Finite Group Databasebeta

← #29657248

Subgroups — #29657248

Order 15616.

1262 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456741C211yesmaximal, characteristic, complemented
2C2×#16840108C211yesmaximal, characteristic, complemented
3#21455634C211yesmaximal, characteristic, complemented
4#21445019C214yesmaximal, complemented
5#21444998C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3498C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic, complemented
11#16840134—21no—
12#16840114—21no—
13#16840108—22no—
14#16840108C2214yescomplemented
15#16840057—21no—
16#16840052—21no—
17#16840047C2212yescomplemented
18#16840042C2212yescomplemented
19#16839967C2214yescomplemented
20#16839900—22no—
21#16839815C2211yescharacteristic, complemented
22#16839805—21no—
23#16839786C2211yescharacteristic, complemented
24#16839763—21no—
25C2×#14519698—22no—
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—24no—
47C2×#11758194—21no—
48C22×#92418—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777C2312yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55#14519710—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418C2312yes—
59C4⋊#92418C2312yes—
60#14519701—22no—
61#14519700—24no—
62#14519698—28no—
63#14519698—22no—
64C22⋊#92418—21no—
65C4×#92418—21no—
66C4×#92418—21no—
67C4×#92418C2312yes—
68H(5,𝔽2)×C61—44no—
69H(5,𝔽2)×C61—44no—
70C2×#41×C61—22no—
71C2×#41×C61—22no—
72C2×#41×C61—21no—
73C2×#41×C61D411yescharacteristic
74C22×D4×C61—22no—
75C22×D4×C61—24no—
76C22×D4×C61—22no—
77C22×D4×C61—22no—
78C22×D4×C61—22no—
79C22×D4×C61—21no—
80C22×D4×C61—21no—
81C22×D4×C61D412yes—
82C4⋊Q8×C61D411yescharacteristic
83#127×C61—22no—
84#127×C61—22no—
85#127×C61—22no—
86#127×C61—21no—
87#127×C61—21no—
88#127×C61D411yescharacteristic
89#127×C61D412yes—
90#127×C61C2312yes—
91#124×C61—22no—
92#124×C61—21no—
93#121×C61—22no—
94#121×C61—24no—
95#121×C61—24no—
96#121×C61—21no—
97#121×C61—22no—
98#121×C61—22no—
99#121×C61—22no—
100#121×C61D412yes—
101#120×C61—24no—
102#120×C61—24no—
103#120×C61—24no—
104C4×Q8×C61D411yescharacteristic
105C4×Q8×C61C2311yescharacteristic
106C4×D4×C61—22no—
107C4×D4×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—22no—
110C4×D4×C61—21no—
111C4×D4×C61D411yescharacteristic
112C4×D4×C61C2312yes—
113C4×D4×C61C2311yescharacteristic
114#117×C61—21no—
115#117×C61D411yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C24×C61—24no—
119C24×C61—22no—
120#41×C61—22no—
121#41×C61—48no—
122#41×C61—44no—
123#41×C61—44no—
124#41×C61—42no—
125#41×C61—42no—
126#41×C61SD1614yescomplemented
127C2×Q8×C61—21no—
128C2×Q8×C61C2×D411yescharacteristic
129C2×D4×C61—21no—
130C2×D4×C61—24no—
131C2×D4×C61—24no—
132C2×D4×C61—22no—
133C2×D4×C61—24no—
134C2×D4×C61—24no—
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—22no—
138C2×D4×C61—21no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61—48no—
143C2×D4×C61—44no—
144C2×D4×C61—416no—
145C2×D4×C61—44no—
146C2×D4×C61—44no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61—82no—
150C2×D4×C61—84no—
151C2×D4×C61—84no—
152C2×D4×C61C2×D412yes—
153C2×D4×C61C2×D411yescharacteristic
154C2×D4×C61C2×D414yes—
155C22×C4×C61—21no—
156C22×C4×C61—24no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—22no—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×Dic61—41no—
164C2×Dic122—22no—
165C2×Dic122—41no—
166C2×Dic122—82no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61C2×D411yescharacteristic
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D412yes—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2411yescharacteristic
187#11758200—84no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—41no—
191C4⋊Dic61—41no—
