Finite Group Databasebeta

← #29721295

Subgroups — #29721295

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840069C211yesmaximal, characteristic, complemented
2C2×#16839818C211yesmaximal, characteristic
3#21454130C211yesmaximal, characteristic
4C4×#14519753C211yesmaximal, characteristic
5#21447258C211yesmaximal, characteristic
6#21447251C211yesmaximal, characteristic
7#21447020C211yesmaximal, characteristic
8#21447013C211yesmaximal, characteristic
9#21446774C211yesmaximal, characteristic, complemented
10#21446585C211yesmaximal, characteristic, complemented
11#21446561C211yesmaximal, characteristic, complemented
12#21446556C211yesmaximal, characteristic, complemented
13#21446511C211yesmaximal, characteristic, complemented
14#21446511C211yesmaximal, characteristic, complemented
15C61×#1805C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×#14519753C2211yescharacteristic
18C2×C42×D61C2211yescharacteristic
19#16840069—24no—
20#16839818—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839647—21no—
37#16839647—21no—
38#16839643—21no—
39#16839636—21no—
40#16839636—21no—
41#16839633—21no—
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44#16839619C2211yescharacteristic
45C4×#11758196—21no—
46C4×C4⋊Dic61C2211yescharacteristic
47C4×#11758198—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#382C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519769—22no—
67#14519769—22no—
68#14519753—22no—
69#14519753—22no—
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×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61—22no—
89C42×D61—22no—
90#14519710—22no—
91#14519710—22no—
92#14519708—22no—
93#14519708—22no—
94C4⋊#92418D414yes—
95#14519701—22no—
96#14519701—22no—
97#14519689—22no—
98#14519689—22no—
99#14519687—22no—
100#14519687—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110C22×Q8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×Q8×C61—22no—
114C4×Q8×C61—22no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121C4⋊C8×C61—22no—
122#103×C61—22no—
123#103×C61—22no—
124#95×C61—21no—
125#95×C61—21no—
126C23×D61C2×D411yescharacteristic
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61#4112yes—
133C22×C4×C61—21no—
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×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—42no—
172C42×C61—42no—
173C42×C61C2×D412yes—
174C42×C61#4112yes—
175#11758198—24no—
176#11758198—42no—
177#11758198—42no—
178#11758198—42no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—42no—
187C4⋊Dic61—42no—
188C4⋊Dic61#4112yes—
189#11758196—24no—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—42no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C2×D412yes—
203C2×#92418C2×D412yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61#13712yes—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
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×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61#12112yes—
233C2×C4×C61#12112yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
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—41no—
261C4×D61—44no—
262C4×D61—44no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—41no—
267C8×C61—41no—
268C8×C61—41no—
269#92418—22no—
270#92418—22no—
271#24707—611nomaximal, Sylow
272C22×C61C2×#12111yescharacteristic
273C22×C61#52511yescharacteristic
274C22×C61#53611yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57511yescharacteristic
278C22×C61#57511yescharacteristic
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61—81no—
292C4×C61#54711yescharacteristic
293C4×C61#54811yescharacteristic
294C4×C61#55211yescharacteristic
295C4×C61#55611yescharacteristic
296Dic61—42no—
297Dic61—41no—
298Dic61—41no—
299Dic61—41no—
300Dic61—81no—
301Dic61—81no—
302Dic61—81no—
303C2×#479—611no—
304C2×#432—611no—
305#2851—611no—
306C4×C22⋊Q8—611no—
307#2077—611no—
308#2076—611no—
309#2036—611no—
310#2035—611no—
311#1989—611no—
312#1949—611no—
313#1944—611no—
314#1943—611no—
315#1935—611no—
316#1935—611no—
317#1805D6111yescharacteristic, complemented
318C2×C61#241511yescharacteristic
319C2×C61#305411yescharacteristic
320C2×C61#305611yescharacteristic
321C2×C61#311111yescharacteristic
322C2×C61#311711yescharacteristic
323C2×C61#323611yescharacteristic
