Finite Group Databasebeta

← #30626560

Subgroups — #30626560

Order 15616.

1920 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462751C211yesmaximal, characteristic, complemented
2#21462744C213yesmaximal, complemented
3#21462737C211yesmaximal, characteristic, complemented
4#21462734C211yesmaximal, characteristic, complemented
5#21462709C213yesmaximal, complemented
6#21462708C213yesmaximal, complemented
7#21462699C213yesmaximal, complemented
8#21462665C211yesmaximal, characteristic, complemented
9#21462663C211yesmaximal, characteristic, complemented
10#21462629C213yesmaximal, complemented
11#41×#92420C211yesmaximal, characteristic, complemented
12#21462562C213yesmaximal, complemented
13#21462561C213yesmaximal, complemented
14#21462542C213yesmaximal, complemented
15#21459442C211yesmaximal, characteristic
16C2×#14519864C2216yes—
17#16840814C2211yescharacteristic
18#16840814C2213yes—
19Q8×#92420C2211yescharacteristic
20Q8×#92420C2211yescharacteristic
21Q8×#92420C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23#16840797C2213yes—
24#16840797C2213yes—
25#16840797C2213yes—
26#16840797C2213yes—
27#16840796C2213yes—
28#16840796C2213yes—
29#16840795C2213yes—
30#16840795C2213yes—
31#16840794C2216yes—
32#16840793C2213yes—
33#16840793C2216yes—
34#16840793C2213yes—
35#16840791C2213yes—
36#16840791C2213yes—
37#16840790C2213yes—
38#16840790C2213yes—
39#16840789C2216yes—
40D4×#92420C2213yes—
41D4×#92420C2213yes—
42D4×#92420C2213yes—
43D4×#92420C2213yes—
44#16840784C2213yes—
45#16840783C2213yes—
46#16840780C2213yes—
47#16840779C2213yes—
48#16840778C2213yes—
49#16840778C2213yes—
50#16840765C2213yes—
51D61⋊(C4⋊C8)C2213yes—
52C2×C4⋊#92420C2213yes—
53#16840736C2216yes—
54#16840735C2216yes—
55#16840735C2216yes—
56C4×#11758213C2213yes—
57C4×#11758217C2213yes—
58C2×C4×#92420C2213yes—
59Q8×Dic122C411yescharacteristic
60#16840477C413yes—
61#16840476C413yes—
62#16840476C413yes—
63#16840432C413yes—
64#16840431C411yescharacteristic
65#16840431C411yescharacteristic
66#16840310C2213yes—
67#16840307C2213yes—
68#16840307C2213yes—
69#41×Dic61C2211yescharacteristic
70#41×Dic61C2211yescharacteristic
71C4×#11758226C2211yescharacteristic
72C4×#11758224C2213yes—
73C61×#550C411yescharacteristic
74#14519864—412no—
75C2×#11758213C2313yes—
76C2×#11758213C2313yes—
77C2×#11758217C2313yes—
78C2×#11758217C2313yes—
79C2×#11758217C2316yes—
80C2×D61⋊C8C2316yes—
81C22×#92420C2313yes—
82C22×#92420C2313yes—
83C22×#92420C2316yes—
84#14519844C2313yes—
85#14519844C2313yes—
86#14519844C2311yescharacteristic
87#14519844C2311yescharacteristic
88C4⋊#92420C2313yes—
89C4⋊#92420C2313yes—
90C4⋊#92420C2313yes—
91C4⋊#92420C2313yes—
92C4⋊#92420C2311yescharacteristic
93C4⋊#92420C2313yes—
94#14519834C2313yes—
95#14519834C2313yes—
96#14519834C2316yes—
97C22⋊#92420C2313yes—
98C22⋊#92420C2313yes—
99C22⋊#92420C2316yes—
