Finite Group Databasebeta

← #29748682

Subgroups — #29748682

Order 15616.

712 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#21450344C211yesmaximal, characteristic, complemented
2#21448894C211yesmaximal, characteristic
3#21447225C211yesmaximal, characteristic
4#21447205C211yesmaximal, characteristic
5#21447055C211yesmaximal, characteristic
6#21446492C211yesmaximal, characteristic, complemented
7#21446462C211yesmaximal, characteristic, complemented
8#21446394C211yesmaximal, characteristic, complemented
9#21446391C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446085C211yesmaximal, characteristic, complemented
12#21446078C211yesmaximal, characteristic
13#21445822C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#2042C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×C8×Dic61C2211yescharacteristic
26#16839696—21no—
27#16839693—21no—
28#16839669C2211yescharacteristic
29#16839665C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839657—21no—
34#16839655—21no—
35#16839655—21no—
36#16839647—21no—
37#16839637—21no—
38#16839631C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C22⋊C4×Dic61—21no—
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#124×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62Q8×Dic61—22no—
63Q8×Dic61—41no—
64D4×Dic61—22no—
65D4×Dic61—41no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519711—22no—
87#14519708—22no—
88#14519706—41no—
89#14519701—22no—
90#14519701—22no—
91#14519698—22no—
92#14519698—22no—
93C8×Dic61—41no—
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
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107#124×C61—22no—
108#124×C61—22no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61C2311yescharacteristic
115#103×C61—22no—
116#103×C61—41no—
117#102×C61—22no—
118#102×C61—41no—
119#95×C61—21no—
120C24×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61#4112yes—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61—81no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—81no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C42×C61—21no—
164C42×C61—21no—
165C22⋊Dic61—42no—
166C22⋊Dic61—42no—
167C22⋊Dic61—81no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—42no—
177C4⋊Dic61#4112yes—
178C4⋊Dic61#4112yes—
179#11758196—42no—
180#11758196—81no—
181#11758196—81no—
182#11758196—81no—
183#11758196—81no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—42no—
189C4×Dic61—81no—
190C4×Dic61—81no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C23×C61—22no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61C22×D411yescharacteristic
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61—41no—
208Q8×C61—41no—
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—41no—
212D4×C61—41no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
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×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×C4×C612−1+411yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
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—
256C2×Dic61—84no—
257C2×Dic61#13512yes—
258C2×Dic61#13712yes—
259C8×C61—41no—
260C8×C61—41no—
261#92418—41no—
262#92418—41no—
263#26763—611nomaximal, Sylow
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61#52511yescharacteristic
272C22×C61#54611yescharacteristic
273C22×C61#54811yescharacteristic
274C22×C61C2×#13511yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57611yescharacteristic
277C22×C61#57711yescharacteristic
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—42no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61#52911yescharacteristic
284C4×C61#53911yescharacteristic
285C4×C61#55211yescharacteristic
286C4×C61#55311yescharacteristic
287Dic61—22no—
288Dic61—22no—
289Dic61—42no—
290Dic61—41no—
291Dic61—41no—
292Dic61—42no—
293Dic61—81no—
294Dic61—81no—
295#2507—611no—
296#2367—611no—
297#2071—611no—
298#2067—611no—
299#2042—611no—
300#2042D6111yescharacteristic
301#1933—611no—
302#1928—611no—
303#1919—611no—
304#1918—611no—
305#1916—611no—
306#1865—611no—
307#1864—611no—
308#1803—611no—
309C4×#102—611no—
310C2×C61—82no—
311C2×C61#246011yescharacteristic
312C2×C61#310711yescharacteristic
313C2×C61#311911yescharacteristic
314C2×C61#330111yescharacteristic
315C2×C61#331411yescharacteristic
316C2×C61#336111yescharacteristic
317C2×C61#336911yescharacteristic
318C2×#124C2×D6111yescharacteristic
319C2×C4×Q8—611no—
