Finite Group Databasebeta

← #29886450

Subgroups — #29886450

Order 15616.

2006 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454082C211yesmaximal, characteristic, complemented
2#21454003C211yesmaximal, characteristic, complemented
3#21453924C213yesmaximal, complemented
4#21453923C213yesmaximal, complemented
5#21453915C213yesmaximal, complemented
6#21453908C211yesmaximal, characteristic, complemented
7#21453813C213yesmaximal, complemented
8#21449122C211yesmaximal, characteristic, complemented
9#21449121C211yesmaximal, characteristic, complemented
10#21449097C211yesmaximal, characteristic, complemented
11#21449077C213yesmaximal, complemented
12#21449062C213yesmaximal, complemented
13#21449054C213yesmaximal, complemented
14#21448324C213yesmaximal, complemented
15C61×#2825C211yesmaximal, characteristic, complemented
16C23×C4×D61C2211yescharacteristic
17C2×#14519745C2213yes—
18C2×D4×Dic61C2213yes—
19C2×D4×Dic61C2213yescomplemented
20C2×#14519737C2213yes—
21C2×#14519737C2213yes—
22C2×#14519733C2213yescomplemented
23C61⋊#401C2211yescharacteristic, complemented
24#16839703C2213yescomplemented
25#16839703C2213yescomplemented
26#16839701C2213yescomplemented
27#16839699C2211yescharacteristic
28#16839698C2213yescomplemented
29#16839697C2213yes—
30#16839696C2213yescomplemented
31#16839696C2213yescomplemented
32#16839694C2213yescomplemented
33#16839691C2213yes—
34#16839691C2213yes—
35#16839690C2213yescomplemented
36#16839690C2213yescomplemented
37#16839687C2213yes—
38#16839687C2213yescomplemented
39#16839686C2211yescharacteristic
40#16839685C2213yes—
41#16839685C2213yescomplemented
42#16839685C2213yescomplemented
43#16839684C2213yes—
44#16839678C2213yes—
45#16839678C2213yes—
46#16839677C2213yes—
47#16839677C2213yes—
48#16839676C2213yes—
49#16839676C2213yes—
50#16839676C2213yescomplemented
51#16839670C2213yes—
52#16839670C2213yes—
53#16839638C2213yes—
54#16839638C2213yes—
55#16839636C2213yes—
56#16839636C2213yes—
57#16839635C2213yes—
58#16839617C2213yes—
59C4⋊C4×Dic61C2213yescomplemented
60#16839613C2213yescomplemented
61C22⋊C4×Dic61C2213yes—
62C22⋊C4×Dic61C2213yescomplemented
63#16839609C2213yes—
64#16839609C2213yes—
65#16839607C2211yescharacteristic, complemented
66C61×#401C2211yescharacteristic, complemented
67C61×#397C2213yescomplemented
68C61×#396C2211yescharacteristic
69C61×#394C2213yescomplemented
70C61×#392C2213yes—
71C61×#392C2211yescharacteristic, complemented
72C61×#386C2213yes—
73C24×D61C2311yescharacteristic
74C22×C22⋊D61C2313yes—
75C22×C22⋊D61C2313yes—
76C23×Dic61C2313yes—
77C23×Dic61C2311yescharacteristic
78C22×C4×D61—46no—
79C22×C4×D61C2311yescharacteristic
80C22×C4×D61C2311yescharacteristic
81C22×C4×D61C2311yescharacteristic
82C22×C4×D61C2311yescharacteristic
83#14519745—212no—
84D4×Dic61—46no—
85D4×Dic61—46no—
86#14519737—46no—
87#14519737—46no—
88#14519733—46no—
89C2×C4⋊Dic61C2313yes—
90C2×C4⋊Dic61C2313yes—
91C2×C4⋊Dic61C2313yes—
92C2×#11758196C2313yes—
93C2×#11758196C2313yes—
94C2×#11758196C2313yes—
95C2×#11758196C2313yes—
96C2×#11758196C2313yes—
97C2×#11758196C2313yes—
98C2×C22⋊Dic61C2313yes—
99C2×C22⋊Dic61C2313yes—
100C2×C22⋊Dic61C2313yes—
101C2×C22⋊Dic61C2313yes—
102C2×C22⋊Dic61C2313yes—
