Finite Group Databasebeta

← #29779461

Subgroups — #29779461

Order 15616.

712 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449658C211yesmaximal, characteristic, complemented
2#21447321C211yesmaximal, characteristic, complemented
3#21447228C211yesmaximal, characteristic, complemented
4#21447136C211yesmaximal, characteristic
5#21446955C211yesmaximal, characteristic
6#21446779C211yesmaximal, characteristic, complemented
7#21446672C211yesmaximal, characteristic
8#21446519C211yesmaximal, characteristic
9#21446519C211yesmaximal, characteristic
10#21446508C211yesmaximal, characteristic, complemented
11#21446397C211yesmaximal, characteristic, complemented
12#21446177C211yesmaximal, characteristic
13#21445694C211yesmaximal, characteristic, complemented
14#21445652C211yesmaximal, characteristic, complemented
15C61×#2512C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519700C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839696—21no—
31#16839682—21no—
32#16839668C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839666—21no—
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839648—21no—
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×#126×C61—21no—
49C61×#397—21no—
50C61×#396—21no—
51C61×#390C2211yescharacteristic
52C61×#390C2211yescharacteristic
53C61×#390C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#388—21no—
56C61×#384C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122C2311yescharacteristic
61C2×#11758200—22no—
62C2×#11758200—22no—
63C2×#11758200—41no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519777—22no—
69#14519777—41no—
70C4×Dic122—22no—
71C4×Dic122—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C4×Dic61—21no—
80#14519708—22no—
81C4⋊#92418—22no—
82#14519700—22no—
83#14519700—22no—
84#14519700—22no—
85#14519698—22no—
86#14519698—22no—
87#14519698—41no—
88#14519688—22no—
89#14519688—41no—
90C4×#92418—41no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×D4×C61C2311yescharacteristic
98#126×C61—44no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#95×C61—21no—
118#95×C61—21no—
119#95×C61D411yescharacteristic
120#95×C61D411yescharacteristic
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—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122—81no—
150C2×Dic122#4112yes—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61—81no—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C22⋊C4×C61—42no—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—81no—
167C22⋊C4×C61—81no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—42no—
175#11758200—44no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—42no—
182C4⋊Dic61#4112yes—
183#11758196—42no—
184#11758196—81no—
185#11758196—81no—
186C4×Dic61—42no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418#4112yes—
198C2×#92418#4112yes—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61C22×D411yescharacteristic
207D4×C61—41no—
208D4×C61—41no—
209D4×C61—41no—
210D4×C61—41no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61AGL(1,ℤ/8)12yes—
237C2×C4×C61#13712yes—
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C2×Dic61—22no—
245C2×Dic61—22no—
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—41no—
255C2×Dic61—41no—
256Dic122—22no—
257Dic122—22no—
258Dic122—41no—
259Dic122—41no—
260Dic122—84no—
261#92418—22no—
262#92418—22no—
263#92418—41no—
264#92418—41no—
265#29101—611nomaximal, Sylow
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61#53211yescharacteristic
274C22×C61#54611yescharacteristic
275C22×C61#54811yescharacteristic
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57711yescharacteristic
279C22×C61#57711yescharacteristic
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61#52911yescharacteristic
290C4×C61#53911yescharacteristic
291C4×C61#55111yescharacteristic
292C4×C61#55611yescharacteristic
293Dic61—42no—
294Dic61—42no—
295Dic61—81no—
296Dic61—81no—
297#2512D6111yescharacteristic
298#2451—611no—
299#2088—611no—
300#2072—611no—
301#2055—611no—
302#2023—611no—
303#1990—611no—
304#1968—611no—
305#1936—611no—
306#1936—611no—
307#1935—611no—
308#1919—611no—
309#1880—611no—
310#1771—611no—
311#1763—611no—
312C2×C61—161no—
313C2×C61#256011yescharacteristic
314C2×C61#317311yescharacteristic
315C2×C61#318211yescharacteristic
316C2×C61#330611yescharacteristic
317C2×C61#332011yescharacteristic
