Finite Group Databasebeta

← #29657586

Subgroups — #29657586

Order 15616.

1206 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457135C211yesmaximal, characteristic, complemented
2#21456374C211yesmaximal, characteristic, complemented
3#21455616C211yesmaximal, characteristic, complemented
4#21445103C212yesmaximal, complemented
5#21444998C212yesmaximal, complemented
6#21444970C212yesmaximal, complemented
7#21444935C212yesmaximal, complemented
8#21444496C211yesmaximal, characteristic, complemented
9#21444270C211yesmaximal, characteristic, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3496C211yesmaximal, characteristic
12#16840319C2211yescharacteristic, complemented
13#16840158C2211yescharacteristic, complemented
14#16840134C2211yescharacteristic, complemented
15#16840114C2211yescharacteristic, complemented
16#16840108C2211yescharacteristic, complemented
17#16840062—22no—
18#16840052C2212yescomplemented
19#16840052C2212yescomplemented
20#16840044—21no—
21#16840042C2212yescomplemented
22#16839989—22no—
23#16839989—22no—
24#16839967C2212yescomplemented
25#16839959—21no—
26C4×#11758200—22no—
27#16839802—21no—
28#16839786C2211yescharacteristic, complemented
29C2×C4⋊#92418C2211yescharacteristic, complemented
30#16839767—21no—
31C2×C4×#92418C2211yescharacteristic, 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—
54C22×#92418—21no—
55C22×#92418—21no—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64#14519710—22no—
65#14519708—22no—
66#14519708—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519700—22no—
73#14519700—22no—
74#14519698—22no—
75#14519698—22no—
76#14519698—22no—
77#14519698—22no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81H(5,𝔽2)×C61—44no—
82H(5,𝔽2)×C61—44no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61—21no—
86C2×#41×C61—21no—
87C22×D4×C61—24no—
88C22×D4×C61—22no—
89C22×D4×C61—22no—
90C22×D4×C61—22no—
91C22×D4×C61D411yescharacteristic
92C23×C4×C61—21no—
93C4⋊Q8×C61D411yescharacteristic
94C4⋊Q8×C61C2311yescharacteristic
95#127×C61D411yescharacteristic
96#127×C61C2311yescharacteristic
97#124×C61—21no—
98#124×C61—21no—
99#123×C61—24no—
100#123×C61—24no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103#121×C61—24no—
104#121×C61—22no—
105#121×C61—24no—
106#121×C61—22no—
107#121×C61—21no—
108#121×C61—21no—
109#121×C61—22no—
110#121×C61—22no—
111#121×C61D412yes—
112#121×C61D412yes—
113#120×C61—24no—
114#120×C61—24no—
115C4×D4×C61—21no—
116C4×D4×C61—21no—
117C4×D4×C61—21no—
118C4×D4×C61—21no—
119C4×D4×C61D412yes—
120C4×D4×C61D412yes—
121C4×D4×C61C2312yes—
122C4×D4×C61C2312yes—
123C2×C4⋊C4×C61—22no—
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C22⋊C4×C61—24no—
126C2×C22⋊C4×C61—21no—
127C2×C42×C61C2311yescharacteristic
128C24×C61—21no—
129C24×C61—24no—
130#41×C61—44no—
131#41×C61—48no—
132#41×C61—44no—
133#41×C61—42no—
134#41×C61—42no—
135#41×C61—42no—
136#41×C61—42no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×D4×C61—24no—
140C2×D4×C61—24no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—24no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—22no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—44no—
151C2×D4×C61—48no—
152C2×D4×C61—416no—
153C2×D4×C61—44no—
154C2×D4×C61—82no—
155C2×D4×C61—82no—
156C2×D4×C61SD1614yescomplemented
157C2×D4×C61C2×D412yes—
158C2×D4×C61C2×D412yes—
159C2×D4×C61C2×D412yes—
160C22×C4×C61—24no—
161C22×C4×C61—21no—
162C22×C4×C61—22no—
163C22×C4×C61—22no—
164C22×C4×C61—22no—
165C22×C4×C61—22no—
166C22×C4×C61—81no—
167C22×C4×C61—81no—
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C2×Dic122—21no—
172C2×Dic122—21no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C2×D412yes—
178C4⋊C4×C61C2×D412yes—
