Finite Group Databasebeta

← #29657146

Subgroups — #29657146

Order 15616.

1314 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456944C211yesmaximal, characteristic, complemented
2C2×#16840134C211yesmaximal, characteristic, complemented
3#21455634C211yesmaximal, characteristic, complemented
4#21444946C214yesmaximal, complemented
5#21444935C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3498C211yesmaximal, characteristic
10C22×#11758200—22no—
11C2×#14519777C2211yescharacteristic, complemented
12#16840158—21no—
13#16840134—22no—
14#16840134C2214yescomplemented
15#16840108—21no—
16#16840078—22no—
17#16840057C2212yescomplemented
18#16840052C2212yescomplemented
19#16840052C2214yescomplemented
20#16840047—21no—
21#16840042—21no—
22#16839815C2211yescharacteristic, complemented
23#16839805—21no—
24#16839786C2211yescharacteristic, complemented
25#16839763—21no—
26C2×C4×#92418C2211yescharacteristic, complemented
27#16839506C2214yescomplemented
28Q8⋊#92418C2212yescomplemented
29D4⋊#92418C2212yescomplemented
30D4⋊#92418C2214yescomplemented
31C2×H(5,𝔽2)×C61—21no—
32C61×#550—21no—
33C61×#550C2212yes—
34D42×C61—22no—
35D42×C61—22no—
36D42×C61—21no—
37D42×C61C2212yes—
38C61×#535—22no—
39C61×#535—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×#127×C61—21no—
43C2×#127×C61C2211yescharacteristic
44C4×#41×C61C2211yescharacteristic
45C22×Dic122—21no—
46C2×#11758200—28no—
47C2×#11758200—48no—
48C2×#11758194—21no—
49C22×#92418—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777C2312yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56#14519710—22no—
57#14519708—22no—
58#14519708—24no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519698—22no—
64#14519698—24no—
65C22⋊#92418—21no—
66C4×#92418—21no—
67C4×#92418—21no—
68C4×#92418C2312yes—
69H(5,𝔽2)×C61—44no—
70H(5,𝔽2)×C61—44no—
71C2×#41×C61—22no—
72C2×#41×C61—22no—
73C2×#41×C61—21no—
74C2×#41×C61D411yescharacteristic
75C22×D4×C61—22no—
76C22×D4×C61—24no—
77C22×D4×C61—22no—
78C22×D4×C61—22no—
79C22×D4×C61—22no—
80C22×D4×C61—21no—
81C22×D4×C61—21no—
82C22×D4×C61D412yes—
83C4⋊Q8×C61D411yescharacteristic
84#127×C61—22no—
85#127×C61—22no—
86#127×C61—22no—
87#127×C61—21no—
88#127×C61—21no—
89#127×C61D411yescharacteristic
90#127×C61D412yes—
91#127×C61C2312yes—
92#124×C61—22no—
93#124×C61—21no—
94#121×C61—22no—
95#121×C61—24no—
96#121×C61—24no—
97#121×C61—21no—
98#121×C61—22no—
99#121×C61—22no—
100#121×C61—22no—
101#121×C61D412yes—
102#120×C61—24no—
103#120×C61—24no—
104#120×C61—24no—
105C4×Q8×C61D411yescharacteristic
106C4×Q8×C61C2311yescharacteristic
107C4×D4×C61—22no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110C4×D4×C61—22no—
111C4×D4×C61—21no—
112C4×D4×C61D411yescharacteristic
113C4×D4×C61C2312yes—
114C4×D4×C61C2311yescharacteristic
115#117×C61—21no—
116#117×C61D411yescharacteristic
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C24×C61—24no—
120C24×C61—22no—
121#41×C61—22no—
122#41×C61—48no—
123#41×C61—44no—
124#41×C61—44no—
125#41×C61—42no—
126#41×C61—42no—
127#41×C61SD1614yescomplemented
128C2×Q8×C61—21no—
129C2×Q8×C61C2×D411yescharacteristic
130C2×D4×C61—21no—
131C2×D4×C61—24no—
132C2×D4×C61—24no—
133C2×D4×C61—22no—
134C2×D4×C61—24no—
135C2×D4×C61—24no—
136C2×D4×C61—22no—
137C2×D4×C61—21no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—48no—
144C2×D4×C61—44no—
145C2×D4×C61—416no—
146C2×D4×C61—44no—
147C2×D4×C61—44no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61—82no—
151C2×D4×C61—84no—
152C2×D4×C61—84no—
153C2×D4×C61C2×D412yes—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D414yes—
156C22×C4×C61—21no—
157C22×C4×C61—24no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—22no—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×Dic61—41no—
165C2×Dic122—22no—
166C2×Dic122—44no—
167C2×Dic122—41no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D412yes—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—22no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2411yescharacteristic