100C22⋊#92420C2316yes—
101C22⋊#92420C2313yes—
102C22⋊#92420C2313yes—
103#14519832C2311yescharacteristic
104#14519832C2313yes—
105#14519832C2311yescharacteristic
106#14519832C2313yes—
107C4×#92420C2311yescharacteristic
108C4×#92420C2313yes—
109C4×#92420C2313yes—
110C4×#92420C2313yes—
111C4×#92420C2313yes—
112C4×#92420C2313yes—
113C2×#11758226C2×C411yescharacteristic
114C2×#11758224C2×C413yes—
115C2×#11758224C2316yes—
116#14519777C2×C413yes—
117#14519776C2×C413yes—
118#14519776C2×C413yes—
119#14519773C2×C413yes—
120#14519773C2×C413yes—
121#14519764C2×C416yes—
122#14519763C2×C416yes—
123#14519763C2×C416yes—
124#14519751C2×C413yes—
125#14519751C2×C413yes—
126#14519749C2×C416yes—
127#14519749C2×C416yes—
128#14519748C2×C413yes—
129#14519748C2×C413yes—
130#14519748C2×C416yes—
131Q8×Dic61C2×C411yescharacteristic
132Q8×Dic61C2×C411yescharacteristic
133Q8×Dic61C2311yescharacteristic
134C4×Dic122C2×C413yes—
135#14519742C2311yescharacteristic
136C4×D244C2×C413yes—
137D4×Dic61C2×C413yes—
138D4×Dic61C2×C413yes—
139D4×Dic61C2313yes—
140D4×Dic61C2313yes—
141D4×Dic61C2313yes—
142#14519739C2313yes—
143C4×C22⋊D61C2×C413yes—
144C4×C22⋊D61C2×C413yes—
145#14519737C2313yes—
146#14519737C2313yes—
147#14519736C2×C413yes—
148#14519736C2×C413yes—
149#14519735C2×C413yes—
150#14519735C2×C413yes—
151#14519733C2×C413yes—
152#14519733C2×C413yes—
153C2×C4×Dic61C2313yes—
154C2×C4×Dic61C2313yes—
155C42×D61C2313yes—
156C2×#41×C61C2×C411yescharacteristic
157C2×#41×C61C2×C411yescharacteristic
158#127×C61C2×C413yes—
159#121×C61C2×C413yes—
160#121×C61C2×C413yes—
161C4×Q8×C61C2×C411yescharacteristic
162C4×D4×C61C2×C413yes—
163#41×C61—24no—
164#41×C61—24no—
165C2×Q8×C61C22×C411yescharacteristic
166C2×D4×C61C22×C413yes—
167C2×D4×C61C22×C413yes—
168C2×D4×C61C22×C413yes—
169C2×D4×C61C22×C416yes—
170C22×C4×C61C22×C413yes—
171C22×C4×C61C22×C413yes—
172C2×C22⋊D61C22×C413yes—
173C2×C22⋊D61C22×C413yes—
174C2×C22⋊D61C22×C416yes—
175C2×C22⋊D61C22×C416yes—
176C22×Dic61C22×C413yes—
177C22×Dic61C22×C413yes—
178C22×Dic61C2413yes—
179C22×Dic61C2413yes—
180#11758226—24no—
181#11758224—46no—
182#11758224—46no—
183#11758224—46no—
184C2×D244C22×C413yes—
185C2×C4×D61C22×C413yes—
186C2×C4×D61C22×C413yes—
187C2×C4×D61C2413yes—
188C2×Dic122C22×C411yescharacteristic
189C2×Dic122C22×C416yes—
190#11758217—26no—
191#11758217—43no—
192#11758217—46no—
193C2×#92420—26no—
194C2×#92420—43no—
195C2×#92420—46no—
196C2×#92420C2411yescharacteristic
197C2×#92420C2411yescharacteristic
198C2×#92420C2411yescharacteristic
199C2×#92420C2413yes—
200C2×#92420C2413yes—
201C2×#92420C2413yes—
202C2×#92420C2413yes—
203C2×#92420C2411yescharacteristic
204#11758213—26no—
205#11758213—43no—
206D61⋊C8—46no—
207C4⋊C4×C61C22×C411yescharacteristic
208C4⋊C4×C61C22×C413yes—
209C22⋊C4×C61C22×C413yes—
210C22⋊C4×C61C22×C413yes—
211C42×C61C22×C413yes—