318C2×C61#336811yescharacteristic
319C2×C61#336911yescharacteristic
320C22×SD16—611no—
321C2×C4⋊Q8—611no—
322C2×#126—21no—
323C2×C4×Q8—611no—
324C2×#106—611no—
325C2×C4⋊C8—611no—
326C2×#103—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#102—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#97—611no—
333C2×#97—611no—
334C2×C4×C8—611no—
335#398—1221no—
336#397—21no—
337#396—21no—
338#395—1221no—
339#391—611no—
340#391—611no—
341#391—1221no—
342#390C2×D6111yescharacteristic
343#390C2×D6111yescharacteristic
344#390C2×D6111yescharacteristic
345#389—611no—
346#389—611no—
347#389C2×D6111yescharacteristic
348#388—21no—
349#387—1221no—
350#384C2×D6111yescharacteristic
351C4×C4⋊C4C2×D6111yescharacteristic
352C4×C4⋊C4C2×D6111yescharacteristic
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#336—611no—
362#336—611no—
363C61#2910111yescharacteristic, complemented, Sylow
364C22×Q8—611no—
365C22×Q8—1221no—
366C22×D4C22×D6111yescharacteristic
367C2×SD16—1222no—
368C2×SD16—1222no—
369C2×SD16—2441no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8—611no—
374C4⋊Q8—1222no—
375C4⋊Q8—2441no—
376#126—44no—
377C4×Q8—1222no—
378C4×Q8—2441no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
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×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—21no—
401C2×C42—1221no—
402C2×C42C22×D6111yescharacteristic
403C2×C42C22×D6111yescharacteristic
404C2×C42C22×D6111yescharacteristic
405#106—1222no—
406C4⋊C8—1222no—
407#103—1222no—
408#103—1222no—
409#103—1222no—
410#102—1222no—
411#102—1222no—
412#102—2441no—
413#97—1222no—
414#97—2441no—
415C4×C8—2441no—
416#95—21no—
417#95—21no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95C22⋊D6111yescharacteristic
425#95C22⋊D6111yescharacteristic
426C24—21no—
427C2×Q8—612no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—4881no—
434C2×Q8—2442no—
435C2×Q8—2442no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456SD16—2444no—
457C2×C8—612no—
458C2×C8—612no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—612no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—4881no—
487C4⋊C4—4881no—
488C4⋊C4—2442no—
489C4⋊C4—2442no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—2442no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C22×C22⋊D6111yescharacteristic
512Q8—1222no—
513Q8—1222no—
514Q8—4884no—
515Q8—2441no—
516Q8—2441no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—1222no—
547C2×C4—1222no—
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—2441no—
558C2×C4#1451980312yes—
559C2×C4#1451980712yes—
560C2×C4#41×D6111yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4#41×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4#1451982311yescharacteristic
565C2×C4#1451982411yescharacteristic
566C8—1222no—
567C8—1222no—
568C8—2441no—
569C8—2441no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#1684030811yescharacteristic
578C22#1684040011yescharacteristic
579C22#1684041911yescharacteristic
580C22C2×#1451980311yescharacteristic
581C22C2×#1451980711yescharacteristic
582C22#1684057011yescharacteristic
583C22#1684057311yescharacteristic
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—4881no—
594C4—4881no—
595C4—2442no—
596C4—2442no—
597C4#1684029511yescharacteristic
598C4#1684035311yescharacteristic
599C4#1684043711yescharacteristic
600C4#1684047911yescharacteristic
601C2—161no—
602C2#2145096111yescharacteristic
603C2#2145643011yescharacteristic
604C2#2145653311yescharacteristic
605C2#2145777411yescharacteristic
606C2#2145798411yescharacteristic
607C2#2145854411yescharacteristic
608C2#2145855511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2512C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2512D61
hypocenterC61#29101
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2512

Complements

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

NH ≅ G/Ntype
1. #21449658C2semidirect
2. #21447321C2semidirect
3. #21447228C2semidirect
6. #21446779C2semidirect
10. #21446508C2semidirect
11. #21446397C2semidirect
13. #21445694C2semidirect
14. #21445652C2semidirect
363. C61#29101semidirect

Maximal subgroups

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

subgroupindex
#214496582
#214473212
#214472282
#214471362
#214469552
#214467792
#214466722
#214465192
#214465082
#214463972
#214461772
#214456942
#214456522
C61×#25122
#2910161