103C2×C22⋊Dic61C2313yes—
104C2×C22⋊Dic61C2313yes—
105C2×#11758198C2313yes—
106C2×#11758198C2313yes—
107C2×#11758198C2313yes—
108C2×#11758198C2313yes—
109C2×#11758198C2313yes—
110C2×#11758198C2313yes—
111C2×C4×Dic61C2313yes—
112C2×C4×Dic61C2313yes—
113C2×C4×Dic61C2313yes—
114#14519684C2313yes—
115#14519684C2313yes—
116#14519684C2313yes—
117#14519684C2313yes—
118#14519684C2313yes—
119#14519684C2313yes—
120#14519684C2313yes—
121#14519684C2313yes—
122#14519684C2313yes—
123#14519684C2313yes—
124C22×D4×C61C2313yes—
125C22×D4×C61C2313yes—
126C23×C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2313yes—
128C2×C22⋊C4×C61C2313yes—
129C2×C22⋊C4×C61C2313yes—
130C2×C22⋊C4×C61C2313yes—
131C2×C22⋊C4×C61C2313yes—
132C2×C22⋊C4×C61C2313yes—
133#95×C61C2311yescharacteristic
134#95×C61C2313yes—
135#95×C61C2313yes—
136#95×C61C2313yes—
137C24×C61C2413yes—
138C24×C61C2411yescharacteristic
139C23×D61—43no—
140C23×D61—43no—
141C23×D61#4114yes—
142C23×D61C2411yescharacteristic
143C23×D61C2411yescharacteristic
144C2×D4×C61—26no—
145C2×D4×C61—43no—
146C2×D4×C61—43no—
147C2×D4×C61—43no—
148C2×D4×C61—43no—
149C2×D4×C61—43no—
150C22×C4×C61—43no—
151C22×C4×C61C2411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×C4×C61C2413yes—
154C22×C4×C61C2413yes—
155C22×C4×C61C2413yes—
156C2×C22⋊D61—26no—
157C2×C22⋊D61—26no—
158C2×C22⋊D61—43no—
159C2×C22⋊D61—43no—
160C2×C22⋊D61—43no—
161C2×C22⋊D61—43no—
162C22×Dic61—26no—
163C22×Dic61—26no—
164C22×Dic61—26no—
165C22×Dic61—26no—
166C22×Dic61C2411yescharacteristic
167C22×Dic61C2413yes—
168C22×Dic61C2413yes—
169C22×Dic61C2413yes—
170C22×Dic61C2413yes—
171C22×Dic61C2411yescharacteristic
172C2×C4×D61—26no—
173C2×C4×D61—26no—
174C2×C4×D61—43no—
175C2×C4×D61—43no—
176C2×C4×D61—416no—
177C4⋊C4×C61—43no—
178C22⋊C4×C61—43no—
179C22⋊C4×C61—43no—
180C22⋊C4×C61—43no—
181C22⋊C4×C61—43no—
182C22⋊C4×C61—43no—
183C22⋊Dic61—26no—
184C22⋊Dic61—43no—
185C22⋊Dic61—43no—
186C22⋊Dic61—43no—
187C22⋊Dic61—43no—
188C22⋊Dic61—43no—
189C22⋊Dic61—43no—
190#11758198—26no—
191#11758198—26no—
192#11758198—43no—
193#11758198—43no—
194#11758198—43no—
195#11758198—43no—
196C4⋊Dic61—43no—
197C4⋊Dic61—43no—
198C4⋊Dic61—43no—
199#11758196—43no—
200#11758196—43no—
201#11758196—43no—
202#11758196—43no—
203#11758196—43no—
204#11758196—43no—
205C4×Dic61—26no—
206C4×Dic61—26no—
207C4×Dic61—26no—
208C23×C61—23no—
209C23×C61—23no—
210C23×C61—23no—
211C23×C61—23no—
212C23×C61—23no—
213C23×C61—23no—
214C23×C61—23no—
215C23×C61—23no—
216C23×C61—23no—
217C23×C61C2×#4112yes—
218C23×C61C2511yescharacteristic
219C22×D61—23no—
220C22×D61—23no—
221C22×D61—23no—
222C22×D61—23no—
223C22×D61—412no—
224C22×D61—44no—
225C22×D61—412no—
226C22×D61C2×#4112yes—
227C22×D61C2×#4112yes—
228D4×C61—412no—
229D4×C61—412no—
230C2×C4×C61—23no—
231C2×C4×C61—23no—
232C2×C4×C61—23no—
233C2×C4×C61—23no—
234C2×C4×C61—23no—
235C2×C4×C61—23no—
236C2×C4×C61—23no—
237C2×C4×C61—23no—
238C2×C4×C61—23no—
239C2×C4×C61—23no—
240C2×C4×C61—23no—
241C2×C4×C61—44no—
242C22⋊D61—412no—
243C22⋊D61—412no—
244C2×Dic61—23no—
245C2×Dic61—23no—
246C2×Dic61—23no—
247C2×Dic61—23no—
248C2×Dic61—23no—
249C2×Dic61—23no—
250C2×Dic61—23no—
251C2×Dic61—23no—
252C2×Dic61—23no—
253C2×Dic61—23no—
