Finite Group Databasebeta

← #29731812

Subgroups — #29731812

Order 15616.

756 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840171C211yesmaximal, characteristic, complemented
2C2×#16840062C211yesmaximal, characteristic
3#21450992C211yesmaximal, characteristic
4#21448915C211yesmaximal, characteristic, complemented
5#21447306C211yesmaximal, characteristic, complemented
6#21447305C211yesmaximal, characteristic, complemented
7#21447205C211yesmaximal, characteristic
8#21447191C211yesmaximal, characteristic
9#21446771C211yesmaximal, characteristic, complemented
10#21446590C211yesmaximal, characteristic, complemented
11#21446586C211yesmaximal, characteristic, complemented
12#21446542C211yesmaximal, characteristic, complemented
13#21446448C211yesmaximal, characteristic
14#21446441C211yesmaximal, characteristic
15C61×#1762C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18#16840171—24no—
19#16840062—24no—
20C2×#14519711C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31#16839674—21no—
32#16839674—21no—
33#16839665C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839659—21no—
36#16839658C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839649—21no—
39#16839646—21no—
40#16839627C2211yescharacteristic
41#16839627C2211yescharacteristic
42#16839622—21no—
43#16839622—21no—
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519777—22no—
65#14519777—22no—
66#14519769—22no—
67#14519769D414yes—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519711—22no—
88#14519711—22no—
89#14519710—22no—
90#14519709—22no—
91#14519709—22no—
92#14519708—22no—
93C4⋊#92418—22no—
94#14519706—22no—
95#14519706—22no—
96#14519698—22no—
97#14519698—22no—
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
109C23×C4×C61—21no—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C42×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#102×C61—22no—
121#102×C61Q814yescomplemented
122#97×C61—22no—
123#95×C61—21no—
124C24×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61C2×Q812yes—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×Dic122—42no—
149C2×Dic122—81no—
150C2×Dic122—81no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61C2×Q812yes—
156C2×C8×C61#4112yes—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61C2×Q812yes—
163C22⋊C4×C61—42no—
164C42×C61—81no—
165C42×C61C2×D412yes—
166C42×C61#4112yes—
167C22⋊Dic61—81no—
168C22⋊Dic61—81no—
169C22⋊Dic61—81no—
170C22⋊Dic61—81no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—41no—
178C4⋊Dic61—41no—
179C4⋊Dic61—42no—
180C4⋊Dic61C2×D412yes—
181C4⋊Dic61C2×D412yes—
182#11758196—24no—
183#11758196—24no—
184#11758196—42no—
185#11758196—81no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
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—22no—
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—84no—
224C2×C4×C61#12112yes—
225C2×C4×C61#12112yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×Q811yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×C4×C612−1+411yescharacteristic
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
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—
256Dic122—84no—
257C8×C61—22no—
258C8×C61—22no—
259#92418—41no—
260#92418—41no—
261#25466—611nomaximal, Sylow
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61C2×#12111yescharacteristic
270C22×C61#54711yescharacteristic
271C22×C61D4×Q811yescharacteristic
272C22×C61#57511yescharacteristic
273C22×C61#57511yescharacteristic
274C22×C61#57611yescharacteristic
275C22×C61#57711yescharacteristic
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61#53311yescharacteristic
282C4×C61#54111yescharacteristic
283C4×C61#54811yescharacteristic
284C4×C61D4×Q811yescharacteristic
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288Dic61—41no—
289Dic61—42no—
290Dic61—42no—
291Dic61—81no—
292Dic61—81no—
293C2×#501—611no—
294C2×#478—611no—
295#2563—611no—
296#2370—611no—
297#2085—611no—
298#2085—611no—
299#2067—611no—
300#2065—611no—
301#1988—611no—
302#1949—611no—
303#1949—611no—
304#1940—611no—
305#1926—611no—
306#1925—611no—
307#1762D6111yescharacteristic
308C2×C61—82no—
309C2×C61#242911yescharacteristic
310C2×C61#304011yescharacteristic
311C2×C61#304511yescharacteristic
312C2×C61#334711yescharacteristic
313C2×C61#334811yescharacteristic
314C2×C61#338811yescharacteristic
315C2×C61#338911yescharacteristic
316C2×C4⋊Q8—611no—
317C2×#125—611no—
