Finite Group Databasebeta

← #30626673

Subgroups — #30626673

Order 15616.

1930 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462749C213yesmaximal, complemented
2#21462746C213yesmaximal, complemented
3#21462741C213yesmaximal, complemented
4#21462737C211yesmaximal, characteristic, complemented
5#21462722C211yesmaximal, characteristic, complemented
6#21462721C211yesmaximal, characteristic, complemented
7#21462716C213yesmaximal, complemented
8#21462661C213yesmaximal, complemented
9#21462660C213yesmaximal, complemented
10#21462630C213yesmaximal, complemented
11#21462625C211yesmaximal, characteristic, complemented
12Q8×#11758213C211yesmaximal, characteristic, complemented
13Q8×#11758217C211yesmaximal, characteristic, complemented
14D61⋊#446C213yesmaximal, complemented
15#21459028C211yesmaximal, characteristic
16#16840814C2213yes—
17#16840814C2216yes—
18#16840814C2213yes—
19#16840813C2213yes—
20#16840813C2213yes—
21#16840813C2213yes—
22#16840813C2213yes—
23#16840812C2213yes—
24#16840812C2213yes—
25#16840812C2213yes—
26#16840812C2213yes—
27Q8×#92420C2211yescharacteristic
28Q8×#92420C2211yescharacteristic
29Q8×#92420C2211yescharacteristic
30Q8×#92420C2211yescharacteristic
31#16840795C2211yescharacteristic
32#16840795C2213yes—
33#16840794C2213yes—
34#16840794C2213yes—
35#16840794C2216yes—
36#16840792C2213yes—
37#16840792C2213yes—
38#16840792C2216yes—
39#16840790C2213yes—
40#16840790C2213yes—
41#16840790C2213yes—
42#16840790C2213yes—
43#16840785C2216yes—
44#16840784C2216yes—
45#16840784C2213yes—
46#16840783C2213yes—
47#16840781C2213yes—
48#16840781C2213yes—
49#16840780C2213yes—
50#16840779C2213yes—
51#16840766C2213yes—
52#16840765C2213yes—
53D61⋊(C4⋊C8)C2213yes—
54D61⋊(C4⋊C8)C2216yes—
55C4×#11758213C2213yes—
56C4×#11758217C2213yes—
57#16840721C2216yes—
58C4×D61⋊C8C2213yes—
59C2×#14519825C411yescharacteristic
60#16840244C413yes—
61#16840244C411yescharacteristic
62#16840242C413yes—
63#16840242C413yes—
64#16840238C411yescharacteristic
65#16840238C413yes—
66C4×#11758226C2211yescharacteristic
67C4×#11758226C2213yes—
68#16840230C2213yes—
69#16840230C2213yes—
70C2×Q8×Dic61C2211yescharacteristic
71C4×Q8×D61C2213yes—
72C4×Q8×D61C2211yescharacteristic
73C61×#519C411yescharacteristic
74C2×#11758213C2311yescharacteristic
75C2×#11758213C2313yes—
76C2×#11758217C2311yescharacteristic
77C2×#11758217C2313yes—
78C2×D61⋊C8C2313yes—
79C2×D61⋊C8C2313yes—
80C2×D61⋊C8C2316yes—
81#14519844C2313yes—
82#14519844C2313yes—
83#14519844C2316yes—
84#14519844C2313yes—
85#14519844C2311yescharacteristic
86C4⋊#92420C2313yes—
87C4⋊#92420C2313yes—
88C4⋊#92420C2313yes—
89C4⋊#92420C2316yes—
90C4⋊#92420C2313yes—
91C4⋊#92420C2313yes—
92C4⋊#92420C2313yes—
93#14519834C2313yes—
94#14519834C2313yes—
95#14519834C2311yescharacteristic
96#14519834C2313yes—
97#14519834C2316yes—
98C22⋊#92420C2311yescharacteristic