254C2×Dic61—23no—
255C2×Dic61—23no—
256C2×Dic61—23no—
257C2×Dic61—44no—
258C2×Dic61—412no—
259C2×Dic61H(5,𝔽2)16yes—
260C4×D61—44no—
261C4×D61—44no—
262C4×D61—44no—
263C4×D61—44no—
264#36676—611nomaximal, Sylow
265C22×C61—26no—
266C22×C61—41no—
267C22×C61—43no—
268C22×C61—43no—
269C22×C61—43no—
270C22×C61—43no—
271C22×C61—43no—
272C22×C61—43no—
273C22×C61—43no—
274C22×C61—43no—
275C22×C61C22×#4111yescharacteristic
276C22×C61C2×H(5,𝔽2)13yes—
277C22×C61#58513yes—
278C2×D61—26no—
279C2×D61—26no—
280C2×D61—416no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—43no—
284C2×D61—43no—
285C4×C61—43no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—43no—
289C4×C61—43no—
290Dic61—43no—
291Dic61—43no—
292Dic61—41no—
293Dic61—43no—
294Dic61—43no—
295Dic61#56014yes—
296#2842—611no—
297#2834—611no—
298#2827—613no—
299#2827—613no—
300#2826—613no—
301#2825—611no—
302#2825D6111yescharacteristic, complemented
303#2812—613no—
304#2401—611no—
305#2401—611no—
306#2397—611no—
307#2395—613no—
308#2392—613no—
309#2392—613no—
310C4×#120—613no—
311C2×C61—42no—
312C2×C61—83no—
313C2×C61#345913yes—
314C2×C61C2×#56011yescharacteristic
315C2×C61#352913yes—
316D61—42no—
317D61—42no—
318C24×C4—611no—
319C2×#120—613no—
320C2×C4×D4—613no—
321C2×C4×D4—613no—
322C2×C4×D4—613no—
323C2×C4×D4—613no—
324C2×#117—613no—
325#401—611no—
326#401C2×D6111yescharacteristic, complemented
327#400—613no—
328#400—613no—
329#400—613no—
330#399—613no—
331#399—613no—
332#399—611no—
333#398—613no—
334#398—613no—
335#398—613no—
336#397—613no—
337#397—613no—
338#397—613no—
339#397—613no—
340#397—613no—
341#397—613no—
342#397C2×D6113yescomplemented
343#396—613no—
344#396—613no—
345#396—613no—
346#396—613no—
347#396—611no—
348#396C2×D6111yescharacteristic
349#394—613no—
350#394—613no—
351#394—613no—
352#394—613no—
353#394—613no—
354#394—613no—
355#394—613no—
356#394C2×D6113yescomplemented
357#392—613no—
358#392—613no—
359#392C2×D6113yes—
360#392C2×D6111yescharacteristic, complemented
361#386—613no—
362#386—613no—
363#386C2×D6113yes—
364#385—613no—
365#385—613no—
366#385—613no—
367#379—613no—
368C4×C4⋊C4—613no—
369C4×C4⋊C4—613no—
370C4×C22⋊C4—613no—
371C4×C22⋊C4—613no—
372C4×C22⋊C4—613no—
373C4×C22⋊C4—613no—
374#376—611no—
375C61#3667611yescharacteristic, complemented, Sylow
376C25—611no—
377C22×D4—613no—
378C22×D4—613no—
379C22×D4C22×D6113yes—
380C22×D4C22×D6113yes—
381C23×C4—613no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—611no—
387C23×C4—2446no—
388C23×C4C22×D6111yescharacteristic
389#120—12212no—
390C4×D4—2446no—
391C4×D4—2446no—
392C4×D4—2446no—
393C4×D4—2446no—
394#117—2446no—
395C2×C4⋊C4—613no—
396C2×C4⋊C4—613no—
397C2×C4⋊C4—613no—
398C2×C4⋊C4—613no—
399C2×C4⋊C4—613no—
400C2×C4⋊C4—613no—
401C2×C4⋊C4—613no—
402C2×C4⋊C4—613no—
403C2×C4⋊C4—613no—
404C2×C4⋊C4C22×D6113yes—
405C2×C22⋊C4—613no—
406C2×C22⋊C4—613no—
407C2×C22⋊C4—613no—
408C2×C22⋊C4—613no—
409C2×C22⋊C4—613no—
410C2×C22⋊C4—613no—
411C2×C22⋊C4—613no—
412C2×C22⋊C4—613no—
413C2×C22⋊C4—613no—
414C2×C22⋊C4—613no—
415C2×C22⋊C4—613no—
416C2×C22⋊C4—613no—
417C2×C22⋊C4—613no—
418C2×C22⋊C4C22×D6113yes—
419C2×C22⋊C4C22×D6113yes—
420C2×C22⋊C4C22×D6113yes—
421C2×C22⋊C4C22×D6113yes—
422C2×C22⋊C4C22×D6113yes—