318C2×C4×D4C2×D6111yescharacteristic
319#501—1224no—
320#478—1224no—
321C2×#107—611no—
322C2×#107—611no—
323C2×#107—611no—
324C2×#106—611no—
325C2×#106—611no—
326C2×#106—611no—
327C2×C4⋊C8—611no—
328C2×C4⋊C8—611no—
329C2×C4⋊C8—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#102C2×D6111yescharacteristic
333C2×#102C2×D6111yescharacteristic
334C2×#97C2×D6111yescharacteristic
335#393—1221no—
336#393—1221no—
337#390—611no—
338#390—611no—
339#389—611no—
340#389—611no—
341#389—1221no—
342#387—1221no—
343#387—1221no—
344#382—611no—
345#382—611no—
346#380—1221no—
347#380—1221no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—1221no—
352C4×C4⋊C4C2×D6111yescharacteristic
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340C2×D6111yescharacteristic
358#340C2×D6111yescharacteristic
359#336—611no—
360#336—611no—
361C61#2546611yescharacteristic, complemented, Sylow
362C22×Q8—1221no—
363C22×D4C22×D6111yescharacteristic
364C23×C4—21no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369C4⋊Q8—1222no—
370C4⋊Q8—1222no—
371#125—614no—
372#125—1222no—
373C4×D4—22no—
374C4×D4—22no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C42—21no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42C22×D6111yescharacteristic
400#107—1222no—
401#107—1222no—
402#107—1222no—
403#106—1222no—
404#106—1222no—
405#106—1222no—
406C4⋊C8—1222no—
407C4⋊C8—1222no—
408C4⋊C8—1222no—
409#102—22no—
410#102—1222no—
411#102—1222no—
412#102Dic12214yescomplemented
413#97—22no—
414#95—21no—
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—4881no—
427C2×Q8—4881no—
428C2×Q8—2442no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×Dic12212yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
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×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4C23×D6111yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8C2×Dic12212yes—
463C2×C8#1175822212yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—612no—
470C4⋊C4—612no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1224no—
474C4⋊C4—1224no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—4881no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2442no—
483C4⋊C4—2442no—
484C4⋊C4C2×Dic12212yes—
485C22⋊C4—42no—
486C22⋊C4—4881no—
487C22⋊C4—4881no—
488C22⋊C4—4881no—
489C22⋊C4—4881no—
490C42—81no—
491C42—1221no—
492C42—1221no—
493C42—1221no—
494C42—1221no—
495C42—4881no—
496C42—2442no—
497C42#1175822212yes—
498C42D4×D6112yes—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C2×D4×D6111yescharacteristic
507Q8—4884no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4#1451975012yes—
551C2×C4#1451975012yes—
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C22×Dic12211yescharacteristic
554C2×C4C2×#1175822211yescharacteristic
555C2×C4C2×#1175822211yescharacteristic
556C2×C4#1451982411yescharacteristic
557C2×C4#1451982411yescharacteristic
558C8—22no—
559C8—22no—
560C8—2441no—
561C8—2441no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1451975011yescharacteristic
570C22#1684041711yescharacteristic
571C22D4×Dic12211yescharacteristic
572C22#1684055811yescharacteristic
573C22#1684055811yescharacteristic
574C22#1684056611yescharacteristic
575C22#1684057111yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4—4881no—
582C4—4881no—
583C4—2441no—
584C4—2441no—
585C4—2441no—
586C4—2441no—
587C4—2442no—
588C4—2442no—
589C4#1684031911yescharacteristic
590C4#1684037411yescharacteristic
591C4#1684042111yescharacteristic
592C4D4×Dic12211yescharacteristic
593C2—82no—
594C2#2144941811yescharacteristic
595C2#2145531211yescharacteristic
596C2#2145535711yescharacteristic
597C2#2145832411yescharacteristic
598C2#2145833911yescharacteristic
599C2#2145876911yescharacteristic
600C2#2145877811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1762D61
hypocenterC61#25466
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
1. C2×#16840171C2semidirect
4. #21448915C2semidirect
5. #21447306C2semidirect
6. #21447305C2semidirect
9. #21446771C2semidirect
10. #21446590C2semidirect
11. #21446586C2semidirect
12. #21446542C2semidirect
121. #102×C61Q8semidirect
361. C61#25466semidirect
412. #102Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168401712
C2×#168400622
#214509922
#214489152
#214473062
#214473052
#214472052
#214471912
#214467712
#214465902
#214465862
#214465422
#214464482
#214464412
C61×#17622
#2546661