99C22⋊#92420C2313yes—
100#14519832C2313yes—
101#14519832C2313yes—
102#14519832C2316yes—
103#14519832C2316yes—
104#14519832C2313yes—
105#14519832C2313yes—
106C4×#92420C2311yescharacteristic
107C4×#92420C2313yes—
108C4×#92420C2313yes—
109C4×#92420C2313yes—
110C4×#92420C2313yes—
111C4×#92420C2313yes—
112#14519825—28no—
113C2×#11758226C2×C413yes—
114C2×#11758226C2×C411yescharacteristic
115C2×#11758222C2×C413yes—
116C2×#11758222C2×C413yes—
117C2×Q8×D61C2×C411yescharacteristic
118C2×Q8×D61C2×C413yes—
119Q8×Dic61C2×C413yes—
120Q8×Dic61C2×C411yescharacteristic
121Q8×Dic61C2311yescharacteristic
122Q8×Dic61C2313yes—
123C4×Dic122C2×C413yes—
124C4×Dic122C2×C416yes—
125C4×Dic122C2×C413yes—
126#14519742C2×C413yes—
127#14519742C2×C413yes—
128#14519742C2313yes—
129#14519742C2313yes—
130C4×D244C2×C416yes—
131C4×D244C2×C413yes—
132C4×D244C2×C413yes—
133#14519739C2×C413yes—
134#14519739C2×C413yes—
135#14519739C2313yes—
136#14519739C2313yes—
137#14519737C2×C413yes—
138#14519737C2×C413yes—
139#14519737C2×C416yes—
140#14519737C2313yes—
141#14519737C2311yescharacteristic
142#14519735C2×C413yes—
143#14519735C2×C416yes—
144#14519735C2×C413yes—
145#14519734C2×C413yes—
146#14519734C2×C416yes—
147#14519734C2×C413yes—
148C2×C4⋊Dic61C2×C413yes—
149C4⋊C4×D61C2×C416yes—
150C4⋊C4×D61C2×C413yes—
151C4⋊C4×D61C2×C413yes—
152C2×C4×Dic61C2313yes—
153C42×D61C2313yes—
154C42×D61C2316yes—
155C42×D61C2313yes—
156C22×Q8×C61C2×C411yescharacteristic
157C4×Q8×C61C2×C413yes—
158C4×Q8×C61C2×C411yescharacteristic
159C4×Q8×C61C2×C413yes—
160C4×Q8×C61C2×C411yescharacteristic
161#117×C61C2×C413yes—
162#117×C61C2×C413yes—
163C2×Q8×C61—24no—
164C2×Q8×C61C22×C413yes—
165C2×Q8×C61C22×C411yescharacteristic
166C22×C4×C61C22×C413yes—
167C2×C22⋊D61C22×C413yes—
168C2×C22⋊D61C22×C411yescharacteristic
169C22×Dic61C22×C413yes—
170C22×Dic61C2411yescharacteristic
171#11758226—212no—
172#11758226M1618yescomplemented
173Q8×D61—24no—
174Q8×D61—212no—
175#11758222—212no—
176#11758222—212no—
177C2×D244C22×C413yes—
178C2×D244C22×C413yes—
179C2×C4×D61C22×C416yes—
180C2×C4×D61C22×C413yes—
181C2×C4×D61C22×C413yes—
182C2×C4×D61C22×C413yes—
183C2×C4×D61C22×C416yes—
184C2×C4×D61C22×C413yes—
185C2×C4×D61C2413yes—
186C2×C4×D61C2411yescharacteristic
187C2×Dic122C22×C413yes—
188C2×Dic122C22×C413yes—
189#11758217—22no—
190#11758217—43no—
191C2×#92420C2411yescharacteristic
192C2×#92420C2411yescharacteristic
193C2×#92420C2411yescharacteristic
194C2×#92420C2413yes—
195C2×#92420C2413yes—
196C2×#92420C2413yes—
197C2×#92420C2413yes—
198C2×#92420C2411yescharacteristic
199#11758213—22no—
200#11758213—43no—
201D61⋊C8—26no—
202D61⋊C8—43no—
203D61⋊C8—46no—
204C4⋊C4×C61C22×C413yes—
205C4⋊C4×C61C22×C413yes—
206C4⋊C4×C61C22×C413yes—
207C4⋊C4×C61C22×C413yes—
208C22⋊C4×C61C22×C413yes—
209C22⋊C4×C61C22×C411yescharacteristic
210C42×C61C22×C416yes—
