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8—2441no—
421C2×Q8C2×C22⋊D6111yescharacteristic
422C2×Q8C2×C22⋊D6111yescharacteristic
423C2×D4D4×D6111yescharacteristic
424C22×C4—21no—
425Q16—41no—
426Q16—2442no—
427Q16—2442no—
428Q16—2442no—
429SD16—1222no—
430C2×C8—21no—
431C2×C8—42no—
432C2×C8—611no—
433C2×C8—611no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8C2×C22⋊D6111yescharacteristic
439C2×C8C2×C22⋊D6111yescharacteristic
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—611no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—2441no—
455C4⋊C4—2441no—
456C4⋊C4—2441no—
457C4⋊C4—2441no—
458C4⋊C4—2441no—
459C4⋊C4D4×D6111yescharacteristic
460C4⋊C4C2×C22⋊D6111yescharacteristic
461C4⋊C4C2×C22⋊D6111yescharacteristic
462C22⋊C4—21no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C42—21no—
468C42—21no—
469C42—41no—
470C42—41no—
471C42—1221no—
472C42—2441no—
473C42—2441no—
474C42C23×D6111yescharacteristic
475C23—21no—
476Q8—21no—
477Q8—21no—
478Q8—21no—
479Q8—21no—
480Q8—41no—
481Q8—41no—
482Q8—41no—
483Q8—41no—
484Q8—612no—
485Q8—1221no—
486Q8—9761no—
487Q8—4881no—
488Q8—2441no—
489Q8—2441no—
490D4—21no—
491D4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4#1451982211yescharacteristic
508C8—21no—
509C8—21no—
510C8—41no—
511C8—1221no—
512C8—1221no—
513C8—2441no—
514C8—2441no—
515C8—2441no—
516C8—2441no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22D4×C22⋊D6111yescharacteristic
521C4—21no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—81no—
528C4—81no—
529C4—1222no—
530C4—4881no—
531C4—4881no—
532C4—2441no—
533C4#135×D6111yescharacteristic
534C4#1684056011yescharacteristic
535C4#1684057611yescharacteristic
536C4#1684057711yescharacteristic
537C2—81no—
538C2#2145737011yescharacteristic
539C2#2145755411yescharacteristic
540C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3342C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenter#3342D61
hypocenterC61#19848
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3342

Complements

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

NH ≅ G/Ntype
3. #21458495C2semidirect
4. #21445464C2semidirect
5. #21445371C2semidirect
6. #21445275C2semidirect
8. #21445210C2semidirect
9. #21445095C2semidirect
11. #21445039C2semidirect
13. #21444670C2semidirect
337. C61#19848semidirect

Maximal subgroups

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

subgroupindex
#214585552
#214585002
#214584952
#214454642
#214453712
#214452752
#214452702
#214452102
#214450952
#214450892
#214450392
#214449912
#214446702
#214446312
C61×#33422
#1984861