Finite Group Databasebeta

← #30626688

Subgroups — #30626688

Order 15616.

1744 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462762C213yesmaximal, complemented
2#21462761C213yesmaximal, complemented
3#21462751C211yesmaximal, characteristic, complemented
4#21462750C213yesmaximal, complemented
5#21462728C213yesmaximal, complemented
6#21462713C211yesmaximal, characteristic, complemented
7#21462712C211yesmaximal, characteristic, complemented
8#21462651C213yesmaximal, complemented
9#21462650C213yesmaximal, complemented
10#21462629C213yesmaximal, complemented
11#41×#92420C211yesmaximal, characteristic, complemented
12Q8×#11758213C211yesmaximal, characteristic, complemented
13Q8×#11758217C211yesmaximal, characteristic, complemented
14#21462607C213yesmaximal, complemented
15#21459435C211yesmaximal, characteristic
16#16840814C2213yes—
17#16840814C2213yes—
18#16840813C2213yes—
19#16840813C2213yes—
20#16840812C2213yes—
21#16840812C2213yes—
22#16840811C2213yes—
23#16840811C2216yes—
24#16840811C2213yes—
25Q8×#92420C2213yes—
26Q8×#92420C2211yescharacteristic
27Q8×#92420C2211yescharacteristic
28Q8×#92420C2211yescharacteristic
29Q8×#92420C2211yescharacteristic
30Q8×#92420C2213yes—
31#16840797C2216yes—
32#16840797C2213yes—
33#16840797C2213yes—
34#16840795C2211yescharacteristic
35#16840795C2213yes—
36#16840795C2216yes—
37#16840793C2213yes—
38#16840793C2213yes—
39#16840791C2213yes—
40#16840791C2213yes—
41D4×#92420C2213yes—
42D4×#92420C2213yes—
43#16840786C2216yes—
44#16840786C2216yes—
45#16840781C2213yes—
46#16840780C2213yes—
47#16840779C2213yes—
48#16840778C2213yes—
49#16840778C2213yes—
50#16840778C2213yes—
51#16840768C2216yes—
52#16840766C2213yes—
53#16840765C2213yes—
54#16840765C2213yes—
55#16840726C2216yes—
56C4×#11758213C2213yes—
57C4×#11758217C2213yes—
58C2×C4×#92420C2213yes—
59#16840488C411yescharacteristic
60#16840481C413yes—
61#16840452C413yes—
62#16840446C413yes—
63Q8×C22⋊D61C411yescharacteristic
64#16840419C411yescharacteristic
65#16840412C413yes—
66#16840314C2213yes—
67#16840293C2213yes—
68#16840291C2213yes—
69#41×Dic61C2211yescharacteristic
70C4×#11758224C2213yes—
71C2×Q8×Dic61C2211yescharacteristic
72C4×Q8×D61C2211yescharacteristic
73C61×#548C411yescharacteristic
74C2×#11758213C2311yescharacteristic
75C2×#11758213C2313yes—
76C2×#11758217C2311yescharacteristic
77C2×#11758217C2313yes—
78C2×#11758217C2316yes—
79C22×#92420C2313yes—
80C22×#92420C2313yes—
81#14519844C2313yes—
82#14519844C2316yes—
83#14519844C2313yes—
84#14519844C2311yescharacteristic
85#14519844C2313yes—
86C4⋊#92420C2313yes—
87C4⋊#92420C2313yes—
88C4⋊#92420C2313yes—
89C4⋊#92420C2316yes—
90C4⋊#92420C2313yes—
91C4⋊#92420C2313yes—
92C4⋊#92420C2313yes—
93#14519834C2311yescharacteristic
94#14519834C2313yes—
