Finite Group Databasebeta

← #29748660

Subgroups — #29748660

Order 15616.

732 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#21450331C211yesmaximal, characteristic, complemented
2#21448889C211yesmaximal, characteristic
3#21447193C211yesmaximal, characteristic
4#21447058C211yesmaximal, characteristic
5#21447058C211yesmaximal, characteristic
6#21446496C211yesmaximal, characteristic, complemented
7#21446451C211yesmaximal, characteristic, complemented
8#21446410C211yesmaximal, characteristic, complemented
9#21446402C211yesmaximal, characteristic, complemented
10#21446379C211yesmaximal, characteristic, complemented
11#21446076C211yesmaximal, characteristic, complemented
12#21446068C211yesmaximal, characteristic
13#21445821C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#2071C211yesmaximal, characteristic
16C2×#14519767—21no—
17C2×#14519764C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839700—21no—
31#16839693—21no—
32#16839667C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839656—21no—
38#16839650—21no—
39#16839648—21no—
40#16839634—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196C2211yescharacteristic
44C22⋊C4×Dic61—21no—
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#102×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C61×#390C2211yescharacteristic
55C61×#384C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519767—44no—
64#14519764—22no—
65#14519764—22no—
66C4×Dic122—22no—
67C4×Dic122—41no—
68D4×Dic61—22no—
69D4×Dic61—41no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196D411yescharacteristic
79C2×#11758196D411yescharacteristic
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519711—22no—
89#14519708—22no—
90C4⋊#92418—41no—
91#14519702—22no—
92#14519702—41no—
93#14519700—22no—
94#14519700—22no—
95#14519698—22no—
96#14519698—41no—
97C8×Dic61—41no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#102×C61—22no—
120#102×C61—22no—
121C24×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61#4112yes—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61—81no—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122#4112yes—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61—42no—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—81no—
167C42×C61—21no—
168C42×C61—21no—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171C22⋊Dic61—42no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—42no—
177C4⋊Dic61#4112yes—
178C4⋊Dic61#4112yes—
179#11758196—22no—
180#11758196—22no—
181#11758196—42no—
182#11758196—42no—
183#11758196—42no—
184#11758196—42no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—42no—
190C4×Dic61—42no—
191C4×Dic61—81no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C23×C61—22no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61C22×D411yescharacteristic
206D4×C61—22no—
207D4×C61—22no—
208D4×C61—41no—
209D4×C61—41no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61#13512yes—
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—84no—
255C2×Dic61AGL(1,ℤ/8)12yes—
256Dic122—22no—
257Dic122—22no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—41no—
261C8×C61—41no—
262#92418—41no—
263#92418—41no—
264#26761—611nomaximal, Sylow
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61#52511yescharacteristic
273C22×C61#54611yescharacteristic
274C22×C61#54811yescharacteristic
275C22×C61C2×#13511yescharacteristic
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61#57711yescharacteristic
278C22×C61#57811yescharacteristic
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61#52911yescharacteristic
285C4×C61#53911yescharacteristic
286C4×C61#55211yescharacteristic
287C4×C61#55311yescharacteristic
288Dic61—22no—
289Dic61—41no—
290Dic61—42no—
291Dic61—41no—
292Dic61—42no—
293Dic61—41no—
294Dic61—42no—
295Dic61—81no—
296#2506—611no—
297#2367—611no—
298#2071D6111yescharacteristic
299#2065—611no—
300#2042—611no—
301#2042—611no—
302#1933—611no—
303#1927—611no—
304#1921—611no—
305#1920—611no—
306#1916—611no—
307#1863—611no—
308#1862—611no—
309#1803—611no—
310C4×#102—611no—
311C2×C61—82no—
312C2×C61#246011yescharacteristic
313C2×C61#310611yescharacteristic
314C2×C61#312011yescharacteristic
315C2×C61#330011yescharacteristic
316C2×C61#330611yescharacteristic
317C2×C61#336511yescharacteristic
