Finite Group Databasebeta

← #29656882

Subgroups — #29656882

Order 15616.

1190 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457013C211yesmaximal, characteristic, complemented
2#21456533C211yesmaximal, characteristic, complemented
3#21455827C211yesmaximal, characteristic, complemented
4#21445054C212yesmaximal, complemented
5#21444998C212yesmaximal, complemented
6#21444970C212yesmaximal, complemented
7#21444935C212yesmaximal, complemented
8#21444524C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3496C211yesmaximal, characteristic
12#16840319C2211yescharacteristic, complemented
13#16840168—21no—
14#16840165C2211yescharacteristic, complemented
15#16840134C2211yescharacteristic, complemented
16#16840126—21no—
17#16840114C2211yescharacteristic, complemented
18#16840108C2211yescharacteristic, complemented
19#16840092—21no—
20#16840083—22no—
21#16840052C2212yescomplemented
22#16840052C2212yescomplemented
23#16840042C2212yescomplemented
24#16840006—22no—
25#16839967C2212yescomplemented
26#16839913—21no—
27#16839862—22no—
28C4×#11758200—22no—
29#16839819C2211yescharacteristic, complemented
30#16839815C2211yescharacteristic, complemented
31C4×#11758194C2211yescharacteristic, complemented
32#16839508C2212yescomplemented
33#16839506C2212yescomplemented
34D4⋊#92418C2212yescomplemented
35D4⋊#92418C2212yescomplemented
36D4⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38C2×H(5,𝔽2)×C61—21no—
39C61×#547—21no—
40C61×#547C2212yes—
41C61×#546—22no—
42C61×#546—22no—
43D42×C61—21no—
44D42×C61C2212yes—
45C61×#537—21no—
46C61×#534—22no—
47C61×#532C2211yescharacteristic
48C61×#532C2211yescharacteristic
49C2×#121×C61—22no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758200—22no—
52C2×#11758200—22no—
53C2×#11758194—21no—
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519769—21no—
57#14519756—21no—
58#14519756—21no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61#14519736—21no—
62#14519710—22no—
63#14519710—22no—
64#14519708—22no—
65#14519708—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519700—22no—
69#14519700—22no—
70#14519698—22no—
71#14519698—22no—
72#14519698—22no—
73#14519698—22no—
74C22⋊#92418—21no—
75C22⋊#92418—21no—
76#14519688—21no—
77#14519688—21no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81H(5,𝔽2)×C61—48no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—21no—
85C2×#41×C61—21no—
86C22×D4×C61—24no—
87C22×D4×C61—22no—
88C22×D4×C61—22no—
89C22×D4×C61—22no—
90C22×D4×C61D411yescharacteristic
91C23×C4×C61—21no—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61C2311yescharacteristic
94#127×C61D411yescharacteristic
95#127×C61C2311yescharacteristic
96#124×C61—21no—
97#124×C61—21no—
98#123×C61—24no—
99#123×C61—24no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102#121×C61—24no—
103#121×C61—22no—
104#121×C61—24no—
105#121×C61—22no—
106#121×C61—21no—
107#121×C61—21no—
108#121×C61—22no—
109#121×C61—22no—
110#121×C61D412yes—
111#121×C61D412yes—
112#120×C61—24no—
113#120×C61—24no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61—21no—
117C4×D4×C61—21no—
118C4×D4×C61D412yes—
119C4×D4×C61D412yes—
120C4×D4×C61C2312yes—
121C4×D4×C61C2312yes—
122C2×C4⋊C4×C61—22no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C22⋊C4×C61—24no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2311yescharacteristic
127C24×C61—21no—
128C24×C61—24no—
129#41×C61—44no—
130#41×C61—44no—
131#41×C61—48no—
132#41×C61—44no—
133#41×C61—44no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×D4×C61—24no—
137C2×D4×C61—24no—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61—24no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—48no—
148C2×D4×C61—44no—