95C22⋊#92420C2311yescharacteristic
96C22⋊#92420C2313yes—
97C22⋊#92420C2316yes—
98C22⋊#92420C2313yes—
99C22⋊#92420C2313yes—
100#14519832C2313yes—
101#14519832C2313yes—
102#14519832C2313yes—
103#14519832C2313yes—
104C4×#92420C2311yescharacteristic
105C4×#92420C2313yes—
106C4×#92420C2313yes—
107C4×#92420C2313yes—
108C4×#92420C2313yes—
109C4×#92420C2313yes—
110C4×#92420C2316yes—
111C4×#92420C2316yes—
112C2×#11758224C2×C413yes—
113C2×Q8×D61C2×C411yescharacteristic
114#14519776C2×C413yes—
115#14519775C2×C413yes—
116#14519770C2×C416yes—
117#14519770C2×C416yes—
118#14519769C2×C413yes—
119#14519768C2×C413yes—
120#14519767C2×C416yes—
121#14519764C2×C413yes—
122#14519759C2×C416yes—
123#14519758C2×C416yes—
124#14519756C2×C413yes—
125#14519756C2×C413yes—
126#14519754C2×C416yes—
127#14519753C2×C413yes—
128#14519748C2×C413yes—
129Q8×Dic61C2×C413yes—
130Q8×Dic61C2×C411yescharacteristic
131Q8×Dic61C2×C411yescharacteristic
132Q8×Dic61C2311yescharacteristic
133Q8×Dic61C2313yes—
134C4×Dic122C2×C413yes—
135C4×Dic122C2×C413yes—
136#14519742C2313yes—
137#14519742C2313yes—
138D4×Dic61C2×C413yes—
139D4×Dic61C2313yes—
140D4×Dic61C2313yes—
141C4×C22⋊D61C2×C413yes—
142C4×C22⋊D61C2×C413yes—
143#14519737C2311yescharacteristic
144#14519737C2313yes—
145#14519736C2×C413yes—
146#14519735C2×C413yes—
147#14519733C2×C413yes—
148#14519733C2×C413yes—
149#14519733C2316yes—
150C2×C4⋊Dic61C2×C413yes—
151C4⋊C4×D61C2×C413yes—
152C2×C4×Dic61C2313yes—
153C2×C4×Dic61C2313yes—
154C42×D61C2313yes—
155C2×#41×C61C2×C411yescharacteristic
156C22×Q8×C61C2×C411yescharacteristic
157#124×C61C2×C413yes—
158C22⋊Q8×C61C2×C413yes—
159#121×C61C2×C413yes—
160C4×Q8×C61C2×C411yescharacteristic
161C4×D4×C61C2×C413yes—
162#41×C61—24no—
163C2×Q8×C61—24no—
164C2×Q8×C61C22×C413yes—
165C2×Q8×C61C22×C411yescharacteristic
166C2×D4×C61C22×C413yes—
167C2×D4×C61C22×C413yes—
168C22×C4×C61C22×C413yes—
169C22×C4×C61C22×C413yes—
170C2×C22⋊D61C22×C411yescharacteristic
171C2×C22⋊D61C22×C413yes—
172C22×Dic61C22×C413yes—
173C22×Dic61C22×C413yes—
174C22×Dic61C2413yes—
175C22×Dic61C2411yescharacteristic
176Q8×D61—24no—
177#11758224—46no—
178C2×C4×D61C22×C413yes—
179C2×C4×D61C22×C413yes—
180C2×C4×D61C2411yescharacteristic
181C2×Dic122C22×C413yes—
182C2×Dic122C22×C413yes—
183#11758217—22no—
184#11758217—43no—
185#11758217—46no—
186C2×#92420—26no—
187C2×#92420—43no—
188C2×#92420C2413yes—
189C2×#92420C2411yescharacteristic
190C2×#92420C2411yescharacteristic
191C2×#92420C2411yescharacteristic
192C2×#92420C2413yes—
193C2×#92420C2413yes—
194C2×#92420C2413yes—
195C2×#92420C2411yescharacteristic
196#11758213—22no—
197#11758213—43no—
198C4⋊C4×C61C22×C413yes—
199C4⋊C4×C61C22×C413yes—
200C22⋊C4×C61C22×C411yescharacteristic
201C22⋊C4×C61C22×C413yes—
202C22⋊C4×C61C22×C416yes—
203C42×C61C22×C413yes—