212C22⋊Dic61C22×C413yes—
213C22⋊Dic61C22×C413yes—
214C22⋊Dic61C22×C416yes—
215C22⋊Dic61C22×C416yes—
216#11758198C22×C413yes—
217#11758198C22×C413yes—
218#11758198C22×C416yes—
219C4⋊Dic61C22×C413yes—
220C4⋊Dic61C22×C413yes—
221C4⋊Dic61C22×C413yes—
222C4⋊Dic61C22×C413yes—
223#11758196C22×C411yescharacteristic
224#11758196C22×C411yescharacteristic
225#11758196C22×C413yes—
226#11758196C22×C413yes—
227C4×Dic61C22×C413yes—
228C4×Dic61C22×C413yes—
229C4×Dic61C22×C413yes—
230C4×Dic61C22×C413yes—
231C4×Dic61C22×C413yes—
232C4×Dic61C2413yes—
233C4×Dic61C2411yescharacteristic
234C4×Dic61C2411yescharacteristic
235C4×Dic61C2411yescharacteristic
236C23×C61C23×C413yes—
237C23×C61C23×C413yes—
238C22×D61C23×C413yes—
239Q8×C61#13114yescomplemented
240D4×C61—26no—
241D4×C61—26no—
242D4×C61—46no—
243D4×C61—43no—
244C2×C4×C61—26no—
245C2×C4×C61—26no—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C23×C413yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C23×C413yes—
251C22⋊D61—26no—
252C22⋊D61—46no—
253C22⋊D61—46no—
254C22⋊D61—43no—
255C2×Dic61—26no—
256C2×Dic61—23no—
257C2×Dic61—23no—
258C2×Dic61—23no—
259C2×Dic61—23no—
260C2×Dic61—43no—
261C2×Dic61C23×C411yescharacteristic
262C2×Dic61C23×C411yescharacteristic
263C2×Dic61C23×C413yes—
264C2×Dic61C23×C413yes—
265C2×Dic61C23×C413yes—
266C2×Dic61C23×C411yescharacteristic
267C2×Dic61C2511yescharacteristic
268D244—26no—
269C4×D61—26no—
270C4×D61—23no—
271C4×D61—23no—
272C4×D61—43no—
273Dic122—22no—
274Dic122—46no—
275#92420—23no—
276#92420—23no—
277#92420—23no—
278#92420—21no—
279#92420—23no—
280#92420—21no—
281#92420—21no—
282#92420H(5,𝔽2)12yes—
283#60328—611nomaximal, Sylow
284C22×C61—23no—
285C22×C61—23no—
286C22×C61—23no—
287C22×C61—23no—
288C22×C61—23no—
289C22×C61—23no—
290C22×C61C24×C411yescharacteristic
291C2×D61—23no—
292C2×D61—23no—
293C2×D61—23no—
294C4×C61—23no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61C2×#13116yes—
299Dic61—23no—
300Dic61—23no—
301Dic61—21no—
302Dic61—21no—
303Dic61—23no—
304Dic61#56812yes—
305Dic61C2×H(5,𝔽2)11yescharacteristic
306Dic61#58511yescharacteristic
307#3483—611no—
308#2991—611no—
309#2990—613no—
310#2988—611no—
311#2988—611no—
312#2981—613no—
313#2981—613no—
314#2979—613no—
315#2971—611no—
316#2971—611no—
317#2965—613no—
318C8×#41—611no—
319#2935—613no—
320#2935—613no—
321#2931—613no—
322C2×C61—43no—
323C2×C61—43no—
324C2×C61C4×H(5,𝔽2)11yescharacteristic
325C2×C61C22×#13111yescharacteristic
326C2×C61C2×#56811yescharacteristic
327D61—43no—
328C2×#131—616no—
329Q82—611no—
330#555—613no—
331#555—613no—
332#555—613no—
333#550—611no—
334#550—611no—
335#550—613no—
336#550#692311yescharacteristic
337#532—613no—
338#532—613no—
339#532—613no—
340C4×#41—611no—
341C4×#41—611no—
342C4×#41—611no—
343C4×#41—613no—
344#446—611no—
345#446—613no—
346C8×Q8—611no—
347C8×Q8—611no—
