Finite Group Databasebeta

← #29670601

Subgroups — #29670601

Order 15616.

574 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458708C211yesmaximal, characteristic, complemented
2#21458600C211yesmaximal, characteristic
3#21458461C211yesmaximal, characteristic
4#21445468C211yesmaximal, characteristic, complemented
5#21445375C211yesmaximal, characteristic, complemented
6#21445279C211yesmaximal, characteristic, complemented
7#21445274C211yesmaximal, characteristic, complemented
8#21445208C211yesmaximal, characteristic
9#21445094C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic
11#21445026C211yesmaximal, characteristic
12#21444976C211yesmaximal, characteristic, complemented
13#21444670C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic
15C61×#3299C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17#16840491C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840163—21no—
20#16840158C2211yescharacteristic
21#16840132—21no—
22#16840114—21no—
23#16840114C2211yescharacteristic
24#16840114C2211yescharacteristic
25#16840112C2211yescharacteristic
26#16840071—21no—
27#16840071—21no—
28#16840066—21no—
29#16840054C2211yescharacteristic
30#16840047C2211yescharacteristic
31#16840047C2211yescharacteristic
32#16840044—21no—
33#16840044C2211yescharacteristic
34#16839967—21no—
35#16839967—21no—
36#16839967C2211yescharacteristic
37#16839967C2211yescharacteristic
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839862—21no—
42Q16×Dic61—21no—
43C4×#11758200—21no—
44C4×#11758200C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#554C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#484—21no—
60C61×#477C2211yescharacteristic
61C61×#467—21no—
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C4×Q16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#103×C61—21no—
67C2×#11758200—21no—
68C2×#11758200—21no—
69C2×#11758200D411yescharacteristic
70#14519777—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519768—21no—
75#14519768—21no—
76#14519768—41no—
77#14519768—41no—
78Q8×Dic61—21no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519711—21no—
84#14519711—21no—
85#14519710—21no—
86#14519708—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—41no—
90#14519708D411yescharacteristic
91C4⋊#92418—21no—
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—41no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700—21no—
108#14519700D411yescharacteristic
109#14519698D411yescharacteristic
110#14519688—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C22×Q8×C61—21no—
116C2×Q16×C61—21no—
117C2×Q16×C61D411yescharacteristic
118C22×C8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#125×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
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×C61—41no—
147C2×Q8×C61—42no—
148C2×Q8×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154C22×C4×C61—41no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122C2×D411yescharacteristic
158Q16×C61—22no—
159Q16×C61—81no—
160C2×C8×C61—21no—
161C2×C8×C61—81no—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61C2411yescharacteristic
182#11758200—22no—
183#11758200—42no—
184#11758200—42no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190C4⋊Dic61—41no—
191C4⋊Dic61—41no—
192C4⋊Dic61—41no—
193C4⋊Dic61C2×D411yescharacteristic
194#11758196—21no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—21no—
198C4×Dic61—41no—
199C4×Dic61—41no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C23×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—41no—
211Q8×C61—41no—
212Q8×C61—82no—
213Q8×C61—81no—
214Q8×C61#13712yes—
215D4×C61—21no—
216D4×C61#13512yes—
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×C61—82no—
227C2×C4×C61—161no—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235Dic122—21no—
236Dic122—21no—
237Dic122—41no—
238Dic122—41no—
239Dic122—41no—
240Dic122—41no—
241C8×C61—21no—
242C8×C61—21no—
243C8×C61—41no—
244#92418—21no—
245#92418—21no—
246#92418—41no—
247#92418—41no—
248#92418—41no—
249#92418—41no—
250#19838—611nomaximal, Sylow
251C22×C61—22no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61D4211yescharacteristic
255C4×C61—22no—
256C4×C61—21no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61C2×#13511yescharacteristic
264C4×C61C2×#13711yescharacteristic
265C4×C61#57711yescharacteristic
266C4×C61#57811yescharacteristic
267Dic61—41no—
268Dic61—41no—
269Dic61—81no—
270Dic61—81no—
271#3383—611no—
272#3372—611no—
273#3359—611no—
274#3299D6111yescharacteristic
275#1718—611no—
276#1700—611no—
277#1684—611no—
278#1684—611no—
279#1673—611no—
280#1653—611no—
281#1651—611no—
282#1643—611no—
283#1636—611no—
284#1584—611no—
285SD16⋊C8—611no—
286C2×C61—42no—
287C2×C61#326611yescharacteristic
288C2×C61#328511yescharacteristic
289C2×C61#328911yescharacteristic
290#557—611no—
291#557—611no—
292#554C2×D6111yescharacteristic
293D4×Q8C2×D6111yescharacteristic