320C2×C4×D4—611no—
321C2×#107—611no—
322C2×#106—611no—
323C2×C4⋊C8—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103C2×D6111yescharacteristic
327C2×#103C2×D6111yescharacteristic
328C2×#102—611no—
329C2×#102—611no—
330C2×#102C2×D6111yescharacteristic
331C2×#102C2×D6111yescharacteristic
332C2×C4×C8—611no—
333#398—1221no—
334#398—1221no—
335#391—611no—
336#390—611no—
337#389—611no—
338#389—611no—
339#389—611no—
340#389—1221no—
341#389C2×D6111yescharacteristic
342#388—1221no—
343#388—1221no—
344#387—1221no—
345#385—1221no—
346#384—611no—
347C4×C4⋊C4—611no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—611no—
350C4×C22⋊C4—1221no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#336—611no—
358#336—611no—
359#336—611no—
360#336C2×D6111yescharacteristic
361C61#2676311yescharacteristic, complemented, Sylow
362C22×Q8C22×D6111yescharacteristic
363C22×D4C22×D6111yescharacteristic
364C23×C4—1221no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369#124—22no—
370#124—22no—
371C4×Q8—1222no—
372C4×Q8—2441no—
373C4×D4—1222no—
374C4×D4—2441no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401#107—1222no—
402#106—1222no—
403C4⋊C8—2441no—
404#103—22no—
405#103—41no—
406#103—1222no—
407#103—1222no—
408#102—22no—
409#102—41no—
410#102—1222no—
411#102—1222no—
412C4×C8—2441no—
413#95—21no—
414#95—611no—
415#95—611no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423C24—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1175822412yes—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4#1175822412yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—4881no—
448C22×C4—2442no—
449C22×C4—2442no—
450C22×C4D4×D6111yescharacteristic
451C22×C4D4×D6111yescharacteristic
452C22×C4C23×D6111yescharacteristic
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—81no—
470C4⋊C4—612no—
471C4⋊C4—612no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—4881no—
481C4⋊C4—4881no—
482C4⋊C4—4881no—
483C4⋊C4—4881no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—4881no—
489C22⋊C4—2442no—
490C22⋊C4—2442no—
491C42—21no—
492C42—21no—
493C42—1221no—
494C42—1221no—
495C42—1221no—
496C42—1221no—
497C42—4881no—
498C42—4881no—
499C42—2442no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C2×D4×D6111yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
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—612no—
532C2×C4—612no—
533C2×C4—1221no—
534C2×C4—1221no—
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—1222no—
544C2×C4—1222no—
545C2×C4—4884no—
546C2×C4—2441no—
547C2×C4—2441no—
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#41×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C2×C4C2×#1175822411yescharacteristic
559C2×C4C2×#1175822411yescharacteristic
560C2×C4#1451982211yescharacteristic
561C2×C4#1451982411yescharacteristic
562C8—41no—
563C8—41no—
564C8—2441no—
565C8—2441no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22#1684026811yescharacteristic
574C22#1684040911yescharacteristic
575C22#1684042011yescharacteristic
576C22#135×D6111yescharacteristic
577C22#137×D6111yescharacteristic
578C22#1684056411yescharacteristic
579C22#1684057211yescharacteristic
580C4—41no—
581C4—42no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—1222no—
586C4—1222no—
587C4—4881no—
588C4—4881no—
589C4—2442no—
590C4—2441no—
591C4—2441no—
592C4—2442no—
593C4#1684028911yescharacteristic
594C4#1684035711yescharacteristic
595C4#1684044911yescharacteristic
596C4#1684045511yescharacteristic
597C2—82no—
598C2#2144976411yescharacteristic
599C2#2145588311yescharacteristic
600C2#2145601111yescharacteristic
601C2#2145772811yescharacteristic
602C2#2145790711yescharacteristic
603C2#2145847611yescharacteristic
604C2#2145855611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2042C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2042D61
hypocenterC61#26763
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
1. #21450344C2semidirect
6. #21446492C2semidirect
7. #21446462C2semidirect
8. #21446394C2semidirect
9. #21446391C2semidirect
10. #21446375C2semidirect
11. #21446085C2semidirect
13. #21445822C2semidirect
361. C61#26763semidirect

Maximal subgroups

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

subgroupindex
#214503442
#214488942
#214472252
#214472052
#214470552
#214464922
#214464622
#214463942
#214463912
#214463752
#214460852
#214460782
#214458222
#102×Dic612
C61×#20422
#2676361