Finite Group Databasebeta

← #29688124

Subgroups — #29688124

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840120C211yesmaximal, characteristic
2#21448563C211yesmaximal, characteristic, complemented
3#21448531C211yesmaximal, characteristic
4#21446778C211yesmaximal, characteristic, complemented
5#21446629C211yesmaximal, characteristic
6#21446518C211yesmaximal, characteristic
7#21446507C211yesmaximal, characteristic, complemented
8#21446499C211yesmaximal, characteristic
9#21446496C211yesmaximal, characteristic, complemented
10#21445951C211yesmaximal, characteristic, complemented
11#21445652C211yesmaximal, characteristic, complemented
12#21445644C211yesmaximal, characteristic
13C4×#14519702C211yesmaximal, characteristic, complemented
14#21445505C211yesmaximal, characteristic, complemented
15C4×#102×C61C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19#16840120C418yescomplemented
20C2×#14519702C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519688C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839668—21no—
32#16839667C2211yescharacteristic
33#16839666—21no—
34#16839665C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839650—21no—
38#16839650—21no—
39#16839649—21no—
40#16839646—21no—
41#16839635—21no—
42#16839635—21no—
43#16839631C2211yescharacteristic
44#16839627C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46#16839608C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×Dic122—21no—
61C22×Dic122C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519776—22no—
65#14519776C2×C414yescomplemented
66#14519764—22no—
67#14519764C2×C414yes—
68C4×Dic122—22no—
69C4×Dic122—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
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#14519702—22no—
88#14519702C2×C414yescomplemented
89#14519700—22no—
90#14519700C2×C414yescomplemented
91#14519698—22no—
92#14519698C2×C414yes—
93#14519688—22no—
94#14519687—22no—
95#14519687C2×C414yescomplemented
96C8×Dic61—22no—
97C8×Dic61—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684D411yescharacteristic
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C23×C4×C61—21no—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×D4×C61—22no—
114C4×D4×C61—22no—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#102×C61—22no—
122#102×C61C2×C414yes—
123C4×C8×C61—22no—
124#95×C61—21no—
125C24×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C22×C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—42no—
136C22×C4×C61—81no—
137C22×C4×C61—81no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—24no—
155C2×Dic122—81no—
156C2×Dic122—81no—
157C2×Dic122C22×C412yes—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C22×C412yes—
163C2×C8×C61#4112yes—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C22×C412yes—
170C22⋊C4×C61—42no—
171C42×C61—81no—
172C42×C61#4112yes—
173C42×C61#4112yes—
174C22⋊Dic61—24no—
175C22⋊Dic61—24no—
176C22⋊Dic61—81no—
177C22⋊Dic61—81no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—81no—
184C4⋊Dic61C22×C412yes—
185#11758196—24no—
186#11758196—42no—
187#11758196—81no—
188#11758196—81no—
189#11758196—81no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—42no—
193C4×Dic61—81no—
194C4×Dic61C22×C412yes—
195C4×Dic61#4112yes—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418C22×C412yes—
201C2×#92418#4112yes—
202C23×C61—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61C22×D411yescharacteristic
210D4×C61—22no—
211D4×C61—22no—
212D4×C61—41no—
213D4×C61—41no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
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—84no—
228C2×C4×C61#12412yes—
229C2×C4×C61#12412yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
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—21no—
247C2×Dic61—21no—
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—41no—
257C2×Dic61—41no—
258C2×Dic61C4×D412yes—
259C2×Dic61C4×D412yes—
260Dic122—22no—
261Dic122—22no—
262Dic122—41no—
263Dic122—41no—
264Dic122—84no—
265C8×C61—22no—
266C8×C61—22no—
267#92418—22no—
268#92418—22no—
269#21654—611nomaximal, Sylow
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61C2×C4×D411yescharacteristic
278C22×C61C2×#12411yescharacteristic
279C22×C61#53211yescharacteristic
280C22×C61#57611yescharacteristic
281C22×C61#57711yescharacteristic
282C22×C61#57711yescharacteristic
283C22×C61#57811yescharacteristic
284C4×C61—22no—
285C4×C61—22no—
286C4×C61—42no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61C4×#4111yescharacteristic
290C4×C61C4×#4111yescharacteristic
291C4×C61#52911yescharacteristic
292C4×C61#52911yescharacteristic
293Dic61—22no—
294Dic61—22no—
295Dic61—41no—
296Dic61—41no—
297Dic61—42no—
298Dic61—42no—
299Dic61—81no—
300Dic61—81no—
301C2×#489—611no—
302#2331—611no—
303#2326—611no—
304#1990—611no—
305#1958—611no—
306#1936—611no—
307#1935—611no—
308#1934—611no—
309#1933—611no—
310#1836—611no—
311#1763—611no—
312#1762—611no—
313C4×#103—611no—
314C4×#102D6111yescharacteristic
315#1726—611no—
316C2×C61—82no—
317C2×C61C4×#12411yescharacteristic
318C2×C61#294611yescharacteristic