179C4⋊C4×C61C2×D412yes—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193#11758200—44no—
194#11758200—82no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—41no—
202#11758194—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—42no—
208C2×#92418—81no—
209C23×C61—22no—
210C23×C61—24no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—48no—
214C23×C61—81no—
215C23×C61—81no—
216C23×C61—84no—
217C23×C61—84no—
218C23×C61#12011yescharacteristic
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—42no—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—24no—
226D4×C61—42no—
227D4×C61—48no—
228D4×C61—48no—
229D4×C61—44no—
230D4×C61—44no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—42no—
236D4×C61—81no—
237D4×C61—84no—
238D4×C61C2×SD1614yes—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—48no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—44no—
251C2×C4×C61—82no—
252C2×C4×C61—81no—
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61C2×SD1612yes—
257C2×C4×C61AGL(1,ℤ/8)12yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×Dic61—41no—
262C2×Dic61—41no—
263Dic122—42no—
264Dic122—41no—
265Dic122—41no—
266#92418—22no—
267#92418—41no—
268#92418—42no—
269#92418—42no—
270#18663—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—24no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—84no—
283C22×C61—168no—
284C22×C61#45012yes—
285C22×C61C2×#12011yescharacteristic
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61#45012yes—
293C4×C61C22×SD1611yescharacteristic
294C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
295C4×C61#57611yescharacteristic
296C4×C61#57711yescharacteristic
297Dic61—81no—
298Dic61—81no—
299#3496D6111yescharacteristic
300#3246—611no—
301#3167—611no—
302#3077—611no—
303#1654—612no—
304#1639—612no—
305#1635—612no—
306#1629—612no—
307#1547—611no—
308#1492—611no—
309C2×D4⋊C8—611no—
310C2×C61—22no—
311C2×C61—44no—
312C2×C61—82no—
313C2×C61—82no—
314C2×C61C2×#45011yescharacteristic
315C2×C61#301211yescharacteristic
316C2×C61#301811yescharacteristic
317C2×H(5,𝔽2)—21no—
318#547—21no—
319#547C2×D6112yes—
320#546—22no—
321#546—22no—
322D42—21no—
323D42C2×D6112yes—
324#537—21no—
325#534—22no—
326#533—611no—
327#532C2×D6111yescharacteristic
328#532C2×D6111yescharacteristic
329C2×#121—22no—
330C2×C4×D4C2×D6111yescharacteristic
331#498—611no—
332#492—611no—
333#487—611no—
334#486—611no—
335#478—1222no—
336#476—612no—
337#476—612no—
338#475—1221no—
339#474—612no—
340#465—1222no—
341#465—1222no—
342#461—612no—
343#460—1221no—
344C4×SD16—1222no—
345#428—1221no—
346#423—611no—
347C2×C4⋊C8—611no—
348#418—1221no—
349C2×C4×C8—611no—
350#332—612no—
351#331—612no—
352D4⋊C8—612no—
353D4⋊C8—612no—
354D4⋊C8—612no—
355D4⋊C8—612no—
356C61#1866311yescharacteristic, complemented, Sylow
357H(5,𝔽2)—44no—
358H(5,𝔽2)—44no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—21no—
362C2×#41—21no—
363C22×D4—24no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—22no—
367C22×D4C22⋊D6111yescharacteristic
368C23×C4—21no—
369C2×SD16—1222no—
370C2×SD16—1222no—
371C2×M16—1221no—
372C22×C8—1221no—
373C22×C8—1221no—
374C4⋊Q8—611no—
375C4⋊Q8—611no—
376C4⋊Q8C22⋊D6111yescharacteristic
377C4⋊Q8C22×D6111yescharacteristic
378#127C22⋊D6111yescharacteristic
379#127C22×D6111yescharacteristic
380#125—1221no—
381#124—21no—
382#124—21no—
383#123—24no—
384#123—24no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389#121—24no—
390#121—24no—
391#121—22no—
392#121—22no—
393#121—21no—
394#121—21no—
395#121—22no—
396#121—22no—
397#121C22⋊D6112yes—
398#121C22⋊D6112yes—
399#120—24no—
400#120—24no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4C22⋊D6112yes—
408C4×D4C22⋊D6112yes—
409C4×D4C22×D6112yes—
410C4×D4C22×D6112yes—
411#117—1221no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4C22⋊D6111yescharacteristic