423C2×C42—613no—
424C2×C42—613no—
425C2×C42—613no—
426#95—613no—
427#95—613no—
428#95—613no—
429#95—613no—
430#95—613no—
431#95—613no—
432#95—613no—
433#95—613no—
434#95—613no—
435#95—613no—
436#95C22×D6111yescharacteristic
437#95C22×D6113yes—
438#95C22×D6113yes—
439#95C22×D6113yes—
440C24—611no—
441C24—611no—
442C24—614no—
443C24—2443no—
444C24—2443no—
445C24C23×D6113yes—
446C24C23×D6111yescharacteristic
447C2×D4—26no—
448C2×D4—43no—
449C2×D4—43no—
450C2×D4—43no—
451C2×D4—43no—
452C2×D4—43no—
453C2×D4—1226no—
454C2×D4—1226no—
455C2×D4—2443no—
456C2×D4—2443no—
457C2×D4—2443no—
458C2×D4—2443no—
459C22×C4—43no—
460C22×C4—611no—
461C22×C4—613no—
462C22×C4—613no—
463C22×C4—611no—
464C22×C4—613no—
465C22×C4—613no—
466C22×C4—1226no—
467C22×C4—1226no—
468C22×C4—1226no—
469C22×C4—1226no—
470C22×C4—1226no—
471C22×C4—1226no—
472C22×C4—2443no—
473C22×C4—2443no—
474C22×C4—24416no—
475C22×C4C23×D6113yes—
476C22×C4C23×D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478C22×C4C23×D6113yes—
479C22×C4C23×D6113yes—
480C4⋊C4—43no—
481C4⋊C4—2443no—
482C4⋊C4—2443no—
483C4⋊C4—2443no—
484C4⋊C4—2443no—
485C4⋊C4—2443no—
486C4⋊C4—2443no—
487C4⋊C4—2443no—
488C4⋊C4—2443no—
489C4⋊C4—2443no—
490C22⋊C4—43no—
491C22⋊C4—43no—
492C22⋊C4—43no—
493C22⋊C4—43no—
494C22⋊C4—43no—
495C22⋊C4—1226no—
496C22⋊C4—1226no—
497C22⋊C4—1226no—
498C22⋊C4—2443no—
499C22⋊C4—2443no—
500C22⋊C4—2443no—
501C22⋊C4—2443no—
502C22⋊C4—2443no—
503C22⋊C4—2443no—
504C22⋊C4—2443no—
505C22⋊C4—2443no—
506C22⋊C4—2443no—
507C22⋊C4—2443no—
508C42—1226no—
509C42—1226no—
510C42—1226no—
511C23—23no—
512C23—23no—
513C23—23no—
514C23—23no—
515C23—23no—
516C23—23no—
517C23—23no—
518C23—23no—
519C23—23no—
520C23—612no—
521C23—612no—
522C23—1223no—
523C23—1223no—
524C23—1223no—
525C23—1223no—
526C23—24412no—
527C23—2444no—
528C23—24412no—
529C23#41×D6112yes—
530C23C24×D6111yescharacteristic
531D4—412no—
532D4—412no—
533D4—24412no—
534D4—24412no—
535C2×C4—23no—
536C2×C4—23no—
537C2×C4—23no—
538C2×C4—23no—
539C2×C4—23no—
540C2×C4—23no—
541C2×C4—23no—
542C2×C4—23no—
543C2×C4—23no—
544C2×C4—23no—
545C2×C4—23no—
546C2×C4—44no—
547C2×C4—616no—
548C2×C4—1223no—
549C2×C4—1223no—
550C2×C4—1223no—
551C2×C4—1223no—
552C2×C4—1223no—
553C2×C4—1223no—
554C2×C4—1223no—
555C2×C4—1223no—
556C2×C4—1223no—
557C2×C4—1223no—
558C2×C4—1223no—
559C2×C4—1223no—
560C2×C4—1223no—
561C2×C4—2444no—
562C2×C4—2444no—
563C2×C4—2444no—
564C2×C4—2444no—
565C2×C4—24412no—
566C2×C4—2444no—
567C22—26no—
568C22—41no—
569C22—43no—
570C22—43no—
571C22—43no—
572C22—43no—
573C22—43no—
574C22—43no—
575C22—43no—
576C22—43no—
577C22—1226no—
578C22—1226no—
579C22—24416no—
580C22—2441no—
581C22—2441no—
582C22—2443no—
583C22—2443no—
584C22C2×#41×D6111yescharacteristic
585C22H(5,𝔽2)×D6113yes—
586C22#1684059713yes—
587C4—43no—
588C4—41no—
589C4—41no—
590C4—43no—
591C4—43no—
592C4—614no—
593C4—2443no—
594C4—2443no—
595C4—2443no—
596C4—2441no—
597C4—2443no—
598C2—42no—
599C2—83no—
600C2—2442no—
601C2—2442no—
602C2#2145923013yes—
603C2#560×D6111yescharacteristic
604C2#2145995413yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2825C2
hypercenter#2825D61
hypocenterC61#36676
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2825