204C22⋊Dic61C22×C413yes—
205C22⋊Dic61C22×C413yes—
206C22⋊Dic61C22×C416yes—
207C22⋊Dic61C22×C416yes—
208#11758198C22×C411yescharacteristic
209#11758198C22×C413yes—
210C4⋊Dic61C22×C413yes—
211C4⋊Dic61C22×C413yes—
212C4⋊Dic61C22×C416yes—
213C4⋊Dic61C22×C413yes—
214C4⋊Dic61C22×C413yes—
215#11758196C22×C416yes—
216#11758196C22×C411yescharacteristic
217#11758196C22×C413yes—
218#11758196C22×C413yes—
219#11758196C22×C413yes—
220#11758196C22×C416yes—
221#11758196C22×C416yes—
222C4×Dic61C22×C413yes—
223C4×Dic61C22×C413yes—
224C4×Dic61C22×C413yes—
225C4×Dic61C22×C413yes—
226C4×Dic61C22×C413yes—
227C4×Dic61C2413yes—
228C4×Dic61C2411yescharacteristic
229C4×Dic61C2413yes—
230C4×Dic61C2413yes—
231C23×C61C23×C413yes—
232C23×C61C23×C411yescharacteristic
233C22×D61C23×C411yescharacteristic
234Q8×C61—26no—
235Q8×C61#13114yes—
236D4×C61—26no—
237D4×C61—43no—
238C2×C4×C61—26no—
239C2×C4×C61—26no—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C23×C413yes—
243C2×C4×C61C23×C413yes—
244C2×C4×C61C23×C413yes—
245C22⋊D61—22no—
246C22⋊D61—43no—
247C2×Dic61—26no—
248C2×Dic61—23no—
249C2×Dic61—21no—
250C2×Dic61—23no—
251C2×Dic61—21no—
252C2×Dic61—43no—
253C2×Dic61C23×C411yescharacteristic
254C2×Dic61C23×C413yes—
255C2×Dic61C23×C413yes—
256C2×Dic61C23×C413yes—
257C2×Dic61C23×C413yes—
258C2×Dic61C23×C411yescharacteristic
259C2×Dic61C2511yescharacteristic
260C4×D61—26no—
261C4×D61—21no—
262C4×D61—21no—
263C4×D61—43no—
264Dic122—26no—
265Dic122—26no—
266#92420—21no—
267#92420—23no—
268#92420—21no—
269#92420—23no—
270#92420—23no—
271#92420—21no—
272#92420—23no—
273#924202−1+412yes—
274#60345—611nomaximal, Sylow
275C22×C61—23no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—23no—
279C22×C61—23no—
280C22×C61—21no—
281C22×C61C24×C411yescharacteristic
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—23no—
288C4×C61—23no—
289C4×C61C2×#13116yes—
290Dic61—23no—
291Dic61—21no—
292Dic61—23no—
293Dic61—23no—
294Dic61—23no—
295Dic61#56812yes—
296Dic61C2×2−1+411yescharacteristic
297Dic61#58511yescharacteristic
298#3482—611no—
299#2993—613no—
300#2993—613no—
301#2991—613no—
302#2991—611no—
303#2986—613no—
304#2982—611no—
305#2982—611no—
306#2969—613no—
307#2969—613no—
308#2965—613no—
309C8×#41—611no—
310Q8×M16—611no—
311Q8×M16—611no—
312#2960—613no—
313C2×C61—41no—
314C2×C61—43no—
315C2×C61C4×2−1+411yescharacteristic
316C2×C61C22×#13111yescharacteristic
317C2×C61C2×#56811yescharacteristic
318D61—41no—
319#557—611no—
320#556—613no—
321#552—613no—
322#552—613no—
323D4×Q8—611no—
324#548—611no—
325#548#692311yescharacteristic
326#547—613no—
327#533—613no—
328#529—613no—
329#529—613no—
330C4×#41—611no—
331C4×#41—613no—
332C2×C4×Q8—611no—
333C2×C4×Q8—611no—
334#446—613no—
335#446—613no—
336#446—613no—
337#446—613no—
