Finite Group Databasebeta

← #30625068

Subgroups — #30625068

Order 15616.

2576 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462358C213yesmaximal, complemented
2#21462355C213yesmaximal, complemented
3#21462350C216yesmaximal, complemented
4#21462341C213yesmaximal, complemented
5#21462320C216yesmaximal, complemented
6#21462309C213yesmaximal, complemented
7#21462304C213yesmaximal, complemented
8#120×#6923C213yesmaximal, complemented
9H(3,𝔽4)×D61C211yesmaximal, characteristic
10C24×#6923C2211yescharacteristic, complemented
11C2×Q8×#6923C2213yescomplemented
12C2×D4×#6923C2213yes—
13C2×D4×#6923C2216yes—
14C22×C22⋊#6923C2213yescomplemented
15(C2×#6923)⋊Q8C2213yescomplemented
16(C2×#6923)⋊Q8C2216yescomplemented
17#16840714C2216yes—
18#16840714C2213yes—
19#16840712C2216yescomplemented
20#16840712C2216yescomplemented
21#16840712C2216yescomplemented
22#16840712C2216yescomplemented
23#16840712C2216yescomplemented
24#16840712C2216yescomplemented
25#16840710C2216yes—
26#16840710C2213yes—
27#16840710C2213yes—
28#16840710C2216yes—
29#16840709C2216yescomplemented
30#16840709C2213yescomplemented
31#16840709C2213yescomplemented
32#16840709C2216yescomplemented
33#16840708C2211yescharacteristic, complemented
34#16840707C2212yescomplemented
35#16840703C2213yescomplemented
36#16840703C2211yescharacteristic, complemented
37C4×C4⋊#6923C2216yescomplemented
38C22⋊C4×#6923C2213yescomplemented
39C22⋊C4×#6923C2216yescomplemented
40C22⋊C4×#6923C2213yescomplemented
41C22⋊C4×#6923C2216yescomplemented
42#16840698C2213yescomplemented
43#16840505C413yescomplemented
44#16840505C413yescomplemented
45#16840504C413yescomplemented
46#16840504C416yescomplemented
47#124×D61C2213yes—
48#124×D61C2216yes—
49#120×D61C2213yes—
50#120×D61C2213yes—
51C61×H(3,𝔽4)C411yescharacteristic, complemented
52C23×#6923—46no—
53C23×#6923C2313yes—
54C23×#6923C2313yes—
55C23×#6923C2313yes—
56C23×#6923C2313yes—
57Q8×#6923—46no—
58D4×#6923—46no—
59D4×#6923—412no—
60C2×C4⋊#6923C2316yes—
61C2×C4⋊#6923C2313yes—
62C2×C4⋊#6923C2313yes—
63C2×C4⋊#6923C2316yes—
64C2×C22⋊#6923—46no—
65C2×C22⋊#6923C2316yes—
66C2×C22⋊#6923C2313yes—
67C2×C22⋊#6923C2316yes—
68C2×C22⋊#6923C2316yes—
69C2×C22⋊#6923C2313yes—
70C2×C22⋊#6923C2313yes—
71C2×C22⋊#6923C2316yes—
72C2×C22⋊#6923C2313yes—
73C2×C4×#6923C2313yes—
74C2×C4×#6923C2316yes—
75C2×C4×#6923C2313yes—
76C2×C4×#6923C2316yes—
77(C2×#6923)⋊C4C2316yes—
78(C2×#6923)⋊C4C2316yes—
79(C2×#6923)⋊C4C2316yes—
80(C2×#6923)⋊C4C2316yes—
81(C2×#6923)⋊C4C2316yes—
82(C2×#6923)⋊C4C2316yes—
83C24×D61C2311yescharacteristic
84C24×D61C2311yescharacteristic