348C8×Q8—611no—
349C8×Q8—611no—
350#436—613no—
351#436—613no—
352#436—613no—
353#436—613no—
354#436—613no—
355#436—613no—
356#436—613no—
357#436—613no—
358#435—613no—
359#435—613no—
360#435—613no—
361#435—616no—
362#435—613no—
363#435—616no—
364C8×D4—613no—
365C8×D4—613no—
366C8×D4—613no—
367C8×D4—613no—
368C8×D4—616no—
369C8×D4—613no—
370C8×D4—613no—
371#432—613no—
372#432—613no—
373#431—613no—
374#431—613no—
375#431—613no—
376#431—613no—
377#423—613no—
378C2×C4⋊C8—613no—
379C2×C4⋊C8—613no—
380#408—616no—
381#408—616no—
382#408—616no—
383C4×M16—613no—
384C4×M16—613no—
385C2×C4×C8—613no—
386C61#6032811yescharacteristic, complemented, Sylow
387C2×#41—611no—
388C2×#41—613no—
389C2×#41—616no—
390C2×#41C2×#692311yescharacteristic
391C2×#41C2×#692311yescharacteristic
392#131—24412no—
393C2×M16—613no—
394C2×M16—613no—
395C2×M16—616no—
396C2×M16—613no—
397C2×M16—613no—
398C22×C8—613no—
399C22×C8—616no—
400C22×C8—613no—
401C22×C8—616no—
402C4⋊Q8—613no—
403C4⋊Q8—613no—
404C4⋊Q8—613no—
405#127—613no—
406#127—613no—
407#127C2×#692313yes—
408#124—616no—
409#124—616no—
410#124—616no—
411#121—613no—
412#121—613no—
413#121—616no—
414#121—616no—
415#121—616no—
416#121—613no—
417#121—613no—
418#121C2×#692313yes—
419#121C2×#692313yes—
420C4×Q8—611no—
421C4×Q8—611no—
422C4×Q8—611no—
423C4×Q8—611no—
424C4×Q8—613no—
425C4×Q8C2×#692311yescharacteristic
426C4×D4—613no—
427C4×D4—613no—
428C4×D4—613no—
429C4×D4—613no—
430C4×D4—613no—
431C4×D4—613no—
432C4×D4—613no—
433C4×D4—613no—
434C4×D4—613no—
435C4×D4—613no—
436C4×D4—613no—
437C4×D4C2×#692313yes—
438#117—613no—
439#117—613no—
440#117—613no—
441#117—613no—
442#117—613no—
443#117—613no—
444C2×C42—613no—
445C2×C42—613no—
446C2×C42—613no—
447C4⋊C8—613no—
448C4⋊C8—613no—
449C4⋊C8—613no—
450C4⋊C8—613no—
451C4⋊C8—611no—
452C4⋊C8—613no—
453C4⋊C8—613no—
454C4⋊C8—613no—
455C4⋊C8—611no—
456C4⋊C8—611no—
457C22⋊C8—616no—
458C22⋊C8—613no—
459C22⋊C8—613no—
460C22⋊C8—613no—
461C22⋊C8—613no—
462C22⋊C8—616no—
463C22⋊C8—613no—
464C22⋊C8—613no—
465C22⋊C8—616no—
466#97—611no—
467#97—611no—
468#97—613no—
469#97—613no—
470C4×C8—611no—
471C4×C8—613no—
472C4×C8—613no—
473C4×C8—613no—
474C4×C8—613no—
475C4×C8—613no—
476#41—24no—
477#41—24no—
478#41—1224no—
479#41—2446no—
480#41—2446no—
481#41—2446no—
482C2×Q8—611no—
483C2×Q8—616no—
484C2×Q8C22×#692311yescharacteristic
485C2×D4—613no—
486C2×D4—613no—
487C2×D4—616no—
488C2×D4—616no—
489C2×D4—613no—
490C2×D4C22×#692313yes—
491C2×D4C22×#692313yes—
492C2×D4C22×#692313yes—
493C2×D4C22×#692316yes—
494C22×C4—613no—
495C22×C4—613no—
496C22×C4—613no—
497C22×C4—613no—
498C22×C4—613no—
499C22×C4—613no—
500C22×C4—613no—
501C22×C4C22×#692313yes—
502C22×C4C22×#692313yes—
503M16—1226no—
504M16—1226no—
505M16—2446no—
506M16—2443no—
507M16—2443no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—613no—