192#11758194—42no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—42no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—24no—
200C23×C61—48no—
201C23×C61—44no—
202C23×C61—82no—
203C23×C61—82no—
204C23×C61—84no—
205C23×C61—84no—
206C23×C61#12011yescharacteristic
207Q8×C61—21no—
208Q8×C61—42no—
209Q8×C61C2×SD1612yes—
210D4×C61—24no—
211D4×C61—21no—
212D4×C61—24no—
213D4×C61—44no—
214D4×C61—48no—
215D4×C61—48no—
216D4×C61—44no—
217D4×C61—42no—
218D4×C61—42no—
219D4×C61—44no—
220D4×C61—42no—
221D4×C61—44no—
222D4×C61—44no—
223D4×C61—42no—
224D4×C61—82no—
225D4×C61—84no—
226D4×C61—84no—
227D4×C61C2×SD1612yes—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—48no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61C2×SD1612yes—
244C2×C4×C61AGL(1,ℤ/8)12yes—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×Dic61—42no—
249C2×Dic61—82no—
250Dic122—44no—
251Dic122—82no—
252#92418—42no—
253#92418—42no—
254#92418—42no—
255#18631—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—22no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—44no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—44no—
265C22×C61—44no—
266C22×C61—42no—
267C22×C61—164no—
268C22×C61—168no—
269C22×C61#45012yes—
270C22×C61C2×#12011yescharacteristic
271C4×C61—22no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—42no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61#45012yes—
278C4×C61#45512yes—
279C4×C61C22×SD1611yescharacteristic
280C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C4×C61#57612yes—
282Dic61—82no—
283#3498D6111yescharacteristic
284#3206—611no—
285C2×#486—611no—
286#3079—611no—
287#1642—614no—
288#1639—614no—
289#1545—611no—
290#1492—611no—
291#1476—611no—
292C2×C61—22no—
293C2×C61—42no—
294C2×C61—82no—
295C2×C61—84no—
296C2×C61C2×#45011yescharacteristic
297C2×C61#300811yescharacteristic
298C2×C61C2×#45511yescharacteristic
299C2×H(5,𝔽2)—21no—
300#550—21no—
301#550C2×D6112yes—
302D42—22no—
303D42—22no—
304D42—21no—
305D42C2×D6112yes—
306#535—22no—
307#535—21no—
308#532—21no—
309#532C2×D6111yescharacteristic
310C2×C4⋊Q8—611no—
311C2×#127—21no—
312C2×#127C2×D6111yescharacteristic
313C4×#41C2×D6111yescharacteristic
314#492—1221no—
315#487—1221no—
316#486—614no—
317#486—1222no—
318#477—1221no—
319#476—1221no—
320#475—612no—
321#474—612no—
322#461—614no—
323#448—1222no—
324#431—611no—
325#428—1221no—
326#423—611no—
327#417—1221no—
328C2×#102—1222no—
329C2×C4×C8—611no—
330#331—614no—
331Q8⋊C8—612no—
332D4⋊C8—612no—
333D4⋊C8—614no—
334C61#1863111yescharacteristic, complemented, Sylow
335H(5,𝔽2)—44no—
336H(5,𝔽2)—44no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—21no—
340C2×#41C22⋊D6111yescharacteristic
341C22×Q8—1221no—
342C22×D4—22no—
343C22×D4—24no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4C22⋊D6112yes—
350C2×SD16—1224no—
351C2×M16—1221no—
352C22×C8—1221no—
353C4⋊Q8—612no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8C22⋊D6111yescharacteristic
358#127—22no—
359#127—22no—
360#127—22no—
361#127—21no—
362#127—21no—
363#127C22⋊D6111yescharacteristic
364#127C22⋊D6112yes—
365#127C22×D6112yes—
366#124—22no—
367#124—21no—
368#121—22no—
369#121—24no—
370#121—24no—
371#121—21no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121C22⋊D6112yes—
376#120—24no—
377#120—24no—
378#120—24no—
379C4×Q8C22⋊D6111yescharacteristic
380C4×Q8C22×D6111yescharacteristic
381C4×D4—22no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—22no—
385C4×D4—21no—
386C4×D4C22⋊D6111yescharacteristic
387C4×D4C22×D6111yescharacteristic
388C4×D4C22×D6112yes—
389#117—21no—
390#117C22⋊D6111yescharacteristic
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C42—21no—
394C2×C42C22×D6111yescharacteristic