211C42×C61C22×C413yes—
212C42×C61C22×C413yes—
213#11758198C22×C411yescharacteristic
214#11758198C22×C413yes—
215#11758198C22×C413yes—
216#11758198C22×C416yes—
217#11758198C22×C413yes—
218C4⋊Dic61C22×C416yes—
219C4⋊Dic61C22×C413yes—
220C4⋊Dic61C22×C413yes—
221#11758196C22×C416yes—
222#11758196C22×C413yes—
223#11758196C22×C413yes—
224#11758196C22×C413yes—
225#11758196C22×C411yescharacteristic
226C4×Dic61C22×C416yes—
227C4×Dic61C22×C413yes—
228C4×Dic61C22×C413yes—
229C4×Dic61C22×C413yes—
230C4×Dic61C22×C413yes—
231C4×Dic61C2413yes—
232C4×Dic61C2411yescharacteristic
233C4×Dic61C2413yes—
234C4×Dic61C2413yes—
235C23×C61C23×C411yescharacteristic
236C22×D61C23×C411yescharacteristic
237C22×D61C23×C413yes—
238Q8×C61—26no—
239Q8×C61C2×M1614yes—
240C2×C4×C61—26no—
241C2×C4×C61C23×C413yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C23×C413yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C23×C413yes—
246C2×C4×C61C23×C413yes—
247C22⋊D61—26no—
248C22⋊D61—22no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—43no—
252C2×Dic61C23×C411yescharacteristic
253C2×Dic61C23×C413yes—
254C2×Dic61C23×C413yes—
255C2×Dic61C23×C413yes—
256C2×Dic61C23×C411yescharacteristic
257C2×Dic61C2511yescharacteristic
258D244—26no—
259D244C2×M16112yes—
260C4×D61—212no—
261C4×D61—26no—
262C4×D61—23no—
263C4×D61—21no—
264C4×D61—23no—
265C4×D61—21no—
266C4×D61—43no—
267C4×D61—46no—
268C4×D61—43no—
269C4×D61C2×M16112yes—
270Dic122—26no—
271Dic122—26no—
272#92420—23no—
273#92420—21no—
274#92420—21no—
275#92420—21no—
276#92420—23no—
277#92420—23no—
278#92420—23no—
279#924202−1+412yescomplemented
280#60343—611nomaximal, Sylow
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61C24×C411yescharacteristic
285C2×D61—23no—
286C2×D61—23no—
287C2×D61—21no—
288C2×D61—21no—
289C2×D61—21no—
290C2×D61C22×M1616yes—
291C4×C61—23no—
292C4×C61—21no—
293C4×C61—23no—
294C4×C61—21no—
295C4×C61—23no—
296C4×C61C22×M1616yes—
297Dic61—21no—
298Dic61—23no—
299Dic61—23no—
300Dic61—23no—
301Dic61C22×M1612yes—
302Dic61C2×2−1+411yescharacteristic
303Dic61#58511yescharacteristic
304C4×2−1+4—611no—
305#2990—613no—
306#2990—613no—
307#2989—613no—
308#2988—611no—
309#2984—611no—
310#2984—611no—
311#2983—613no—
312#2970—613no—
313#2970—613no—
314#2965—613no—
315C8×#41—611no—
316Q8×M16—611no—
317Q8×M16—611no—
318C2×#446—613no—
319C2×C61—41no—
320C2×C61C4×2−1+411yescharacteristic
321C2×C61C23×M1611yescharacteristic
322C2×C61#357311yescharacteristic
323D61—26no—
324D61—41no—
325C2×2−1+4—611no—
326#520—611no—
327#520—613no—
328#520—613no—
329#520—613no—
330#519—613no—
331#519—611no—
332#519#692311yescharacteristic
333C4×#41—611no—
334C4×#41—613no—
335C4×#41—613no—
336C4×#41—613no—
337C2×C4×Q8—611no—
338C2×C4×Q8—613no—
339C2×C4×Q8—611no—
340#446—613no—
341#446—613no—
342#446—613no—
343#446—613no—
344#446—613no—
345#446—613no—
346#446—613no—