Complements

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

NH ≅ G/Ntype
1. #21454082C2semidirect
2. #21454003C2semidirect
3. #21453924C2semidirect
4. #21453923C2semidirect
5. #21453915C2semidirect
6. #21453908C2semidirect
7. #21453813C2semidirect
8. #21449122C2semidirect
9. #21449121C2semidirect
10. #21449097C2semidirect
11. #21449077C2semidirect
12. #21449062C2semidirect
13. #21449054C2semidirect
14. #21448324C2semidirect
15. C61×#2825C2semidirect
19. C2×D4×Dic61C22semidirect
22. C2×#14519733C22semidirect
23. C61⋊#401C22semidirect
24. #16839703C22semidirect
25. #16839703C22semidirect
26. #16839701C22semidirect
28. #16839698C22semidirect
30. #16839696C22semidirect
31. #16839696C22semidirect
32. #16839694C22semidirect
35. #16839690C22semidirect
36. #16839690C22semidirect
38. #16839687C22semidirect
41. #16839685C22semidirect
42. #16839685C22semidirect
50. #16839676C22semidirect
59. C4⋊C4×Dic61C22semidirect
60. #16839613C22semidirect
62. C22⋊C4×Dic61C22semidirect
65. #16839607C22semidirect
66. C61×#401C22semidirect
67. C61×#397C22semidirect
69. C61×#394C22semidirect
71. C61×#392C22semidirect
302. #2825D61semidirect
326. #401C2×D61semidirect
342. #397C2×D61semidirect
356. #394C2×D61semidirect
360. #392C2×D61semidirect
375. C61#36676semidirect

Maximal subgroups

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

subgroupindex
#214540822
#214540032
#214539242
#214539232
#214539152
#214539082
#214538132
#214491222
#214491212
#214490972
#214490772
#214490622
#214490542
#214483242
C61×#28252
#3667661