319C2×C61#294711yescharacteristic
320C2×C61#313611yescharacteristic
321C2×C61#313711yescharacteristic
322C2×C61#317311yescharacteristic
323C2×C61#317511yescharacteristic
324C2×C4⋊Q8—611no—
325C2×#124—611no—
326C2×C4×Q8—611no—
327C2×C4×D4C2×D6111yescharacteristic
328#489—618no—
329C2×#103—611no—
330C2×#103—611no—
331C2×#103—611no—
332C2×#103—611no—
333C2×#102—611no—
334C2×#102—611no—
335C2×#102C2×D6111yescharacteristic
336C2×#102C2×D6111yescharacteristic
337C2×#97—611no—
338C2×#97—611no—
339C2×#97—611no—
340C2×C4×C8—611no—
341C2×C4×C8—611no—
342C2×C4×C8C2×D6111yescharacteristic
343#391—611no—
344#391—1221no—
345#391—1221no—
346#390—611no—
347#389—611no—
348#389—611no—
349#388—1221no—
350#388—1221no—
351#387—1221no—
352#387—1221no—
353#385—1221no—
354#385—1221no—
355#384—611no—
356#382—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4C2×D6111yescharacteristic
359#376—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340C2×D6111yescharacteristic
367#340C2×D6111yescharacteristic
368C61#2165411yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×Q8—1221no—
371C22×D4C22×D6111yescharacteristic
372C23×C4—21no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377C4⋊Q8—614no—
378C4⋊Q8—1222no—
379#124—614no—
380#124—1222no—
381C4×Q8—1222no—
382C4×Q8—1222no—
383C4×D4—22no—
384C4×D4—22no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—21no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42C22×D6111yescharacteristic
408#103—614no—
409#103—614no—
410#103—1222no—
411#103—1222no—
412#102—22no—
413#102—614no—
414#102—1222no—
415#102C4×D6114yes—
416#97—614no—
417#97—1222no—
418#97—1222no—
419C4×C8—22no—
420C4×C8—1222no—
421C4×C8—1222no—
422#95—21no—
423#95—611no—
424#95—611no—
425#95—611no—
426#95—611no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434C24—21no—
435C2×Q8—612no—
436C2×Q8—1224no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—4881no—
443C2×Q8—4881no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C2×C4×D6112yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4D4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466C22×C4C23×D6111yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—612no—
472C2×C8—612no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8C2×C4×D6112yes—
478C2×C8#1175822212yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—612no—
485C4⋊C4—1224no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—4881no—
492C4⋊C4—4881no—
493C4⋊C4—4881no—
494C4⋊C4—4881no—
495C4⋊C4—2442no—
496C4⋊C4C2×C4×D6112yes—
497C22⋊C4—42no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C22⋊C4—4881no—
501C22⋊C4—4881no—
502C42—81no—
503C42—612no—
504C42—612no—
505C42—1221no—
506C42—1221no—
507C42—4881no—
508C42—2442no—
509C42#1175822212yes—
510C42#1175822412yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D6111yescharacteristic
519Q8—1222no—
520Q8—1222no—
521Q8—4884no—
522Q8—2441no—
523Q8—2441no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
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—84no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
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#1451976312yes—
567C2×C4#1451976312yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4#41×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C2×C4C2×#1175822411yescharacteristic
572C2×C4C2×#1175822211yescharacteristic
573C2×C4C2×#1175822211yescharacteristic
574C8—22no—
575C8—22no—
576C8—1222no—
577C8—1222no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C4×D4×D6111yescharacteristic
586C22C2×#1451976311yescharacteristic
587C22#1684030911yescharacteristic
588C22#1684056311yescharacteristic
589C22#1684056811yescharacteristic
590C22#1684056911yescharacteristic
591C22#1684057611yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—1222no—
598C4—1222no—
599C4—4881no—
600C4—4881no—
601C4—2441no—
602C4—2442no—
603C4—2441no—
604C4—2442no—
605C4C4×#1175822411yescharacteristic
606C4C4×#1175822211yescharacteristic
607C4#1684029011yescharacteristic
608C4#1684029211yescharacteristic
609C2—82no—
610C2C4×#1451976311yescharacteristic
611C2#2145464411yescharacteristic
612C2#2145465511yescharacteristic
613C2#2145617111yescharacteristic
614C2#2145618811yescharacteristic
615C2#2145643211yescharacteristic
616C2#2145645411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC4×#102D61
hypocenterC61#21654
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #21448563C2semidirect
4. #21446778C2semidirect
7. #21446507C2semidirect
9. #21446496C2semidirect
10. #21445951C2semidirect
11. #21445652C2semidirect
13. C4×#14519702C2semidirect
14. #21445505C2semidirect
19. #16840120C4semidirect
65. #14519776C2×C4semidirect
88. #14519702C2×C4semidirect
90. #14519700C2×C4semidirect
95. #14519687C2×C4semidirect
368. C61#21654semidirect

Maximal subgroups

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

subgroupindex
C2×#168401202
#214485632
#214485312
#214467782
#214466292
#214465182
#214465072
#214464992
#214464962
#214459512
#214456522
#214456442
C4×#145197022
#214455052
C4×#102×C612
#2165461