Finite Group Databasebeta

← #29657239

Subgroups — #29657239

Order 15616.

1282 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456937C211yesmaximal, characteristic, complemented
2C2×#16840108C211yesmaximal, characteristic, complemented
3#21455306C211yesmaximal, characteristic, complemented
4#21445012C214yesmaximal, complemented
5#21444998C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3498C211yesmaximal, characteristic
10C2×#14519807—21no—
11C2×#14519777C2211yescharacteristic, complemented
12#16840165—21no—
13#16840108—21no—
14#16840108—22no—
15#16840108C2214yescomplemented
16#16840073—22no—
17#16840042C2214yescomplemented
18#16839972—21no—
19#16839972C2212yescomplemented
20#16839967—21no—
21#16839967C2212yescomplemented
22#16839815C2211yescharacteristic, complemented
23#16839786C2211yescharacteristic, complemented
24#16839770—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—
46#14519807—44no—
47C2×#11758202—22no—
48C2×#11758200—22no—
49C2×#11758194—21no—
50C22×#92418—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777C2312yes—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57#14519710—24no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519700—22no—
62#14519700—22no—
63#14519698—28no—
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×C61D814yes—
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—42no—
167C2×Dic122—42no—
168C2×Dic122—41no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D412yes—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—22no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2411yescharacteristic
189#11758202—44no—
190#11758200—44no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—41no—
194C4⋊Dic61—41no—
195#11758194—42no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—42no—
200C23×C61—21no—
201C23×C61—22no—
202C23×C61—24no—
203C23×C61—48no—
204C23×C61—44no—
205C23×C61—82no—
206C23×C61—82no—
207C23×C61—84no—
208C23×C61—84no—
209C23×C61#12011yescharacteristic
210Q8×C61—21no—
211Q8×C61—42no—
212Q8×C61C2×D812yes—
213D4×C61—24no—
214D4×C61—21no—
215D4×C61—24no—
216D4×C61—44no—
217D4×C61—48no—
218D4×C61—48no—
219D4×C61—44no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—44no—
223D4×C61—42no—
224D4×C61—44no—
225D4×C61—44no—
226D4×C61—42no—
227D4×C61—82no—
228D4×C61—84no—
229D4×C61—84no—
230D4×C61C2×D812yes—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—24no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—48no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61C2×D812yes—
247C2×C4×C61AGL(1,ℤ/8)12yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×Dic61—42no—
252C2×Dic61—82no—
253Dic122—42no—
254Dic122—42no—
255Dic122—82no—
256#92418—42no—
257#92418—42no—
258#92418—42no—
259#18630—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—22no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—44no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61—42no—
271C22×C61—164no—
272C22×C61—168no—
273C22×C61#44712yes—
274C22×C61C2×#12011yescharacteristic
275C4×C61—22no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—42no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61#44712yes—
282C4×C61#45512yes—
283C4×C61C22×D811yescharacteristic
284C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
285C4×C61#57712yes—
286Dic61—82no—
287#3498D6111yescharacteristic
288#3226—611no—
289C2×#486—611no—
290#3039—611no—
291#1641—614no—
292#1639—614no—
293#1545—611no—
294#1492—611no—
295#1476—611no—
296C2×C61—22no—
297C2×C61—42no—
298C2×C61—82no—
299C2×C61—84no—
300C2×C61C2×#44711yescharacteristic
301C2×C61#300911yescharacteristic
302C2×C61C2×#45511yescharacteristic
303C2×H(5,𝔽2)—21no—
304C2×#137—1221no—
305#550—21no—
306#550C2×D6112yes—
307D42—22no—
308D42—22no—
309D42—21no—
310D42C2×D6112yes—
311#535—22no—
312#535—21no—
313#532—21no—
314#532C2×D6111yescharacteristic
315C2×C4⋊Q8—611no—
316C2×#127—21no—
317C2×#127C2×D6111yescharacteristic
318C4×#41C2×D6111yescharacteristic
319#500—1221no—
320#486—614no—
321#486—1222no—
322#486—1221no—
323#480—1222no—
324#474—614no—
325#462—612no—
326#462—1221no—
327#461—612no—
328#461—1221no—
329#431—611no—
330#423—611no—
331#419—1221no—
332C2×#102—1222no—
333C2×C4×C8—611no—
334#331—614no—
335Q8⋊C8—612no—
336D4⋊C8—612no—
337D4⋊C8—614no—
338C61#1863011yescharacteristic, complemented, Sylow
339H(5,𝔽2)—44no—
340H(5,𝔽2)—44no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—21no—
344C2×#41C22⋊D6111yescharacteristic
345C22×Q8—1221no—
346C22×D4—22no—
347C22×D4—24no—
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—21no—
352C22×D4—21no—
353C22×D4C22⋊D6112yes—
354#137—2444no—
355C2×Q16—1222no—