347#446—613no—
348#446—613no—
349#446—616no—
350#446—613no—
351C8×Q8—611no—
352C8×Q8—611no—
353C8×Q8—611no—
354C8×Q8—611no—
355#436—611no—
356#436—613no—
357#435—613no—
358#435—613no—
359#435—616no—
360#435—613no—
361#435—613no—
362#435—616no—
363C8×D4—613no—
364C8×D4—613no—
365C8×D4—613no—
366C8×D4—613no—
367#432—613no—
368#432—616no—
369#432—613no—
370#432—616no—
371#431—613no—
372#431—613no—
373#431—613no—
374#431—613no—
375#423—613no—
376#423—613no—
377C2×C4⋊C8—613no—
378C2×C4⋊C8—616no—
379C4×M16—613no—
380C4×M16—613no—
381C2×#97—616no—
382C2×C4×C8—613no—
383C61#6034311yescharacteristic, complemented, Sylow
3842−1+4—1228no—
385C2×#41—611no—
386C2×#41—613no—
387C2×#41—613no—
388C2×#41—613no—
389C22×Q8—613no—
390C22×Q8—611no—
391C22×Q8C2×#692311yescharacteristic
392C2×M16—611no—
393C2×M16—611no—
394C2×M16—613no—
395C2×M16—613no—
396C22×C8—613no—
397C22×C8—613no—
398C22×C8—616no—
399C4×Q8—611no—
400C4×Q8—613no—
401C4×Q8—613no—
402C4×Q8—611no—
403C4×Q8—613no—
404C4×Q8—613no—
405C4×Q8—613no—
406C4×Q8—613no—
407C4×Q8—616no—
408C4×Q8—613no—
409C4×Q8—613no—
410C4×Q8C2×#692313yes—
411C4×Q8C2×#692311yescharacteristic
412C4×Q8C2×#692313yes—
413C4×Q8C2×#692311yescharacteristic
414C4×D4—613no—
415C4×D4—613no—
416C4×D4—613no—
417C4×D4—613no—
418C4×D4—613no—
419C4×D4—611no—
420C4×D4—613no—
421C4×D4—613no—
422C4×D4—616no—
423C4×D4—613no—
424C4×D4—613no—
425C4×D4—616no—
426#117—613no—
427#117—613no—
428#117—616no—
429#117—613no—
430#117—613no—
431#117—616no—
432#117C2×#692313yes—
433#117C2×#692313yes—
434C2×C4⋊C4—613no—
435C2×C4⋊C4—613no—
436C2×C4⋊C4—616no—
437C2×C4⋊C4—613no—
438C2×C42—613no—
439C2×C42—613no—
440C2×C42—616no—
441C2×C42—613no—
442C4⋊C8—613no—
443C4⋊C8—613no—
444C4⋊C8—613no—
445C4⋊C8—616no—
446C4⋊C8—613no—
447C4⋊C8—613no—
448C4⋊C8—613no—
449C4⋊C8—613no—
450C4⋊C8—613no—
451C4⋊C8—616no—
452C4⋊C8—611no—
453C4⋊C8—613no—
454C22⋊C8—611no—
455C22⋊C8—613no—
456C22⋊C8—611no—
457C22⋊C8—613no—
458C22⋊C8—613no—
459C22⋊C8—616no—
460C22⋊C8—613no—
461#97—613no—
462#97—613no—
463#97—616no—
464#97—616no—
465#97—613no—
466#97—613no—
467C4×C8—611no—
468C4×C8—613no—
469C4×C8—613no—
470C4×C8—613no—
471C4×C8—613no—
472C4×C8—613no—
473#41—618no—
474#41—12212no—
475#41—12212no—
476#41—12212no—
477C2×Q8—24no—
478C2×Q8—613no—
479C2×Q8—613no—
480C2×Q8—12212no—
481C2×Q8—1224no—
482C2×Q8C22×#692313yes—
483C2×Q8C22×#692311yescharacteristic
484C2×D4—611no—
485C2×D4—613no—
486C2×D4—613no—
487C2×D4—613no—
488C22×C4—613no—
489C22×C4—611no—
490C22×C4—613no—
491C22×C4—613no—
492C22×C4—611no—
493C22×C4—616no—
494C22×C4—613no—
495C22×C4—613no—
496C22×C4—616no—
497C22×C4—613no—
498C22×C4C22×#692313yes—
499M16—1222no—
500M16—1222no—
501M16—2443no—
502M16—2443no—
503C2×C8—611no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—611no—
507C2×C8—613no—
508C2×C8—613no—
509C2×C8—613no—
510C2×C8—613no—
