Finite Group Databasebeta

← #29670481

Subgroups — #29670481

Order 15616.

572 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#21458502C211yesmaximal, characteristic
2#21458497C211yesmaximal, characteristic, complemented
3#21458102C211yesmaximal, characteristic
4#21445463C211yesmaximal, characteristic, complemented
5#21445396C211yesmaximal, characteristic
6#21445249C211yesmaximal, characteristic, complemented
7#21445247C211yesmaximal, characteristic, complemented
8#21445198C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic
10#21445060C211yesmaximal, characteristic, complemented
11#21445040C211yesmaximal, characteristic, complemented
12#21444989C211yesmaximal, characteristic
13#21444663C211yesmaximal, characteristic
14#21444636C211yesmaximal, characteristic, complemented
15Q8×SD16×C61C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17#16840482C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840115—21no—
20#16840114—21no—
21#16840114C2211yescharacteristic
22#16840113C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840100—21no—
26#16840072—21no—
27#16840054—21no—
28#16840054C2211yescharacteristic
29#16840047C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840018—21no—
32#16839982—21no—
33#16839972—21no—
34#16839972C2211yescharacteristic
35#16839972C2211yescharacteristic
36#16839969C2211yescharacteristic
37#16839918—21no—
38Q8×#92418C2211yescharacteristic
39C8×Dic122C2211yescharacteristic
40C4×#11758202—21no—
41C4×#11758202C2211yescharacteristic
42#16839854—21no—
43#16839840C2211yescharacteristic
44#16839771—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#498—21no—
59C61×#498C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202D411yescharacteristic
71C2×#11758194—21no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519770—41no—
76#14519768—21no—
77#14519768—41no—
78#14519764—21no—
79#14519756—21no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519736—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—41no—
88#14519710D411yescharacteristic
89#14519708—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—41no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700—21no—
108#14519700D411yescharacteristic
109#14519698—21no—
110C22⋊#92418—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C22×Q8×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—41no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131#106×C61—21no—
132#106×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×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×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122C2×D411yescharacteristic
158SD16×C61—22no—
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×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—41no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—41no—
179C42×C61C2411yescharacteristic
180C22⋊Dic61—41no—
181C22⋊Dic61—41no—
182#11758202—22no—
183#11758202—42no—
184#11758202—42no—
185#11758202—81no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—41no—
191C4⋊Dic61—41no—
192C4⋊Dic61C2×D411yescharacteristic
193#11758196—21no—
194#11758196—41no—
195#11758196—41no—
196C4×Dic61—21no—
197C4×Dic61—41no—
198C4×Dic61—41no—
199#11758194—81no—
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—41no—
210Q8×C61—41no—
211Q8×C61—82no—
212Q8×C61—81no—
213Q8×C61—161no—
214Q8×C61#13512yes—
215Q8×C61#13712yes—
216D4×C61—21no—
217D4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—82no—
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#19830—611nomaximal, Sylow
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61D4211yescharacteristic
255C4×C61—22no—
256C4×C61—22no—
257C4×C61—21no—
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—
271Q8×SD16D6111yescharacteristic
272#3364—611no—
273#3363—611no—
274#3328—611no—
275#1718—611no—
276#1703—611no—
277#1680—611no—
278#1680—611no—
279#1672—611no—
280#1649—611no—
281#1648—611no—
282#1645—611no—
283#1638—611no—
284#1583—611no—
285#1579—611no—
286C2×C61—81no—
287C2×C61#327011yescharacteristic
288C2×C61#328211yescharacteristic
289C2×C61#328911yescharacteristic
290Q82C2×D6111yescharacteristic
291#557—611no—
292#556—611no—