414C2×C22⋊C4—24no—
415C2×C22⋊C4—21no—
416C2×C42C22×D6111yescharacteristic
417#107—1222no—
418#106—1222no—
419#106—1222no—
420#106—1222no—
421C4⋊C8—612no—
422C4⋊C8—612no—
423C4⋊C8—1221no—
424C4⋊C8—1221no—
425#103—1222no—
426#103—1222no—
427#102—1222no—
428#102—1222no—
429#102—1222no—
430#102—1222no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—1221no—
434C24—21no—
435C24—24no—
436#41—44no—
437#41—48no—
438#41—44no—
439#41—42no—
440#41—42no—
441#41—42no—
442#41—42no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—44no—
459C2×D4—48no—
460C2×D4—416no—
461C2×D4—44no—
462C2×D4—82no—
463C2×D4—82no—
464C2×D4#1175820014yescomplemented
465C2×D4C2×C22⋊D6112yes—
466C2×D4C2×C22⋊D6112yes—
467C2×D4C2×C22⋊D6112yes—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479SD16—4882no—
480SD16—2444no—
481M16—2442no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—4881no—
487C2×C8—2442no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—2441no—
497C4⋊C4—2441no—
498C4⋊C4C2×C22⋊D6112yes—
499C4⋊C4C2×C22⋊D6112yes—
500C4⋊C4C2×C22⋊D6112yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—2441no—
510C42—2441no—
511C42C2×C22⋊D6111yescharacteristic
512C42C2×C22⋊D6111yescharacteristic
513C42C2×C22⋊D6111yescharacteristic
514C42C23×D6111yescharacteristic
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—48no—
520C23—81no—
521C23—81no—
522C23—84no—
523C23—84no—
524C23#1451974611yescharacteristic
525Q8—41no—
526Q8—41no—
527Q8—42no—
528Q8—2442no—
529Q8—2441no—
530Q8—2441no—
531D4—22no—
532D4—22no—
533D4—22no—
534D4—24no—
535D4—42no—
536D4—48no—
537D4—48no—
538D4—44no—
539D4—44no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—42no—
545D4—81no—
546D4—84no—
547D4C2×#1175820014yes—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4#1451974611yescharacteristic
565C2×C4#1451974611yescharacteristic
566C2×C4#1451974611yescharacteristic
567C2×C4C2×#1175820012yes—
568C2×C4#1451980312yes—
569C2×C4C22×C22⋊D6111yescharacteristic
570C2×C4C22×C22⋊D6111yescharacteristic
571C2×C4C22×C22⋊D6111yescharacteristic
572C8—1222no—
573C8—2441no—
574C8—2442no—
575C8—2442no—
576C22—21no—
577C22—21no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—84no—
588C22—168no—
589C22#1683991312yes—
590C22C2×#1451974611yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—4881no—
598C4—4881no—
599C4#1683991312yes—
600C4C22×#1175820011yescharacteristic
601C4C2×#1451980311yescharacteristic
602C4#1684056511yescharacteristic
603C4#1684057311yescharacteristic
604C2—22no—
605C2—44no—
606C2—82no—
607C2—82no—
608C2C2×#1683991311yescharacteristic
609C2#2145512311yescharacteristic
610C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18663
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. #21457135C2semidirect
2. #21456374C2semidirect
3. #21455616C2semidirect
4. #21445103C2semidirect
5. #21444998C2semidirect
6. #21444970C2semidirect
7. #21444935C2semidirect
8. #21444496C2semidirect
9. #21444270C2semidirect
10. C2×D4⋊#92418C2semidirect
12. #16840319C22semidirect
13. #16840158C22semidirect
14. #16840134C22semidirect
15. #16840114C22semidirect
16. #16840108C22semidirect
18. #16840052C22semidirect
19. #16840052C22semidirect
21. #16840042C22semidirect
24. #16839967C22semidirect
28. #16839786C22semidirect
29. C2×C4⋊#92418C22semidirect
31. C2×C4×#92418C22semidirect
32. #16839508C22semidirect
33. #16839506C22semidirect
34. D4⋊#92418C22semidirect
35. D4⋊#92418C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
156. C2×D4×C61SD16semidirect
356. C61#18663semidirect
464. C2×D4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214571352
#214563742
#214556162
#214451032
#214449982
#214449702
#214449352
#214444962
#214442702
C2×D4⋊#924182
C61×#34962
#1866361