85C2×Q8×D61C2313yes—
86C2×D4×D61C2316yes—
87C2×D4×D61C2313yes—
88#14519766C2×C413yescomplemented
89#14519766C2×C416yescomplemented
90#14519765C2×C413yescomplemented
91#14519765C2×C416yescomplemented
92#14519764C2×C413yescomplemented
93#14519764C2×C416yescomplemented
94#14519763C2×C416yescomplemented
95#14519763C2×C416yescomplemented
96#14519763C2×C416yescomplemented
97#14519763C2×C416yescomplemented
98#14519763C2×C416yescomplemented
99#14519763C2×C416yescomplemented
100#14519747C2×C416yescomplemented
101#14519747C2×C413yescomplemented
102#14519747C2×C416yescomplemented
103#14519747C2×C413yescomplemented
104#14519746C2×C413yescomplemented
105#14519746C2×C413yescomplemented
106#14519745C2×C413yescomplemented
107#14519745C2×C416yescomplemented
108#14519745C2×C416yescomplemented
109#14519745C2×C413yescomplemented
110C22⋊C4×D61C2316yes—
111C22⋊C4×D61C2313yes—
112C22⋊C4×D61C2313yes—
113C22⋊C4×D61C2316yes—
114C42×D61C2312yes—
115C42×D61C2311yescharacteristic
116#124×C61C2×C413yescomplemented
117#124×C61C2×C416yescomplemented
118#120×C61C2×C413yescomplemented
119#120×C61C2×C413yescomplemented
120C24×C61C22×C411yescharacteristic
121C24×C61C22×C411yescharacteristic, complemented
122C23×D61—43no—
123C23×D61—43no—
124C23×D61—43no—
125C23×D61—43no—
126C23×D61C22×C413yes—
127C23×D61C22×C413yescomplemented
128C23×D61C2413yes—
129C23×D61C2413yes—
130C22×#6923—212no—
131C22×#6923—26no—
132C22×#6923—26no—
133C22×#6923—43no—
134C22×#6923—46no—
135C22×#6923—43no—
136C22×#6923—46no—
137C22×#6923—46no—
138C22×#6923—416no—
139C22×#6923C2416yes—
140C22×#6923C2411yescharacteristic
141C22×#6923C2413yes—
142C22×#6923C2413yes—
143C22×#6923C2413yes—
144C2×Q8×C61C22×C413yescomplemented
145C2×D4×C61C22×C416yes—
146C2×D4×C61C22×C413yes—
147C2×C22⋊D61C22×C416yes—
148C2×C22⋊D61C22×C413yes—
149C2×C22⋊D61C22×C416yes—
150C2×C22⋊D61C22×C413yes—
151Q8×D61—43no—
152Q8×D61—43no—
153D4×D61—43no—
154D4×D61—46no—
155D4×D61—43no—
156D4×D61—46no—
157C2×D244C22×C416yes—
158C2×D244C22×C413yes—
159C2×C4×D61C2413yes—
160C2×C4×D61C2416yes—
161C2×Dic122C22×C416yescomplemented
162C2×Dic122C22×C413yescomplemented
163C22⋊#6923—212no—
164C22⋊#6923—46no—
165C22⋊#6923—46no—
166C22⋊#6923—43no—
167C22⋊#6923—43no—
168C22⋊#6923—43no—
169C22⋊#6923—43no—
170C22⋊#6923—46no—
171C4⋊#6923—212no—
172C4⋊#6923—46no—
173C4⋊#6923—43no—
174C4⋊#6923—43no—
175C4×#6923—26no—
176C4×#6923—43no—
177C4×#6923—46no—
178C4×#6923—46no—
179C22⋊C4×C61C22×C416yes—
180C22⋊C4×C61C22×C413yes—
181C22⋊C4×C61C22×C416yescomplemented
182C22⋊C4×C61C22×C413yescomplemented