395#107—1222no—
396#106—1222no—
397C4⋊C8—612no—
398C4⋊C8—612no—
399C4⋊C8—1221no—
400#103—1222no—
401#103—1224no—
402#102—1228no—
403#102—1222no—
404C22⋊C8—1221no—
405C4×C8—612no—
406C4×C8—1221no—
407C4×C8—1221no—
408C24—24no—
409C24—22no—
410#41—22no—
411#41—48no—
412#41—44no—
413#41—44no—
414#41—42no—
415#41—42no—
416#41#1175820014yescomplemented
417C2×Q8—21no—
418C2×Q8—1222no—
419C2×Q8—4882no—
420C2×Q8—2441no—
421C2×Q8C2×C22⋊D6111yescharacteristic
422C2×D4—21no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—48no—
436C2×D4—44no—
437C2×D4—416no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—82no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4C2×C22⋊D6112yes—
446C2×D4C2×C22⋊D6111yescharacteristic
447C2×D4C2×C22⋊D6114yes—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—2441no—
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457SD16—4884no—
458M16—2442no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—2442no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C4⋊C4C2×C22⋊D6112yes—
471C4⋊C4C2×C22⋊D6111yescharacteristic
472C4⋊C4C2×C22⋊D6111yescharacteristic
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42C2×C22⋊D6111yescharacteristic
484C42C2×C22⋊D6111yescharacteristic
485C42C2×C22⋊D6111yescharacteristic
486C42C23×D6111yescharacteristic
487C23—21no—
488C23—24no—
489C23—22no—
490C23—48no—
491C23—44no—
492C23—82no—
493C23—82no—
494C23—84no—
495C23—84no—
496C23#1451974611yescharacteristic
497Q8—21no—
498Q8—42no—
499Q8—4882no—
500Q8—2444no—
501Q8C2×#1175820012yes—
502D4—24no—
503D4—21no—
504D4—24no—
505D4—44no—
506D4—48no—
507D4—48no—
508D4—44no—
509D4—44no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—44no—
514D4—42no—
515D4—44no—
516D4—82no—
517D4—84no—
518D4—84no—
519D4C2×#1175820012yes—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—4882no—
533C2×C4—2442no—
534C2×C4#1451974611yescharacteristic
535C2×C4#1451974611yescharacteristic
536C2×C4#1451974611yescharacteristic
537C2×C4C2×#1175820012yes—
538C2×C4#1451980312yes—
539C2×C4C22×C22⋊D6111yescharacteristic
540C2×C4C22×C22⋊D6111yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C8—2442no—
543C8—2442no—
544C8—2442no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—41no—
549C22—41no—
550C22—44no—
551C22—42no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—164no—
557C22—168no—
558C22#1683991312yes—
559C22C2×#1451974611yescharacteristic
560C4—21no—
561C4—22no—
562C4—21no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—4882no—
567C4#1683991312yes—
568C4#1683994012yes—
569C4C22×#1175820011yescharacteristic
570C4C2×#1451980311yescharacteristic
571C4#1684056512yes—
572C2—22no—
573C2—42no—
574C2—84no—
575C2—82no—
576C2C2×#1683991311yescharacteristic
577C2#2145508711yescharacteristic
578C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3498C2
socleC22×C61C2×#120
hypercenter#3498D61
hypocenterC61#18631
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. #21456741C2semidirect
2. C2×#16840108C2semidirect
3. #21455634C2semidirect
4. #21445019C2semidirect
5. #21444998C2semidirect
6. #21444487C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
10. C2×#14519777C22semidirect
14. #16840108C22semidirect
17. #16840047C22semidirect
18. #16840042C22semidirect
19. #16839967C22semidirect
21. #16839815C22semidirect
23. #16839786C22semidirect
26. C2×C4×#92418C22semidirect
27. #16839506C22semidirect
28. Q8⋊#92418C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
126. #41×C61SD16semidirect
334. C61#18631semidirect
416. #41#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567412
C2×#168401082
#214556342
#214450192
#214449982
#214444872
#214442702
#214442012
C61×#34982
#1863161