513C2×C8—613no—
514C2×C8—613no—
515C2×C8—613no—
516C2×C8—1226no—
517C2×C8—2446no—
518C2×C8—2446no—
519C2×C8—2443no—
520C4⋊C4—611no—
521C4⋊C4—611no—
522C4⋊C4—613no—
523C4⋊C4—613no—
524C4⋊C4—613no—
525C4⋊C4—613no—
526C4⋊C4—613no—
527C4⋊C4—613no—
528C4⋊C4C22×#692311yescharacteristic
529C4⋊C4C22×#692313yes—
530C22⋊C4—616no—
531C22⋊C4—616no—
532C22⋊C4—613no—
533C22⋊C4—613no—
534C22⋊C4—616no—
535C22⋊C4—613no—
536C22⋊C4—613no—
537C22⋊C4C22×#692313yes—
538C22⋊C4C22×#692313yes—
539C42—611no—
540C42—613no—
541C42—611no—
542C42—611no—
543C42—613no—
544C42—613no—
545C42—613no—
546C42—613no—
547C42—613no—
548C42C22×#692313yes—
549C23—613no—
550C23C23×#692313yes—
551C23C23×#692313yes—
552Q8—1222no—
553Q8—2446no—
554Q8#1451986414yescomplemented
555D4—26no—
556D4—26no—
557D4—46no—
558D4—43no—
559D4—1226no—
560D4—1226no—
561D4—2446no—
562D4—2446no—
563D4—2443no—
564C2×C4—26no—
565C2×C4—26no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—613no—
570C2×C4—611no—
571C2×C4—613no—
572C2×C4—613no—
573C2×C4—1223no—
574C2×C4—1223no—
575C2×C4—1223no—
576C2×C4—1223no—
577C2×C4—1223no—
578C2×C4—1223no—
579C2×C4—1226no—
580C2×C4—1226no—
581C2×C4—2443no—
582C2×C4—2443no—
583C2×C4C23×#692311yescharacteristic
584C2×C4C23×#692311yescharacteristic
585C2×C4C23×#692311yescharacteristic
586C2×C4C23×#692313yes—
587C2×C4C23×#692313yes—
588C8—612no—
589C8—1223no—
590C8—1221no—
591C8—1223no—
592C8—1223no—
593C8—1223no—
594C8—1221no—
595C8—1221no—
596C22—23no—
597C22—23no—
598C22—23no—
599C22—23no—
600C22—23no—
601C22—23no—
602C22—1223no—
603C22—1223no—
604C22—1223no—
605C22C24×#692311yescharacteristic
606C4—23no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—611no—
611C4—611no—
612C4—612no—
613C4—1223no—
614C4—1223no—
615C4—1221no—
616C4—1223no—
617C4—1221no—
618C4C2×#1451986416yes—
619C2—43no—
620C2—43no—
621C2—2443no—
622C2H(5,𝔽2)×#692311yescharacteristic
623C2C22×#1451986411yescharacteristic
624C2D61⋊#56811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#550C4
prime radical#21459442C2
cosocleC2×Dic61C25
hypercenter#550#6923
hypocenterC61#60328
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #21462751C2semidirect
2. #21462744C2semidirect
3. #21462737C2semidirect
4. #21462734C2semidirect
5. #21462709C2semidirect
6. #21462708C2semidirect
7. #21462699C2semidirect
8. #21462665C2semidirect
9. #21462663C2semidirect
10. #21462629C2semidirect
11. #41×#92420C2semidirect
12. #21462562C2semidirect
13. #21462561C2semidirect
14. #21462542C2semidirect
239. Q8×C61#131semidirect
386. C61#60328semidirect
554. Q8#14519864semidirect

Maximal subgroups

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

subgroupindex
#214627512
#214627442
#214627372
#214627342
#214627092
#214627082
#214626992
#214626652
#214626632
#214626292
#41×#924202
#214625622
#214625612
#214625422
#214594422
#6032861