183C42×C61C22×C412yescomplemented
184C42×C61C22×C411yescharacteristic, complemented
185C22⋊Dic61C22×C416yes—
186C22⋊Dic61C22×C413yes—
187C22⋊Dic61C22×C413yescomplemented
188C22⋊Dic61C22×C416yescomplemented
189#11758198C22×C416yes—
190#11758198C22×C416yes—
191#11758198C22×C416yes—
192#11758198C22×C416yescomplemented
193#11758198C22×C416yescomplemented
194#11758198C22×C416yescomplemented
195C4×Dic61C22×C416yescomplemented
196C4×Dic61C22×C413yescomplemented
197C23×C61—43no—
198C23×C61—43no—
199C23×C61C23×C413yes—
200C23×C61C23×C413yes—
201C22×D61—26no—
202C22×D61—23no—
203C22×D61—26no—
204C22×D61—23no—
205C22×D61—26no—
206C22×D61—23no—
207C22×D61—23no—
208C22×D61—26no—
209C22×D61—46no—
210C22×D61—43no—
211C22×D61—43no—
212C22×D61—46no—
213C22×D61—44no—
214C22×D61—412no—
215C22×D61—82no—
216C22×D61—86no—
217C22×D61C23×C413yes—
218C22×D61C23×C413yes—
219C22×D61C2511yescharacteristic
220C2×#6923—26no—
221C2×#6923—23no—
222C2×#6923—26no—
223C2×#6923—26no—
224C2×#6923—23no—
225C2×#6923—23no—
226C2×#6923—26no—
227C2×#6923—26no—
228C2×#6923—26no—
229C2×#6923—412no—
230C2×#6923—412no—
231C2×#6923—86no—
232C2×#6923—86no—
233C2×#6923H(5,𝔽2)16yes—
234Q8×C61—43no—
235D4×C61—43no—
236D4×C61—46no—
237C2×C4×C61C23×C413yes—
238C2×C4×C61C23×C416yes—
239C22⋊D61—43no—
240C22⋊D61—43no—
241C22⋊D61—46no—
242C22⋊D61—46no—
243C2×Dic61C23×C413yes—
244C2×Dic61C23×C416yes—
245D244—43no—
246D244—46no—
247C4×D61—23no—
248C4×D61—26no—
249C4×D61—23no—
250C4×D61—26no—
251Dic122—46no—
252Dic122—43no—
253#34512—611nomaximal, Sylow
254C22×C61—23no—
255C22×C61—26no—
256C22×C61—26no—
257C22×C61—23no—
258C22×C61—44no—
259C22×C61—82no—
260C22×C61C24×C411yescharacteristic
261C2×D61—23no—
262C2×D61—26no—
263C2×D61—26no—
264C2×D61—23no—
265C2×D61—412no—
266C2×D61—43no—
267C2×D61—43no—
268C2×D61—43no—
269C2×D61—43no—
270C2×D61—46no—
271C2×D61—46no—
272C2×D61—86no—
273C2×D61C2×H(5,𝔽2)13yes—
274C2×D61#58513yes—
275#6923—26no—
276#6923—41no—
277#6923—43no—
278#6923—43no—
279#6923—46no—
280C4×C61—23no—
281C4×C61—26no—
282Dic61—23no—
283Dic61—26no—
284C2×H(3,𝔽4)—611no—
285#2375—613no—
286#2372—613no—
287#2367—616no—
288#2358—613no—
289#2337—616no—
290#2326—613no—
291#2321—613no—
292C4×#120—613no—
293C2×C61—43no—
294C2×C61—43no—
295C2×C61C4×H(5,𝔽2)13yes—
296D61—43no—
297D61—43no—
298D61#353211yescharacteristic
299D61#354013yes—
300C24×C4—611no—
301H(3,𝔽4)—616no—
302H(3,𝔽4)—613no—
303H(3,𝔽4)—613no—
304H(3,𝔽4)—613no—
305H(3,𝔽4)#692311yescharacteristic, complemented
306C2×#124—613no—
307C2×#124—616no—
308C2×#120—613no—
309C2×#120—613no—
310C2×C4×Q8—613no—