149C2×D4×C61—416no—
150C2×D4×C61—44no—
151C2×D4×C61—84no—
152C2×D4×C61C2×D412yes—
153C2×D4×C61C2×D412yes—
154C2×D4×C61C2×D412yes—
155C22×C4×C61—24no—
156C22×C4×C61—21no—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—22no—
161C22×C4×C61—82no—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C2×D412yes—
172C4⋊C4×C61C2×D412yes—
173C4⋊C4×C61C2×D412yes—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2411yescharacteristic
186C22⋊Dic61—41no—
187#11758200—44no—
188#11758200—44no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—41no—
196#11758194—42no—
197C2×#92418—21no—
198C2×#92418—22no—
199C2×#92418—21no—
200C2×#92418—22no—
201C23×C61—24no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—48no—
206C23×C61—81no—
207C23×C61—81no—
208C23×C61—84no—
209C23×C61—84no—
210C23×C61#12011yescharacteristic
211Q8×C61—42no—
212Q8×C61—42no—
213D4×C61—24no—
214D4×C61—22no—
215D4×C61—24no—
216D4×C61—42no—
217D4×C61—42no—
218D4×C61—44no—
219D4×C61—44no—
220D4×C61—48no—
221D4×C61—48no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—81no—
225D4×C61—84no—
226D4×C61AGL(1,ℤ/8)14yes—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—41no—
233C2×C4×C61—44no—
234C2×C4×C61—48no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—44no—
238C2×C4×C61—82no—
239C2×C4×C61—81no—
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×Dic61—41no—
247C2×Dic61—41no—
248Dic122—42no—
249Dic122—42no—
250#92418—22no—
251#92418—41no—
252#92418—42no—
253#92418—42no—
254#18599—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—24no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—84no—
268C22×C61—168no—
269C22×C61C2×#12011yescharacteristic
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61C2×AGL(1,ℤ/8)12yes—
278C4×C61#57712yes—
279Dic61—81no—
280Dic61—81no—
281#3496D6111yescharacteristic
282#3235—611no—
283#3182—611no—
284#3101—611no—
285#1647—612no—
286#1639—612no—
287#1635—612no—
288#1629—612no—
289#1553—611no—
290#1490—611no—
291#1484—611no—
292C2×C61—41no—
293C2×C61—44no—
294C2×C61—82no—
295C2×C61—82no—
296C2×C61#300311yescharacteristic
297C2×C61#300911yescharacteristic
298C2×C61#301811yescharacteristic
299C2×H(5,𝔽2)—21no—
300#547—21no—
301#547C2×D6112yes—
302#546—22no—
303#546—22no—
304D42—21no—
305D42C2×D6112yes—
306#537—21no—
307#534—22no—
308#533—611no—
309#532C2×D6111yescharacteristic
310#532C2×D6111yescharacteristic
311C2×#121—22no—
312C2×C4×D4C2×D6111yescharacteristic
313#501—1221no—
314#500—611no—
315#492—611no—
316#490—1221no—
317#487—611no—
318#486—611no—
319#483—1221no—
320#482—1222no—
321#476—612no—
322#476—612no—
323#474—612no—
324#468—1222no—
325#461—612no—
326#450—1221no—
327#440—1222no—
328C4×SD16—1222no—
329#432—611no—
330#431—611no—
331C4×M16—611no—
332#332—612no—
333#331—612no—
334D4⋊C8—612no—
335D4⋊C8—612no—
336D4⋊C8—612no—
337D4⋊C8—612no—
338C61#1859911yescharacteristic, complemented, Sylow
339H(5,𝔽2)—48no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—21no—
343C2×#41—21no—
344C22×D4—24no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4C22⋊D6111yescharacteristic
349C23×C4—21no—
350C2×SD16—1222no—
351C2×SD16—1222no—
352C2×M16—1221no—
353C4⋊Q8—611no—
354C4⋊Q8—611no—
355C4⋊Q8C22⋊D6111yescharacteristic
356C4⋊Q8C22×D6111yescharacteristic
357#127C22⋊D6111yescharacteristic
358#127C22×D6111yescharacteristic
359#125—1221no—
360#124—21no—
361#124—21no—
362#123—24no—
363#123—24no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—1221no—
367C22⋊Q8—1221no—
368#121—24no—
369#121—24no—
370#121—22no—
371#121—22no—
372#121—21no—
373#121—21no—
374#121—22no—
375#121—22no—
376#121C22⋊D6112yes—
377#121C22⋊D6112yes—
378#120—24no—
379#120—24no—
380C4×Q8—1221no—
