Finite Group Databasebeta

← #29779438

Subgroups — #29779438

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450396C211yesmaximal, characteristic, complemented
2#21449671C211yesmaximal, characteristic
3#21447318C211yesmaximal, characteristic, complemented
4#21447222C211yesmaximal, characteristic
5#21447132C211yesmaximal, characteristic
6#21446960C211yesmaximal, characteristic, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446513C211yesmaximal, characteristic
9#21446511C211yesmaximal, characteristic, complemented
10#21446506C211yesmaximal, characteristic, complemented
11#21446406C211yesmaximal, characteristic, complemented
12#21446155C211yesmaximal, characteristic
13#21445693C211yesmaximal, characteristic, complemented
14#21445646C211yesmaximal, characteristic
15C61×#1968C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519754—21no—
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519706C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519687C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26#16839704—21no—
27#16839702—21no—
28#16839685—21no—
29#16839674—21no—
30#16839669C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839649—21no—
37#16839647—21no—
38#16839637—21no—
39#16839632C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C22×Dic122—21no—
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519777—22no—
64#14519777—41no—
65#14519754—44no—
66D4×Dic61—22no—
67D4×Dic61—41no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
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×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519706—22no—
87#14519701—22no—
88#14519701—41no—
89#14519698—22no—
90#14519698—22no—
91#14519687—22no—
92#14519687—41no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C2×SD16×C61—22no—
106C2×SD16×C61—22no—
107C2×SD16×C61—41no—
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61C2311yescharacteristic
115#106×C61—22no—
116#103×C61—22no—
117#102×C61—22no—
118C4×C8×C61—41no—
119#95×C61—21no—
120#95×C61—21no—
121C24×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61#4112yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61—81no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×Dic122—81no—
154SD16×C61—44no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61#4112yes—
160C2×C8×C61#4112yes—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—42no—
167C42×C61—21no—
168C42×C61—21no—
169C22⋊Dic61—42no—
170C22⋊Dic61—42no—
171C22⋊Dic61—81no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—42no—
180C4⋊Dic61—42no—
181C4⋊Dic61—42no—
182C4⋊Dic61—81no—
183C4⋊Dic61#4112yes—
184#11758196—22no—
185#11758196—22no—
186#11758196—42no—
187#11758196—81no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—81no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C23×C61—22no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61C22×D411yescharacteristic
207Q8×C61—41no—
208Q8×C61—41no—
209Q8×C61—41no—
210Q8×C61—41no—
211D4×C61—22no—
212D4×C61—22no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61AGL(1,ℤ/8)12yes—
229C2×C4×C61#13712yes—
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—84no—
261Dic122—84no—
262C8×C61—22no—
263C8×C61—22no—
264#92418—41no—
265#92418—41no—
266#29099—611nomaximal, Sylow
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61#53211yescharacteristic
275C22×C61#54611yescharacteristic
276C22×C61#54811yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57711yescharacteristic
280C22×C61#57711yescharacteristic
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61#52911yescharacteristic
287C4×C61#53911yescharacteristic
288C4×C61#55111yescharacteristic
289C4×C61#55611yescharacteristic
290Dic61—41no—
291Dic61—41no—
292Dic61—42no—
293Dic61—41no—
294Dic61—41no—
295Dic61—81no—
296Dic61—81no—
297Dic61—81no—
298#2511—611no—
299#2452—611no—
300#2088—611no—
301#2071—611no—
302#2055—611no—
303#2024—611no—
304#1990—611no—
305#1968D6111yescharacteristic
306#1936—611no—
307#1935—611no—
308#1935—611no—
309#1920—611no—
310#1877—611no—
311#1771—611no—
312#1762—611no—
313C2×C61—82no—
314C2×C61#256011yescharacteristic
315C2×C61#317311yescharacteristic
316C2×C61#318211yescharacteristic
317C2×C61#331011yescharacteristic
318C2×C61#331611yescharacteristic
319C2×C61#337211yescharacteristic
320C2×C61#337311yescharacteristic
321C22×SD16C2×D6111yescharacteristic