311C2×C4×D4—613no—
312C2×C4×D4—616no—
313C22×C22⋊C4—613no—
314#391—613no—
315#391—616no—
316#390—616no—
317#390—613no—
318#388—616no—
319#388—616no—
320#388—616no—
321#388—616no—
322#388—616no—
323#388—616no—
324#386—616no—
325#386—613no—
326#386—616no—
327#386—613no—
328#385—613no—
329#385—616no—
330#385—616no—
331#385—613no—
332#384—611no—
333#383—612no—
334#379—611no—
335#379—613no—
336C4×C4⋊C4—616no—
337C4×C22⋊C4—613no—
338C4×C22⋊C4—613no—
339C4×C22⋊C4—616no—
340C4×C22⋊C4—616no—
341#376—613no—
342C61#3451211yescharacteristic, complemented, Sylow
343C25—611no—
344C25—611no—
345C22×Q8—613no—
346C22×D4—613no—
347C22×D4—616no—
348C23×C4—613no—
349C23×C4—613no—
350C23×C4—613no—
351C23×C4—613no—
352C23×C4—2446no—
353#124—613no—
354#124—616no—
355#124—613no—
356#124—616no—
357#124—616no—
358#124—616no—
359#124—616no—
360#124—616no—
361#124—616no—
362#124—616no—
363#124—613no—
364#124—616no—
365#124C2×#692313yescomplemented
366#124C2×#692316yescomplemented
367#120—613no—
368#120—613no—
369#120—616no—
370#120—613no—
371#120—616no—
372#120—613no—
373#120—613no—
374#120—616no—
375#120—613no—
376#120—616no—
377#120C2×#692313yescomplemented
378#120C2×#692313yescomplemented
379C4×Q8—2446no—
380C4×D4—2446no—
381C4×D4—24412no—
382C2×C4⋊C4—616no—
383C2×C4⋊C4—613no—
384C2×C4⋊C4—613no—
385C2×C4⋊C4—616no—
386C2×C22⋊C4—616no—
387C2×C22⋊C4—613no—
388C2×C22⋊C4—616no—
389C2×C22⋊C4—613no—
390C2×C22⋊C4—616no—
391C2×C22⋊C4—616no—
392C2×C22⋊C4—613no—
393C2×C22⋊C4—616no—
394C2×C22⋊C4—613no—
395C2×C22⋊C4—613no—
396C2×C22⋊C4—616no—
397C2×C22⋊C4—613no—
398C2×C22⋊C4—2446no—
399C2×C42—611no—
400C2×C42—612no—
401C2×C42—613no—
402C2×C42—616no—
403C2×C42—616no—
404C2×C42—613no—
405#95—616no—
406#95—616no—
407#95—616no—
408#95—616no—
409#95—616no—
410#95—616no—
411C24—613no—
412C24—613no—
413C24—613no—
414C24—613no—
415C24—2443no—
416C24—2443no—
417C24—2443no—
418C24—2443no—
419C24C22×#692311yescharacteristic
420C24C22×#692311yescharacteristic, complemented
421C2×Q8—613no—
422C2×Q8—616no—
423C2×Q8—2443no—
424C2×Q8—2443no—
425C2×Q8C22×#692313yescomplemented
426C2×D4—613no—
427C2×D4—616no—
428C2×D4—613no—
429C2×D4—616no—
430C2×D4—613no—
431C2×D4—616no—
432C2×D4—2443no—
433C2×D4—2446no—
434C2×D4—2443no—
435C2×D4—2446no—
436C2×D4C22×#692313yes—
437C2×D4C22×#692316yes—
438C22×C4—613no—
439C22×C4—613no—
440C22×C4—616no—
441C22×C4—616no—
442C22×C4—611no—
443C22×C4—613no—
444C22×C4—613no—
445C22×C4—1226no—
446C22×C4—12212no—
447C22×C4—1226no—
448C22×C4—2443no—
449C22×C4—2446no—
450C22×C4—2446no—
451C22×C4—2443no—
452C22×C4—2446no—
453C22×C4—24416no—