381C4×Q8—1221no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4C22⋊D6112yes—
387C4×D4C22⋊D6112yes—
388C4×D4C22×D6112yes—
389C4×D4C22×D6112yes—
390#117—1221no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4C22⋊D6111yescharacteristic
393C2×C22⋊C4—24no—
394C2×C22⋊C4—21no—
395C2×C42C22×D6111yescharacteristic
396#107—1222no—
397#107—1222no—
398#106—1222no—
399#106—1222no—
400C4⋊C8—612no—
401C4⋊C8—612no—
402#103—1222no—
403#103—1222no—
404#102—1222no—
405#102—1222no—
406#102—1222no—
407#102—1222no—
408C22⋊C8—1221no—
409C22⋊C8—1221no—
410#97—1221no—
411#97—1221no—
412C4×C8—611no—
413C4×C8—611no—
414C4×C8—1221no—
415C24—21no—
416C24—24no—
417#41—44no—
418#41—44no—
419#41—48no—
420#41—44no—
421#41—44no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—48no—
438C2×D4—44no—
439C2×D4—416no—
440C2×D4—44no—
441C2×D4—84no—
442C2×D4C2×C22⋊D6112yes—
443C2×D4C2×C22⋊D6112yes—
444C2×D4C2×C22⋊D6112yes—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—82no—
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455SD16—2444no—
456SD16—2444no—
457M16—2442no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1222no—
461C2×C8—1222no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4C2×C22⋊D6112yes—
473C4⋊C4C2×C22⋊D6112yes—
474C4⋊C4C2×C22⋊D6112yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—2441no—
484C42—2441no—
485C42C2×C22⋊D6111yescharacteristic
486C42C2×C22⋊D6111yescharacteristic
487C42C2×C22⋊D6111yescharacteristic
488C42C23×D6111yescharacteristic
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—48no—
494C23—81no—
495C23—81no—
496C23—84no—
497C23—84no—
498C23#1451974611yescharacteristic
499Q8—42no—
500Q8—42no—
501Q8—2442no—
502Q8—2442no—
503D4—24no—
504D4—22no—
505D4—24no—
506D4—42no—
507D4—42no—
508D4—44no—
509D4—44no—
510D4—48no—
511D4—48no—
512D4—42no—
513D4—42no—
514D4—81no—
515D4—84no—
516D4#1451980314yes—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4#1451974611yescharacteristic
533C2×C4#1451974611yescharacteristic
534C2×C4#1451974611yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C2×C4C22×C22⋊D6111yescharacteristic
537C2×C4C22×C22⋊D6111yescharacteristic
538C8—1222no—
539C8—2441no—
540C8—2442no—
541C8—2442no—
542C22—21no—
543C22—21no—
544C22—24no—
545C22—21no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—84no—
555C22—168no—
556C22C2×#1451974611yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—4881no—
565C4—4881no—
566C4C2×#1451980312yes—
567C4#1684057312yes—
568C2—41no—
569C2—44no—
570C2—82no—
571C2—82no—
572C2#2145504611yescharacteristic
573C2#2145509611yescharacteristic
574C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18599
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #21457013C2semidirect
2. #21456533C2semidirect
3. #21455827C2semidirect
4. #21445054C2semidirect
5. #21444998C2semidirect
6. #21444970C2semidirect
7. #21444935C2semidirect
8. #21444524C2semidirect
9. #21444262C2semidirect
10. #21444236C2semidirect
12. #16840319C22semidirect
14. #16840165C22semidirect
15. #16840134C22semidirect
17. #16840114C22semidirect
18. #16840108C22semidirect
21. #16840052C22semidirect
22. #16840052C22semidirect
23. #16840042C22semidirect
25. #16839967C22semidirect
29. #16839819C22semidirect
30. #16839815C22semidirect
31. C4×#11758194C22semidirect
32. #16839508C22semidirect
33. #16839506C22semidirect
34. D4⋊#92418C22semidirect
35. D4⋊#92418C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
338. C61#18599semidirect

Maximal subgroups

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

subgroupindex
#214570132
#214565332
#214558272
#214450542
#214449982
#214449702
#214449352
#214445242
#214442622
#214442362
C61×#34962
#1859961