188#11758200—416no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—41no—
192C4⋊Dic61—41no—
193#11758194—42no—
194C2×#92418—24no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C23×C61—21no—
199C23×C61—22no—
200C23×C61—24no—
201C23×C61—48no—
202C23×C61—44no—
203C23×C61—82no—
204C23×C61—82no—
205C23×C61—84no—
206C23×C61—84no—
207C23×C61#12011yescharacteristic
208Q8×C61—21no—
209Q8×C61—42no—
210Q8×C61C2×SD1612yescomplemented
211D4×C61—24no—
212D4×C61—21no—
213D4×C61—24no—
214D4×C61—44no—
215D4×C61—48no—
216D4×C61—48no—
217D4×C61—44no—
218D4×C61—42no—
219D4×C61—42no—
220D4×C61—44no—
221D4×C61—42no—
222D4×C61—44no—
223D4×C61—44no—
224D4×C61—42no—
225D4×C61—82no—
226D4×C61—84no—
227D4×C61—84no—
228D4×C61C2×SD1612yescomplemented
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—24no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—48no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61C2×SD1612yes—
245C2×C4×C61AGL(1,ℤ/8)12yes—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×Dic61—42no—
250C2×Dic61—82no—
251Dic122—44no—
252Dic122—82no—
253#92418—24no—
254#92418—42no—
255#92418—42no—
256#18621—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—22no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—44no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—44no—
266C22×C61—44no—
267C22×C61—42no—
268C22×C61—164no—
269C22×C61—168no—
270C22×C61#45012yes—
271C22×C61C2×#12011yescharacteristic
272C4×C61—22no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—42no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61#45012yes—
279C4×C61#45312yes—
280C4×C61C22×SD1611yescharacteristic
281C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C4×C61#57712yes—
283Dic61—82no—
284#3498D6111yescharacteristic
285#3227—611no—
286C2×#492—611no—
287#3079—611no—
288#1631—614no—
289#1629—614no—
290#1545—611no—
291#1492—611no—
292#1476—611no—
293C2×C61—22no—
294C2×C61—42no—
295C2×C61—82no—
296C2×C61—84no—
297C2×C61C2×#45011yescharacteristic
298C2×C61#300911yescharacteristic
299C2×C61C2×#45311yescharacteristic
300C2×H(5,𝔽2)—21no—
301C22×SD16—1222no—
302#550—21no—
303#550C2×D6112yes—
304D42—22no—
305D42—22no—
306D42—21no—
307D42C2×D6112yes—
308#535—22no—
309#535—21no—
310#532—21no—
311#532C2×D6111yescharacteristic
312C2×C4⋊Q8—611no—
313C2×#127—21no—
314C2×#127C2×D6111yescharacteristic
315C4×#41C2×D6111yescharacteristic
316#498—1221no—
317#492—614no—
318#492—1222no—
319#486—1221no—
320#481—1222no—
321#477—612no—
322#476—612no—
323#476—614no—
324#475—1221no—
325#474—1221no—
326#431—611no—
327#428—1221no—
328#423—611no—
329#417—1221no—
330C2×C4×C8—611no—
331#331—614no—
332Q8⋊C8—612no—
333D4⋊C8—612no—
334D4⋊C8—614no—
335C61#1862111yescharacteristic, complemented, Sylow
336H(5,𝔽2)—44no—
337H(5,𝔽2)—44no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—21no—
341C2×#41C22⋊D6111yescharacteristic
342C22×Q8—1221no—
343C22×D4—22no—
344C22×D4—24no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—21no—
349C22×D4—21no—
350C22×D4C22⋊D6112yes—
351C2×SD16—1228no—
352C2×SD16—2448no—
353C2×M16—1221no—
354C22×C8—1221no—
355C4⋊Q8—612no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8C22⋊D6111yescharacteristic
360#127—22no—
361#127—22no—
362#127—22no—
363#127—21no—
364#127—21no—
365#127C22⋊D6111yescharacteristic
366#127C22⋊D6112yes—
367#127C22×D6112yes—
368#124—22no—
369#124—21no—
370#121—22no—
371#121—24no—
372#121—24no—
373#121—21no—
374#121—22no—
375#121—22no—
376#121—22no—
377#121C22⋊D6112yes—
378#120—24no—
379#120—24no—
380#120—24no—
381C4×Q8C22⋊D6111yescharacteristic
382C4×Q8C22×D6111yescharacteristic
383C4×D4—22no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—22no—
387C4×D4—21no—
388C4×D4C22⋊D6111yescharacteristic
389C4×D4C22×D6111yescharacteristic
390C4×D4C22×D6112yes—
391#117—21no—
392#117C22⋊D6111yescharacteristic
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C42—21no—
396C2×C42C22×D6111yescharacteristic