318C2×C61#336711yescharacteristic
319C2×#125—1221no—
320C2×#124—611no—
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C2×#107—611no—
324C2×#106—611no—
325C2×C4⋊C8—611no—
326C2×#103—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#102C2×D6111yescharacteristic
333C2×#102C2×D6111yescharacteristic
334C2×C4×C8—611no—
335#400—1221no—
336#398—1221no—
337#391—611no—
338#390C2×D6111yescharacteristic
339#389—611no—
340#389—611no—
341#389—611no—
342#389—611no—
343#388—1221no—
344#388—1221no—
345#387—1221no—
346#385—1221no—
347#384C2×D6111yescharacteristic
348C4×C4⋊C4—611no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C22⋊C4—1221no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340C2×D6111yescharacteristic
357#340C2×D6111yescharacteristic
358#336—611no—
359#336—611no—
360#336—611no—
361#336C2×D6111yescharacteristic
362C61#2676111yescharacteristic, complemented, Sylow
363C22×Q8—611no—
364C22×D4C22×D6111yescharacteristic
365C23×C4—1221no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8C22×D6111yescharacteristic
369C22×C8C22×D6111yescharacteristic
370#125—2444no—
371#124—1222no—
372#124—1222no—
373C4×Q8—1222no—
374C4×Q8—2441no—
375C4×D4—1222no—
376C4×D4—2441no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42C22×D6111yescharacteristic
403#107—1222no—
404#106—1222no—
405C4⋊C8—2441no—
406#103—1222no—
407#103—1222no—
408#103—1222no—
409#103—2441no—
410#102—22no—
411#102—22no—
412#102—1222no—
413#102—2441no—
414C4×C8—2441no—
415#95—611no—
416#95—611no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425C24—21no—
426C2×Q8—612no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4#1175822412yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—4881no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4D4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454C22×C4C23×D6111yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—612no—
476C4⋊C4—612no—
477C4⋊C4—1222no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C22⋊C4—42no—
489C22⋊C4—81no—
490C22⋊C4—1224no—
491C22⋊C4—1224no—
492C22⋊C4—2442no—
493C42—21no—
494C42—21no—
495C42—1221no—
496C42—1221no—
497C42—1221no—
498C42—1221no—
499C42—4881no—
500C42—2442no—
501C42—2442no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×D4×D6111yescharacteristic
510Q8—1222no—
511Q8—1222no—
512Q8—2441no—
513Q8—2441no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—612no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—4884no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4#1451979612yes—
558C2×C4#41×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C2×C4#41×D6111yescharacteristic
561C2×C4C2×#1175822411yescharacteristic
562C2×C4#1451982311yescharacteristic
563C2×C4#1451982411yescharacteristic
564C8—41no—
565C8—41no—
566C8—2441no—
567C8—2441no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22#1684027211yescharacteristic
576C22#1684040411yescharacteristic
577C22#1684042011yescharacteristic
578C22C2×#1451979611yescharacteristic
579C22AGL(1,ℤ/8)×D6111yescharacteristic
580C22#1684056811yescharacteristic
581C22#1684057811yescharacteristic
582C4—22no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—1222no—
588C4—4881no—
589C4—2441no—
590C4—2441no—
591C4—2442no—
592C4—2442no—
593C4—2442no—
594C4—2441no—
595C4#1684028911yescharacteristic
596C4#1684035611yescharacteristic
597C4#1684044811yescharacteristic
598C4#1684046311yescharacteristic
599C2—82no—
600C2#2144977211yescharacteristic
601C2#2145587611yescharacteristic
602C2#2145603011yescharacteristic
603C2#2145770411yescharacteristic
604C2#2145778311yescharacteristic
605C2#2145852011yescharacteristic
606C2#2145853411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2071C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2071D61
hypocenterC61#26761
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
1. #21450331C2semidirect
6. #21446496C2semidirect
7. #21446451C2semidirect
8. #21446410C2semidirect
9. #21446402C2semidirect
10. #21446379C2semidirect
11. #21446076C2semidirect
13. #21445821C2semidirect
362. C61#26761semidirect

Maximal subgroups

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

subgroupindex
#214503312
#214488892
#214471932
#214470582
#214464962
#214464512
#214464102
#214464022
#214463792
#214460762
#214460682
#214458212
#102×Dic612
C61×#20712
#2676161