454C4⋊C4—12212no—
455C4⋊C4—2443no—
456C4⋊C4—2443no—
457C4⋊C4—2446no—
458C22⋊C4—616no—
459C22⋊C4—613no—
460C22⋊C4—616no—
461C22⋊C4—613no—
462C22⋊C4—616no—
463C22⋊C4—616no—
464C22⋊C4—616no—
465C22⋊C4—616no—
466C22⋊C4—616no—
467C22⋊C4—616no—
468C22⋊C4—12212no—
469C22⋊C4—2446no—
470C22⋊C4—2443no—
471C22⋊C4—2443no—
472C22⋊C4—2446no—
473C22⋊C4—2446no—
474C22⋊C4—2443no—
475C22⋊C4—2443no—
476C22⋊C4C22×#692316yes—
477C22⋊C4C22×#692313yes—
478C22⋊C4C22×#692316yescomplemented
479C22⋊C4C22×#692313yescomplemented
480C42—613no—
481C42—616no—
482C42—1226no—
483C42—2446no—
484C42—2446no—
485C42—2443no—
486C42C22×#692311yescharacteristic, complemented
487C42C22×#692312yescomplemented
488C23—43no—
489C23—43no—
490C23—611no—
491C23—613no—
492C23—613no—
493C23—1226no—
494C23—1223no—
495C23—1223no—
496C23—1226no—
497C23—1223no—
498C23—1226no—
499C23—1223no—
500C23—1226no—
501C23—4882no—
502C23—4886no—
503C23—2446no—
504C23—2443no—
505C23—2443no—
506C23—2446no—
507C23—2444no—
508C23—24412no—
509C23C23×#692313yes—
510C23C23×#692313yes—
511Q8—43no—
512Q8—2443no—
513Q8—2446no—
514D4—43no—
515D4—46no—
516D4—2443no—
517D4—2446no—
518D4—2443no—
519D4—2443no—
520D4—2446no—
521D4—2446no—
522C2×C4—616no—
523C2×C4—613no—
524C2×C4—616no—
525C2×C4—1223no—
526C2×C4—1226no—
527C2×C4—1223no—
528C2×C4—1226no—
529C2×C4—1226no—
530C2×C4—1223no—
531C2×C4—1226no—
532C2×C4—1226no—
533C2×C4—1226no—
534C2×C4—1223no—
535C2×C4—1226no—
536C2×C4—1223no—
537C2×C4—1226no—
538C2×C4—4886no—
539C2×C4—4886no—
540C2×C4—24412no—
541C2×C4—24412no—
542C2×C4C23×#692313yes—
543C2×C4C23×#692316yes—
544C22—23no—
545C22—26no—
546C22—26no—
547C22—23no—
548C22—44no—
549C22—82no—
550C22—613no—
551C22—613no—
552C22—1226no—
553C22—1223no—
554C22—1223no—
555C22—1226no—
556C22—4886no—
557C22—2443no—
558C22—2443no—
559C22—2443no—
560C22—2446no—
561C22—2443no—
562C22—2446no—
563C22—24412no—
564C22C24×#692311yescharacteristic
565C4—23no—
566C4—26no—
567C4—1226no—
568C4—1223no—
569C4—1226no—
570C4—2443no—
571C4—2446no—
572C4—2441no—
573C4—2443no—
574C2—43no—
575C2—43no—
576C2—611no—
577C2—613no—
578C2—2443no—
579C2—2443no—
580C2H(5,𝔽2)×#692313yes—
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×H(3,𝔽4)C4
prime radicalH(3,𝔽4)×D61C2
cosocleC22×D61C25
hypercenterH(3,𝔽4)#6923
hypocenterC61#34512
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #21462358C2semidirect
2. #21462355C2semidirect
3. #21462350C2semidirect
4. #21462341C2semidirect
5. #21462320C2semidirect
6. #21462309C2semidirect
7. #21462304C2semidirect
8. #120×#6923C2semidirect
10. C24×#6923C22semidirect