294#500—611no—
295#499—1221no—
296#498—611no—
297#491—1221no—
298#487—611no—
299#487—611no—
300#487—611no—
301#487—1221no—
302#487C2×D6111yescharacteristic
303#484—21no—
304#479—1221no—
305#479—1221no—
306#479—1221no—
307#477—611no—
308#477C2×D6111yescharacteristic
309#475—611no—
310#475—611no—
311#475—611no—
312#475—1221no—
313#467—21no—
314#462C2×D6111yescharacteristic
315#461—611no—
316#461—611no—
317#461—1221no—
318#461—1221no—
319#451—21no—
320#446—611no—
321C8×Q8—611no—
322C8×Q8—611no—
323#440—1221no—
324C4×Q16—1221no—
325C4×Q16C2×D6111yescharacteristic
326C4×SD16—611no—
327C4×SD16—1221no—
328C8×D4C2×D6111yescharacteristic
329C2×#103—21no—
330#335—611no—
331#335—611no—
332#333—611no—
333#332—611no—
334#332—611no—
335#332—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339D4⋊C8—611no—
340#322—611no—
341C61#1983811yescharacteristic, complemented, Sylow
342C22×Q8—21no—
343C2×Q16—21no—
344C2×Q16C22⋊D6111yescharacteristic
345C2×SD16—611no—
346C2×SD16—1221no—
347C2×SD16—1221no—
348C22×C8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—611no—
351C4⋊Q8—611no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8C22×D6111yescharacteristic
355C4⋊Q8C22×D6111yescharacteristic
356#125—21no—
357#125—1221no—
358#125—1221no—
359#125—2441no—
360#125—2441no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—41no—
364C22⋊Q8—41no—
365C4×Q8—611no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8—1221no—
369C4×Q8—1221no—
370C4×Q8C22×D6111yescharacteristic
371C4×Q8C22×D6111yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C22×D6111yescharacteristic
375C2×C4⋊C4—21no—
376#107—21no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#107C22⋊D6111yescharacteristic
381#106—611no—
382#106—1221no—
383#106—1221no—
384#106—1221no—
385#106—2441no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8C22×D6111yescharacteristic
393#103—21no—
394#103—21no—
395#103—21no—
396#103—21no—
397#103—611no—
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—
408#103—2441no—
409#103C22⋊D6111yescharacteristic
410#103C22⋊D6111yescharacteristic
411#102—611no—
412C22⋊C8—21no—
413#97—1221no—
414C4×C8—611no—
415C4×C8—611no—
416C4×C8—1221no—
417C4×C8—1221no—
418C4×C8C22×D6111yescharacteristic
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—41no—
422C2×Q8—42no—
423C2×Q8—81no—
424C2×Q8—611no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8C2×C22⋊D6111yescharacteristic
428C2×Q8C2×C22⋊D6111yescharacteristic
429C2×D4D4×D6111yescharacteristic
430C22×C4—21no—
431C22×C4—41no—
432C22×C4—41no—
433Q16—22no—
434Q16—81no—
435SD16—1222no—
436SD16—2442no—
437SD16—2442no—
438C2×C8—21no—
439C2×C8—81no—
440C2×C8—611no—
441C2×C8—611no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8C2×C22⋊D6111yescharacteristic
447C2×C8C2×C22⋊D6111yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—611no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C4⋊C4D4×D6111yescharacteristic
470C4⋊C4C2×C22⋊D6111yescharacteristic
471C4⋊C4C2×C22⋊D6111yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C42—21no—
476C42—21no—
477C42—1221no—
478C42—2441no—
479C42—2441no—
480C42C23×D6111yescharacteristic
481C23—21no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—41no—
486Q8—41no—
487Q8—82no—
488Q8—81no—
489Q8—1221no—
490Q8—1221no—
491Q8—2441no—
492Q8—2441no—
493Q8—2441no—
494Q8—2441no—
495Q8#1451980712yes—
496D4—21no—
497D4#1451980012yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—82no—
508C2×C4—161no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4#1451982211yescharacteristic
516C8—21no—
517C8—21no—
518C8—41no—
519C8—1221no—
520C8—1221no—
521C8—2441no—
522C8—2441no—
523C8—2441no—
524C8—2441no—
525C22—22no—
526C22—41no—
527C22—41no—
528C22D4×C22⋊D6111yescharacteristic
529C4—22no—
530C4—21no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—81no—
536C4—81no—
537C4—4881no—
538C4—4881no—
539C4—2441no—
540C4—2441no—
541C4C2×#1451980011yescharacteristic
542C4C2×#1451980711yescharacteristic
543C4#1684057311yescharacteristic
544C4#1684057511yescharacteristic
545C2—42no—
546C2#2145732511yescharacteristic
547C2#2145756011yescharacteristic
548C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3299C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenter#3299D61
hypocenterC61#19838
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3299

Complements

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

NH ≅ G/Ntype
1. #21458708C2semidirect
4. #21445468C2semidirect
5. #21445375C2semidirect
6. #21445279C2semidirect
7. #21445274C2semidirect
9. #21445094C2semidirect
12. #21444976C2semidirect
13. #21444670C2semidirect
341. C61#19838semidirect

Maximal subgroups

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

subgroupindex
#214587082
#214586002
#214584612
#214454682
#214453752
#214452792
#214452742
#214452082
#214450942
#214450822
#214450262
#214449762
#214446702
#214446312
C61×#32992
#1983861