338#446—613no—
339#446—613no—
340C8×Q8—613no—
341C8×Q8—611no—
342C8×Q8—611no—
343C8×Q8—611no—
344C8×Q8—611no—
345C8×Q8—613no—
346C8×Q8—613no—
347C8×Q8—616no—
348C8×Q8—613no—
349#436—611no—
350#436—613no—
351#436—616no—
352#436—613no—
353#436—613no—
354#436—616no—
355#435—613no—
356#435—613no—
357#435—613no—
358#435—613no—
359C8×D4—613no—
360C8×D4—613no—
361#433—616no—
362#433—616no—
363#431—613no—
364#431—613no—
365#431—613no—
366#431—613no—
367#431—613no—
368#431—613no—
369#424—616no—
370#423—613no—
371#423—613no—
372#423—613no—
373#405—616no—
374C4×M16—613no—
375C4×M16—613no—
376C2×C4×C8—613no—
377C61#6034511yescharacteristic, complemented, Sylow
378C2×#41—613no—
379C2×#41C2×#692311yescharacteristic
380C22×Q8—611no—
381C22×Q8C2×#692311yescharacteristic
382C2×M16—611no—
383C2×M16—611no—
384C2×M16—613no—
385C2×M16—616no—
386C2×M16—613no—
387C22×C8—613no—
388C22×C8—613no—
389C4⋊Q8—613no—
390C4⋊Q8—613no—
391#126—616no—
392#126—616no—
393#125—613no—
394#125—616no—
395#125—613no—
396#124—613no—
397#124C2×#692313yes—
398#123—616no—
399#123—616no—
400C22⋊Q8—613no—
401C22⋊Q8—613no—
402C22⋊Q8—613no—
403C22⋊Q8—616no—
404C22⋊Q8C2×#692313yes—
405#121—613no—
406#121C2×#692313yes—
407C4×Q8—611no—
408C4×Q8—613no—
409C4×Q8—611no—
410C4×Q8—613no—
411C4×Q8—611no—
412C4×Q8—613no—
413C4×Q8—613no—
414C4×Q8—613no—
415C4×Q8—613no—
416C4×Q8C2×#692311yescharacteristic
417C4×D4—613no—
418C4×D4—613no—
419C4×D4—613no—
420C4×D4—613no—
421C4×D4—613no—
422C4×D4—611no—
423C4×D4—613no—
424C4×D4C2×#692313yes—
425#117—613no—
426#117—613no—
427#117—613no—
428#117—613no—
429#117—616no—
430C2×C4⋊C4—613no—
431C2×C4⋊C4—613no—
432C2×C42—613no—
433C2×C42—613no—
434C2×C42—613no—
435C4⋊C8—613no—
436C4⋊C8—613no—
437C4⋊C8—613no—
438C4⋊C8—616no—
439C4⋊C8—613no—
440C4⋊C8—613no—
441C4⋊C8—613no—
442C4⋊C8—613no—
443C4⋊C8—616no—
444C4⋊C8—613no—
445C4⋊C8—611no—
446C4⋊C8—613no—
447C22⋊C8—611no—
448C22⋊C8—613no—
449C22⋊C8—616no—
450C22⋊C8—613no—
451C22⋊C8—613no—
452C22⋊C8—611no—
453C22⋊C8—613no—
454#97—613no—
455#97—613no—
456#97—613no—
457#97—613no—
458C4×C8—611no—
459C4×C8—613no—
460C4×C8—613no—
461C4×C8—613no—
462C4×C8—613no—
463C4×C8—613no—
464C4×C8—616no—
465C4×C8—616no—
466#41—24no—
467#41—2446no—
468C2×Q8—24no—
469C2×Q8—613no—
470C2×Q8—613no—
471C2×Q8—1224no—
472C2×Q8C22×#692313yes—
473C2×Q8C22×#692311yescharacteristic
474C2×D4—611no—
475C2×D4—613no—
476C2×D4C22×#692313yes—
477C2×D4C22×#692313yes—
478C22×C4—611no—
479C22×C4—613no—
480C22×C4—613no—
481C22×C4—613no—
482C22×C4—611no—
483C22×C4—613no—
484C22×C4—613no—
485C22×C4C22×#692313yes—
486C22×C4C22×#692313yes—
487M16—1222no—
488M16—1222no—
489M16—2443no—
490M16—2443no—
491M16—2446no—
492C2×C8—611no—
493C2×C8—613no—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—613no—