322C2×C4⋊Q8—611no—
323C2×C22⋊Q8—1221no—
324C2×C4×D4—611no—
325C2×#106C2×D6111yescharacteristic
326C2×C4⋊C8—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103C2×D6111yescharacteristic
330C2×#102—611no—
331C2×#102—611no—
332C2×#102C2×D6111yescharacteristic
333C2×#97—611no—
334C2×#97—611no—
335C2×C4×C8C2×D6111yescharacteristic
336#400—1221no—
337#400—1221no—
338#396—1221no—
339#393—1221no—
340#391—611no—
341#391—611no—
342#391C2×D6111yescharacteristic
343#390—611no—
344#390C2×D6111yescharacteristic
345#389—611no—
346#389—611no—
347#389—611no—
348#387—1221no—
349#387—1221no—
350#385—1221no—
351#384—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#336—611no—
363#336—611no—
364C61#2909911yescharacteristic, complemented, Sylow
365C22×Q8—1221no—
366C22×Q8C22×D6111yescharacteristic
367C22×D4C22×D6111yescharacteristic
368C23×C4—1221no—
369C2×SD16—22no—
370C2×SD16—22no—
371C2×SD16—41no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C4⋊Q8—1222no—
377C4⋊Q8—2441no—
378C22⋊Q8—2444no—
379C4×D4—1222no—
380C4×D4—2441no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—1221no—
403C2×C42C22×D6111yescharacteristic
404#106—22no—
405C4⋊C8—1222no—
406#103—22no—
407#103—1222no—
408#103—2441no—
409#102—22no—
410#102—1222no—
411#102—1222no—
412#97—1222no—
413#97—2441no—
414C4×C8—41no—
415#95—21no—
416#95—21no—
417#95—611no—
418#95—611no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427C24—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—4881no—
435C2×Q8—2442no—
436C2×Q8—2442no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4#1175822212yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—4881no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460SD16—44no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8#1175822412yes—
472C2×C8#1175822412yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—612no—
478C4⋊C4—1221no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—2442no—
490C4⋊C4—2442no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—4881no—
496C22⋊C4—2442no—
497C22⋊C4—2442no—
498C42—21no—
499C42—21no—
500C42—1221no—
501C42—1221no—
502C42—1221no—
503C42—1221no—
504C42—4881no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C2×D4×D6111yescharacteristic
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—4884no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1222no—
548C2×C4—4884no—
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—2441no—
559C2×C4—2441no—
560C2×C4#1451980412yes—
561C2×C4#1451980612yes—
562C2×C4#41×D6111yescharacteristic
563C2×C4C2×#1175822411yescharacteristic
564C2×C4C2×#1175822411yescharacteristic
565C2×C4C2×#1175822211yescharacteristic
566C2×C4#1451982211yescharacteristic
567C2×C4#1451982411yescharacteristic
568C8—22no—
569C8—22no—
570C8—2441no—
571C8—2441no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22#1684031011yescharacteristic
580C22#1684040111yescharacteristic
581C22#1684042111yescharacteristic
582C22C2×#1451980411yescharacteristic
583C22C2×#1451980611yescharacteristic
584C22#1684056911yescharacteristic
585C22#1684056911yescharacteristic
586C4—22no—
587C4—22no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—4881no—
592C4—4881no—
593C4—4881no—
594C4—2441no—
595C4—2441no—
596C4—2441no—
597C4—2441no—
598C4—2442no—
599C4#1684029211yescharacteristic
600C4#1684035711yescharacteristic
601C4#1684043811yescharacteristic
602C4#1684048111yescharacteristic
603C2—82no—
604C2#2145096811yescharacteristic
605C2#2145643311yescharacteristic
606C2#2145653511yescharacteristic
607C2#2145784611yescharacteristic
608C2#2145793611yescharacteristic
609C2#2145859811yescharacteristic
610C2#2145860911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1968C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1968D61
hypocenterC61#29099
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
1. #21450396C2semidirect
3. #21447318C2semidirect
6. #21446960C2semidirect
7. #21446780C2semidirect
9. #21446511C2semidirect
10. #21446506C2semidirect
11. #21446406C2semidirect
13. #21445693C2semidirect
364. C61#29099semidirect

Maximal subgroups

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

subgroupindex
#214503962
#214496712
#214473182
#214472222
#214471322
#214469602
#214467802
#214465132
#214465112
#214465062
#214464062
#214461552
#214456932
#214456462
C61×#19682
#2909961