397#107—1222no—
398#106—1222no—
399#106—1224no—
400C4⋊C8—612no—
401C4⋊C8—612no—
402C4⋊C8—1221no—
403#103—1222no—
404#102—1222no—
405#102—1224no—
406C22⋊C8—1221no—
407C4×C8—612no—
408C4×C8—1221no—
409C4×C8—1221no—
410C24—24no—
411C24—22no—
412#41—22no—
413#41—48no—
414#41—44no—
415#41—44no—
416#41—42no—
417#41—42no—
418#41#1175820014yescomplemented
419C2×Q8—21no—
420C2×Q8—1222no—
421C2×Q8—2444no—
422C2×Q8—2441no—
423C2×Q8C2×C22⋊D6111yescharacteristic
424C2×D4—21no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—48no—
438C2×D4—44no—
439C2×D4—416no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—82no—
445C2×D4—84no—
446C2×D4—84no—
447C2×D4C2×C22⋊D6112yes—
448C2×D4C2×C22⋊D6111yescharacteristic
449C2×D4C2×C22⋊D6114yes—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—2441no—
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459SD16—24416no—
460M16—2442no—
461C2×C8—1224no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1222no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4C2×C22⋊D6111yescharacteristic
473C4⋊C4C2×C22⋊D6112yes—
474C4⋊C4C2×C22⋊D6111yescharacteristic
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C22⋊C4—22no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42C2×C22⋊D6111yescharacteristic
486C42C2×C22⋊D6111yescharacteristic
487C42C2×C22⋊D6111yescharacteristic
488C42C23×D6111yescharacteristic
489C23—21no—
490C23—24no—
491C23—22no—
492C23—48no—
493C23—44no—
494C23—82no—
495C23—82no—
496C23—84no—
497C23—84no—
498C23#1451974611yescharacteristic
499Q8—21no—
500Q8—42no—
501Q8—4882no—
502Q8—2444no—
503Q8C2×#1175820012yescomplemented
504D4—24no—
505D4—21no—
506D4—24no—
507D4—44no—
508D4—48no—
509D4—48no—
510D4—44no—
511D4—42no—
512D4—42no—
513D4—44no—
514D4—42no—
515D4—44no—
516D4—44no—
517D4—42no—
518D4—82no—
519D4—84no—
520D4—84no—
521D4C2×#1175820012yescomplemented
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—4882no—
535C2×C4—2442no—
536C2×C4#1451974611yescharacteristic
537C2×C4#1451974611yescharacteristic
538C2×C4#1451974611yescharacteristic
539C2×C4C2×#1175820012yes—
540C2×C4#1451980312yes—
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C8—1224no—
545C8—2442no—
546C8—2442no—
547C22—21no—
548C22—21no—
549C22—22no—
550C22—41no—
551C22—41no—
552C22—44no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—164no—
559C22—168no—
560C22#1683991312yes—
561C22C2×#1451974611yescharacteristic
562C4—21no—
563C4—22no—
564C4—21no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—4882no—
569C4#1683991312yes—
570C4#1683993012yes—
571C4C22×#1175820011yescharacteristic
572C4C2×#1451980311yescharacteristic
573C4#1684057312yes—
574C2—22no—
575C2—42no—
576C2—84no—
577C2—82no—
578C2C2×#1683991311yescharacteristic
579C2#2145509611yescharacteristic
580C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3498C2
socleC22×C61C2×#120
hypercenter#3498D61
hypocenterC61#18621
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3498

Complements

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

NH ≅ G/Ntype
1. #21456944C2semidirect
2. C2×#16840134C2semidirect
3. #21455634C2semidirect
4. #21444946C2semidirect
5. #21444935C2semidirect
6. #21444487C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
11. C2×#14519777C22semidirect
14. #16840134C22semidirect
17. #16840057C22semidirect
18. #16840052C22semidirect
19. #16840052C22semidirect
22. #16839815C22semidirect
24. #16839786C22semidirect
26. C2×C4×#92418C22semidirect
27. #16839506C22semidirect
28. Q8⋊#92418C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
127. #41×C61SD16semidirect
210. Q8×C61C2×SD16semidirect
228. D4×C61C2×SD16semidirect
335. C61#18621semidirect
418. #41#11758200semidirect
503. Q8C2×#11758200semidirect
521. D4C2×#11758200semidirect

Maximal subgroups

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

subgroupindex
#214569442
C2×#168401342
#214556342
#214449462
#214449352
#214444872
#214442702
#214442012
C61×#34982
#1862161