293D4×Q8C2×D6111yescharacteristic
294#500—611no—
295#498—21no—
296#498C2×D6111yescharacteristic
297#488—1221no—
298#487—611no—
299#487—611no—
300#487—611no—
301#487—611no—
302#487—1221no—
303#485—1221no—
304#479—1221no—
305#477—611no—
306#477—1221no—
307#476—21no—
308#476C2×D6111yescharacteristic
309#475—21no—
310#475—611no—
311#475C2×D6111yescharacteristic
312#474—611no—
313#470—1221no—
314#464—1221no—
315#462—611no—
316#462—611no—
317#462—611no—
318#462—1221no—
319#451—1221no—
320C8×Q8—611no—
321C8×Q8—611no—
322C8×Q8C2×D6111yescharacteristic
323C4×Q16—611no—
324C4×Q16—1221no—
325C4×Q16—1221no—
326C4×SD16—21no—
327C4×SD16C2×D6111yescharacteristic
328#436—611no—
329#419—1221no—
330#334—611no—
331#334—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#1983011yescharacteristic, complemented, Sylow
342C22×Q8—21no—
343C2×Q16—611no—
344C2×Q16—1221no—
345C2×Q16—1221no—
346C2×Q16—1221no—
347C2×SD16C22⋊D6111yescharacteristic
348C2×M16—1221no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—21no—
352C4⋊Q8—41no—
353C4⋊Q8—611no—
354C4⋊Q8—611no—
355C4⋊Q8—1221no—
356C4⋊Q8C22×D6111yescharacteristic
357C4⋊Q8C22×D6111yescharacteristic
358#126—2441no—
359#125—1221no—
360#125—2441no—
361#124—1221no—
362C22⋊Q8—21no—
363C22⋊Q8—41no—
364C22⋊Q8—1221no—
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×D4—21no—
374C4×D4C22×D6111yescharacteristic
375#117—1221no—
376#107—611no—
377#107—1221no—
378#107—1221no—
379#107—2441no—
380#106—21no—
381#106—21no—
382#106—21no—
383#106—21no—
384#106—1221no—
385#106C22⋊D6111yescharacteristic
386C4⋊C8—21no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8C22×D6111yescharacteristic
393#103—21no—
394#103—611no—
395#103—611no—
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—2441no—
408#103C22⋊D6111yescharacteristic
409#102—21no—
410#102—1221no—
411#102C22⋊D6111yescharacteristic
412C22⋊C8—1221no—
413C4×C8—21no—
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—41no—
423C2×Q8—42no—
424C2×Q8—81no—
425C2×Q8—611no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8D4×D6111yescharacteristic
429C2×Q8C2×C22⋊D6111yescharacteristic
430C2×D4C2×C22⋊D6111yescharacteristic
431C22×C4—21no—
432C22×C4—41no—
433Q16—1222no—
434Q16—4881no—
435Q16—2442no—
436Q16—2442no—
437SD16—22no—
438M16—4881no—
439C2×C8—21no—
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⋊C4D4×D6111yescharacteristic
468C4⋊C4C2×C22⋊D6111yescharacteristic
469C4⋊C4C2×C22⋊D6111yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—41no—
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C42—21no—
475C42—21no—
476C42—41no—
477C42—1221no—
478C42—2441no—
479C42—2441no—
480C42C23×D6111yescharacteristic
481C23—21no—
482Q8—21no—
483Q8—21no—
484Q8—41no—
485Q8—41no—
486Q8—82no—
487Q8—81no—
488Q8—161no—
489Q8—1221no—
490Q8—1221no—
491Q8—2441no—
492Q8—2441no—
493Q8—2441no—
494Q8—2441no—
495Q8#1451979812yes—
496Q8#1451980812yes—
497D4—21no—
498D4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—82no—
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—41no—
526C22—41no—
527C22—41no—
528C22D4×C22⋊D6111yescharacteristic
529C4—22no—
530C4—21no—
531C4—22no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—81no—
536C4—81no—
537C4—4881no—
538C4—4881no—
539C4—2441no—
540C4—2441no—
541C4C2×#1451979811yescharacteristic
542C4C2×#1451980811yescharacteristic
543C4#1684056811yescharacteristic
544C4#1684057711yescharacteristic
545C2—81no—
546C2#2145739011yescharacteristic
547C2#2145752611yescharacteristic
548C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×SD16×C61C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenterQ8×SD16D61
hypocenterC61#19830
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
2. #21458497C2semidirect
4. #21445463C2semidirect
6. #21445249C2semidirect
7. #21445247C2semidirect
8. #21445198C2semidirect
10. #21445060C2semidirect
11. #21445040C2semidirect
14. #21444636C2semidirect
341. C61#19830semidirect

Maximal subgroups

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

subgroupindex
#214585022
#214584972
#214581022
#214454632
#214453962
#214452492
#214452472
#214451982
#214450682
#214450602
#214450402
#214449892
#214446632
#214446362
Q8×SD16×C612
#1983061