324C2×C61#324011yescharacteristic
325D61—42no—
326C2×#125—611no—
327C2×C22⋊Q8—611no—
328C2×C4×Q8C2×D6111yescharacteristic
329C22×C42—611no—
330#479—1224no—
331#432—1224no—
332C2×#107—611no—
333C2×#107—611no—
334C2×#106—611no—
335C2×#106—611no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8C2×D6111yescharacteristic
338C2×#103—611no—
339C2×#103—611no—
340C2×#103C2×D6111yescharacteristic
341C2×#103C2×D6111yescharacteristic
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344C2×#97—611no—
345C2×#97—611no—
346#391—611no—
347#389—611no—
348#389—611no—
349#389—611no—
350#387—1221no—
351#387—1221no—
352#387—1221no—
353#385—1221no—
354#385—1221no—
355#385—1221no—
356#382C2×D6111yescharacteristic
357#380—611no—
358#380—611no—
359#380—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—1221no—
362C4×C22⋊C4—1221no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340C2×D6111yescharacteristic
370#340C2×D6111yescharacteristic
371C61#2470711yescharacteristic, complemented, Sylow
372C22×Q8C22×D6111yescharacteristic
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—611no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380#125—1222no—
381#125—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C4×Q8—22no—
385C4×Q8—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C42—21no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1222no—
409C2×C42—1222no—
410C2×C42—1221no—
411C2×C42C22×D6111yescharacteristic
412#107—1222no—
413#107—1222no—
414#106—1222no—
415#106—1222no—
416C4⋊C8—22no—
417C4⋊C8—614no—
418#103—22no—
419#103—22no—
420#103—1222no—
421#103—1222no—
422C22⋊C8—1222no—
423C22⋊C8—1222no—
424#97—1222no—
425#97—1222no—
426#95—21no—
427#95—21no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437C24—611no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8#1175822412yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—2441no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4D4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—612no—
477C2×C8—612no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—612no—
490C4⋊C4—1224no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C22⋊C4—1224no—
504C22⋊C4—2442no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C42—42no—
508C42—1221no—
509C42—1221no—
510C42—1222no—
511C42—1222no—
512C42—2442no—
513C42#1175822212yes—
514C42D4×D6112yes—
515C23—612no—
516C23—1221no—
517C23—1221no—
518C23—1221no—
519C23—1221no—
520C23—1221no—
521C23—1221no—
522C23C2×D4×D6111yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4#1451975012yes—
571C2×C4#1451975012yes—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C2×C4#1451982411yescharacteristic
577C2×C4#1451982511yescharacteristic
578C8—41no—
579C8—41no—
580C8—1222no—
581C8—1222no—
582C22—1222no—
583C22—1222no—
584C22—1222no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22—2441no—
589C22C2×#1451975011yescharacteristic
590C22#1684027211yescharacteristic
591C22#1684033511yescharacteristic
592C22#137×D6111yescharacteristic
593C22#1684055911yescharacteristic
594C22#1684055911yescharacteristic
595C22#1684056111yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—4881no—
603C4—4881no—
604C4—4881no—
605C4—2441no—
606C4—2442no—
607C4—2441no—
608C4—2441no—
609C4#1684041411yescharacteristic
610C4#1684041811yescharacteristic
611C4#1684044711yescharacteristic
612C4#1684048011yescharacteristic
613C2—2442no—
614C2#2144925011yescharacteristic
615C2#2145542711yescharacteristic
616C2#2145544711yescharacteristic
617C2#2145592711yescharacteristic
618C2#2145598911yescharacteristic
619C2#2145702311yescharacteristic
620C2#2145706411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1805D61
hypocenterC61#24707
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. C2×#16840069C2semidirect
9. #21446774C2semidirect
10. #21446585C2semidirect
11. #21446561C2semidirect
12. #21446556C2semidirect
13. #21446511C2semidirect
14. #21446511C2semidirect
15. C61×#1805C2semidirect
317. #1805D61semidirect
371. C61#24707semidirect

Maximal subgroups

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

subgroupindex
C2×#168400692
C2×#168398182
#214541302
C4×#145197532
#214472582
#214472512
#214470202
#214470132
#214467742
#214465852
#214465612
#214465562
#214465112
C61×#18052
#2470761