498C2×C8—613no—
499C2×C8—613no—
500C2×C8—1226no—
501C2×C8—2443no—
502C4⋊C4—613no—
503C4⋊C4—616no—
504C4⋊C4—613no—
505C4⋊C4—616no—
506C4⋊C4—611no—
507C4⋊C4—613no—
508C4⋊C4—613no—
509C4⋊C4—613no—
510C4⋊C4—616no—
511C4⋊C4—616no—
512C4⋊C4—613no—
513C4⋊C4—613no—
514C4⋊C4C22×#692313yes—
515C4⋊C4C22×#692313yes—
516C22⋊C4—613no—
517C22⋊C4—616no—
518C22⋊C4—613no—
519C22⋊C4—616no—
520C22⋊C4—611no—
521C22⋊C4—613no—
522C22⋊C4C22×#692311yescharacteristic
523C22⋊C4C22×#692313yes—
524C22⋊C4C22×#692316yes—
525C42—611no—
526C42—613no—
527C42—613no—
528C42—613no—
529C42—613no—
530C42—613no—
531C42—613no—
532C42—613no—
533C42—613no—
534C42C22×#692313yes—
535C23—611no—
536C23C23×#692313yes—
537C23C23×#692311yescharacteristic
538Q8—26no—
539Q8—1226no—
540Q8—1226no—
541Q8#1451986414yes—
542D4—26no—
543D4—43no—
544D4—1222no—
545D4—2443no—
546C2×C4—26no—
547C2×C4—26no—
548C2×C4—611no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—613no—
552C2×C4—613no—
553C2×C4—613no—
554C2×C4—613no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1223no—
558C2×C4—1223no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1226no—
562C2×C4—1226no—
563C2×C4—2443no—
564C2×C4—2443no—
565C2×C4C23×#692311yescharacteristic
566C2×C4C23×#692311yescharacteristic
567C2×C4C23×#692313yes—
568C2×C4C23×#692313yes—
569C2×C4C23×#692313yes—
570C8—612no—
571C8—1223no—
572C8—1221no—
573C8—1223no—
574C8—1221no—
575C8—1221no—
576C8—1223no—
577C8—1223no—
578C22—23no—
579C22—21no—
580C22—21no—
581C22—23no—
582C22—23no—
583C22—21no—
584C22—1221no—
585C22—1221no—
586C22—1221no—
587C22C24×#692311yescharacteristic
588C4—21no—
589C4—21no—
590C4—23no—
591C4—23no—
592C4—611no—
593C4—611no—
594C4—612no—
595C4—1221no—
596C4—1223no—
597C4—1223no—
598C4—1223no—
599C4—1223no—
600C4C2×#1451986416yes—
601C2—41no—
602C2—43no—
603C2—2441no—
604C22−1+4×#692311yescharacteristic
605C2C22×#1451986411yescharacteristic
606C2D61⋊#56811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#548C4
prime radical#21459435C2
cosocleC2×Dic61C25
hypercenter#548#6923
hypocenterC61#60345
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #21462762C2semidirect
2. #21462761C2semidirect
3. #21462751C2semidirect
4. #21462750C2semidirect
5. #21462728C2semidirect
6. #21462713C2semidirect
7. #21462712C2semidirect
8. #21462651C2semidirect
9. #21462650C2semidirect
10. #21462629C2semidirect
11. #41×#92420C2semidirect
12. Q8×#11758213C2semidirect
13. Q8×#11758217C2semidirect
14. #21462607C2semidirect
377. C61#60345semidirect

Maximal subgroups

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

subgroupindex
#214627622
#214627612
#214627512
#214627502
#214627282
#214627132
#214627122
#214626512
#214626502
#214626292
#41×#924202
Q8×#117582132
Q8×#117582172
#214626072
#214594352
#6034561