11. C2×Q8×#6923C22semidirect
14. C22×C22⋊#6923C22semidirect
15. (C2×#6923)⋊Q8C22semidirect
16. (C2×#6923)⋊Q8C22semidirect
19. #16840712C22semidirect
20. #16840712C22semidirect
21. #16840712C22semidirect
22. #16840712C22semidirect
23. #16840712C22semidirect
24. #16840712C22semidirect
29. #16840709C22semidirect
30. #16840709C22semidirect
31. #16840709C22semidirect
32. #16840709C22semidirect
33. #16840708C22semidirect
34. #16840707C22semidirect
35. #16840703C22semidirect
36. #16840703C22semidirect
37. C4×C4⋊#6923C22semidirect
38. C22⋊C4×#6923C22semidirect
39. C22⋊C4×#6923C22semidirect
40. C22⋊C4×#6923C22semidirect
41. C22⋊C4×#6923C22semidirect
42. #16840698C22semidirect
43. #16840505C4semidirect
44. #16840505C4semidirect
45. #16840504C4semidirect
46. #16840504C4semidirect
51. C61×H(3,𝔽4)C4semidirect
88. #14519766C2×C4semidirect
89. #14519766C2×C4semidirect
90. #14519765C2×C4semidirect
91. #14519765C2×C4semidirect
92. #14519764C2×C4semidirect
93. #14519764C2×C4semidirect
94. #14519763C2×C4semidirect
95. #14519763C2×C4semidirect
96. #14519763C2×C4semidirect
97. #14519763C2×C4semidirect
98. #14519763C2×C4semidirect
99. #14519763C2×C4semidirect
100. #14519747C2×C4semidirect
101. #14519747C2×C4semidirect
102. #14519747C2×C4semidirect
103. #14519747C2×C4semidirect
104. #14519746C2×C4semidirect
105. #14519746C2×C4semidirect
106. #14519745C2×C4semidirect
107. #14519745C2×C4semidirect
108. #14519745C2×C4semidirect
109. #14519745C2×C4semidirect
116. #124×C61C2×C4semidirect
117. #124×C61C2×C4semidirect
118. #120×C61C2×C4semidirect
119. #120×C61C2×C4semidirect
121. C24×C61C22×C4semidirect
127. C23×D61C22×C4semidirect
144. C2×Q8×C61C22×C4semidirect
161. C2×Dic122C22×C4semidirect
162. C2×Dic122C22×C4semidirect
181. C22⋊C4×C61C22×C4semidirect
182. C22⋊C4×C61C22×C4semidirect
183. C42×C61C22×C4semidirect
184. C42×C61C22×C4semidirect
187. C22⋊Dic61C22×C4semidirect
188. C22⋊Dic61C22×C4semidirect
192. #11758198C22×C4semidirect
193. #11758198C22×C4semidirect
194. #11758198C22×C4semidirect
195. C4×Dic61C22×C4semidirect
196. C4×Dic61C22×C4semidirect
305. H(3,𝔽4)#6923semidirect
342. C61#34512semidirect
365. #124C2×#6923semidirect
366. #124C2×#6923semidirect
377. #120C2×#6923semidirect
378. #120C2×#6923semidirect
420. C24C22×#6923semidirect
425. C2×Q8C22×#6923semidirect
478. C22⋊C4C22×#6923semidirect
479. C22⋊C4C22×#6923semidirect
486. C42C22×#6923semidirect
487. C42C22×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623582
#214623552
#214623502
#214623412
#214623202
#214623092
#214623042
#120×#69232
H(3,𝔽4)×D612
#3451261