356C2×SD16—1222no—
357C2×M16—1221no—
358C22×C8—1221no—
359C4⋊Q8—612no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362C4⋊Q8—1221no—
363C4⋊Q8C22⋊D6111yescharacteristic
364#127—22no—
365#127—22no—
366#127—22no—
367#127—21no—
368#127—21no—
369#127C22⋊D6111yescharacteristic
370#127C22⋊D6112yes—
371#127C22×D6112yes—
372#124—22no—
373#124—21no—
374#121—22no—
375#121—24no—
376#121—24no—
377#121—21no—
378#121—22no—
379#121—22no—
380#121—22no—
381#121C22⋊D6112yes—
382#120—24no—
383#120—24no—
384#120—24no—
385C4×Q8C22⋊D6111yescharacteristic
386C4×Q8C22×D6111yescharacteristic
387C4×D4—22no—
388C4×D4—21no—
389C4×D4—21no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4C22⋊D6111yescharacteristic
393C4×D4C22×D6111yescharacteristic
394C4×D4C22×D6112yes—
395#117—21no—
396#117C22⋊D6111yescharacteristic
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C42—21no—
400C2×C42C22×D6111yescharacteristic
401#107—1224no—
402C4⋊C8—612no—
403C4⋊C8—612no—
404C4⋊C8—1221no—
405#103—1222no—
406#103—1222no—
407#102—1228no—
408#102—1224no—
409C22⋊C8—1221no—
410C4×C8—612no—
411C4×C8—1221no—
412C4×C8—1221no—
413C24—24no—
414C24—22no—
415#41—22no—
416#41—48no—
417#41—44no—
418#41—44no—
419#41—42no—
420#41—42no—
421#41#1175819914yes—
422C2×Q8—21no—
423C2×Q8—1222no—
424C2×Q8—2442no—
425C2×Q8—2442no—
426C2×Q8—2441no—
427C2×Q8C2×C22⋊D6111yescharacteristic
428C2×D4—21no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—48no—
442C2×D4—44no—
443C2×D4—416no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—82no—
449C2×D4—84no—
450C2×D4—84no—
451C2×D4C2×C22⋊D6112yes—
452C2×D4C2×C22⋊D6111yescharacteristic
453C2×D4C2×C22⋊D6114yes—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—2441no—
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463Q16—2444no—
464SD16—2444no—
465M16—2442no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8—1222no—
469C2×C8—2442no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C4⋊C4C2×C22⋊D6112yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—22no—
485C22⋊C4—21no—
486C22⋊C4—22no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42C2×C22⋊D6111yescharacteristic
491C42C2×C22⋊D6111yescharacteristic
492C42C2×C22⋊D6111yescharacteristic
493C42C23×D6111yescharacteristic
494C23—21no—
495C23—24no—
496C23—22no—
497C23—48no—
498C23—44no—
499C23—82no—
500C23—82no—
501C23—84no—
502C23—84no—
503C23#1451974611yescharacteristic
504Q8—21no—
505Q8—42no—
506Q8—4882no—
507Q8—2442no—
508Q8—2442no—
509Q8C2×#1175819912yes—
510D4—24no—
511D4—21no—
512D4—24no—
513D4—44no—
514D4—48no—
515D4—48no—
516D4—44no—
517D4—44no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—44no—
522D4—42no—
523D4—44no—
524D4—82no—
525D4—84no—
526D4—84no—
527D4C2×#1175819912yes—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—4882no—
541C2×C4—2442no—
542C2×C4#1451974611yescharacteristic
543C2×C4#1451974611yescharacteristic
544C2×C4#1451974611yescharacteristic
545C2×C4C2×#1175819912yes—
546C2×C4#1451980312yes—
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4C22×C22⋊D6111yescharacteristic
549C2×C4C22×C22⋊D6111yescharacteristic
550C8—2442no—
551C8—2442no—
552C8—2442no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—41no—
557C22—41no—
558C22—44no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—164no—
565C22—168no—
566C22#1683989612yes—
567C22C2×#1451974611yescharacteristic
568C4—21no—
569C4—22no—
570C4—21no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—4882no—
575C4#1683989612yes—
576C4#1683994012yes—
577C4C22×#1175819911yescharacteristic
578C4C2×#1451980311yescharacteristic
579C4#1684057312yes—
580C2—22no—
581C2—42no—
582C2—84no—
583C2—82no—
584C2C2×#1683989611yescharacteristic
585C2#2145509611yescharacteristic
586C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3498C2
socleC22×C61C2×#120
hypercenter#3498D61
hypocenterC61#18630
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. #21456937C2semidirect
2. C2×#16840108C2semidirect
3. #21455306C2semidirect
4. #21445012C2semidirect
5. #21444998C2semidirect
6. #21444487C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
11. C2×#14519777C22semidirect
15. #16840108C22semidirect
17. #16840042C22semidirect
19. #16839972C22semidirect
21. #16839967C22semidirect
22. #16839815C22semidirect
23. #16839786C22semidirect
26. C2×C4×#92418C22semidirect
27. #16839506C22semidirect
28. Q8⋊#92418C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
338. C61#18630semidirect

Maximal subgroups

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

subgroupindex
#214569372
C2×#168401082
#214553062
#214450122
#214449982
#214444872
#214442702
#214442012
C61×#34982
#1863061