511C2×C8—1226no—
512C2×C8—2443no—
513C2×C8—2446no—
514C4⋊C4—613no—
515C4⋊C4—616no—
516C4⋊C4—613no—
517C4⋊C4—616no—
518C4⋊C4—613no—
519C4⋊C4—613no—
520C4⋊C4—611no—
521C4⋊C4—613no—
522C4⋊C4C22×#692313yes—
523C4⋊C4C22×#692313yes—
524C4⋊C4C22×#692313yes—
525C4⋊C4C22×#692313yes—
526C22⋊C4—613no—
527C22⋊C4—611no—
528C22⋊C4—616no—
529C22⋊C4—613no—
530C22⋊C4—613no—
531C22⋊C4C22×#692313yes—
532C22⋊C4C22×#692311yescharacteristic
533C42—611no—
534C42—613no—
535C42—613no—
536C42—613no—
537C42—613no—
538C42—613no—
539C42—613no—
540C42—616no—
541C42—613no—
542C42C22×#692316yes—
543C42C22×#692313yes—
544C42C22×#692313yes—
545C23—613no—
546C23—611no—
547C23C23×#692311yescharacteristic
548Q8—26no—
549Q8—1226no—
550Q8—1226no—
551Q8D61⋊M1614yes—
552D4—6112no—
553D4—1226no—
554D4—1222no—
555D4—1226no—
556C2×C4—26no—
557C2×C4—611no—
558C2×C4—611no—
559C2×C4—6112no—
560C2×C4—611no—
561C2×C4—613no—
562C2×C4—613no—
563C2×C4—613no—
564C2×C4—1223no—
565C2×C4—1223no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—12212no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1226no—
572C2×C4—2443no—
573C2×C4—2446no—
574C2×C4—2443no—
575C2×C4—2443no—
576C2×C4C23×#692313yes—
577C2×C4C23×#692311yescharacteristic
578C2×C4C23×#692313yes—
579C2×C4C23×#692311yescharacteristic
580C2×C4C23×#692313yes—
581C2×C4C23×#692313yes—
582C8—612no—
583C8—1223no—
584C8—1221no—
585C8—1223no—
586C8—1223no—
587C8—1223no—
588C8—1221no—
589C8—1221no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—616no—
594C22—1223no—
595C22—1223no—
596C22—1221no—
597C22—1221no—
598C22—1221no—
599C22C24×#692311yescharacteristic
600C4—23no—
601C4—21no—
602C4—23no—
603C4—21no—
604C4—23no—
605C4—611no—
606C4—611no—
607C4—612no—
608C4—1221no—
609C4—1223no—
610C4—1223no—
611C4—1223no—
612C4C2×D61⋊M1616yes—
613C2—41no—
614C2—1226no—
615C2—2441no—
616C22−1+4×#692311yescharacteristic
617C2C22×D61⋊M1611yescharacteristic
618C2#2146281511yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#519C4
prime radicalC2×#14519825C4
cosocleC2×Dic61C25
hypercenter#519#6923
hypocenterC61#60343
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #21462749C2semidirect
2. #21462746C2semidirect
3. #21462741C2semidirect
4. #21462737C2semidirect
5. #21462722C2semidirect
6. #21462721C2semidirect
7. #21462716C2semidirect
8. #21462661C2semidirect
9. #21462660C2semidirect
10. #21462630C2semidirect
11. #21462625C2semidirect
12. Q8×#11758213C2semidirect
13. Q8×#11758217C2semidirect
14. D61⋊#446C2semidirect
172. #11758226M16semidirect
279. #924202−1+4semidirect
383. C61#60343semidirect

Maximal subgroups

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

subgroupindex
#214627492
#214627462
#214627412
#214627372
#214627222
#214627212
#214627162
#214626612
#214626602
#214626302
#214626252
Q8×#117582132
Q8×#117582172
D61⋊#4462
#214590282
#6034361