Finite Group Databasebeta

← #29721376

Subgroups — #29721376

Order 15616.

746 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839822C211yesmaximal, characteristic
2#21451641C211yesmaximal, characteristic
3#21448495C211yesmaximal, characteristic
4#21447248C211yesmaximal, characteristic
5#21447243C211yesmaximal, characteristic
6#21447021C211yesmaximal, characteristic
7#21447014C211yesmaximal, characteristic
8#21446772C211yesmaximal, characteristic, complemented
9#21446589C211yesmaximal, characteristic, complemented
10#21446560C211yesmaximal, characteristic, complemented
11#21446550C211yesmaximal, characteristic, complemented
12#21446506C211yesmaximal, characteristic, complemented
13#21446492C211yesmaximal, characteristic, complemented
14#21445834C211yesmaximal, characteristic, complemented
15C2×C61×#477C211yesmaximal, characteristic, complemented
16C2×#14519753C2211yescharacteristic
17C2×#14519734C2211yescharacteristic
18#16839822—24no—
19C2×#14519711C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519689C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28#16839700—21no—
29#16839697—21no—
30#16839681—21no—
31#16839673—21no—
32#16839669C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839657—21no—
37#16839653—21no—
38#16839652—21no—
39#16839646—21no—
40#16839628C2211yescharacteristic
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#477—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519753—22no—
65#14519753—22no—
66#14519734—22no—
67#14519734—22no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196D411yescharacteristic
77C2×#11758196D411yescharacteristic
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198D411yescharacteristic
83C2×#11758198D411yescharacteristic
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86#14519711—22no—
87#14519709—22no—
88C4⋊#92418—22no—
89#14519701—22no—
90#14519701—22no—
91#14519689—22no—
92#14519689—22no—
93#14519687—22no—
94C8×Dic61—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106C22×Q8×C61—21no—
107C22×Q8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C4⋊Q8×C61—22no—
111C4⋊Q8×C61—22no—
112C4×Q8×C61—22no—
113C4×Q8×C61—22no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#107×C61—22no—
122#107×C61—22no—
123C4⋊C8×C61D414yes—
124#103×C61—22no—
125#103×C61—22no—
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—42no—
133C2×Q8×C61—81no—
134C2×Q8×C61—81no—
135C2×Q8×C61#4112yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
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×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C2×D412yes—
161C2×C8×C61C2×D412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—42no—
169C4⋊C4×C61—42no—
170C42×C61—42no—
171C42×C61C2×D412yes—
172C42×C61#4112yes—
173#11758198—24no—
174#11758198—22no—
175#11758198—22no—
176#11758198—81no—
177#11758198—81no—
178#11758198—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61#4112yes—
187#11758196—24no—
188#11758196—22no—
189#11758196—22no—
190#11758196—81no—
191#11758196—81no—
192#11758196—81no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—81no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C23×C61C22×D411yescharacteristic
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210Q8×C61—22no—
211Q8×C61—22no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—84no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
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×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61#12112yes—
232C2×C4×C61#12112yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C612−1+411yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
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—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—41no—
264C8×C61—22no—
265C8×C61—22no—
266#92418—41no—
267#92418—41no—
268#24714—611nomaximal, Sylow
269C22×C61C2×#12111yescharacteristic
270C22×C61#52511yescharacteristic
271C22×C61#53611yescharacteristic
272C22×C61#57611yescharacteristic
273C22×C61#57711yescharacteristic
274C22×C61#57811yescharacteristic
275C22×C61#57811yescharacteristic
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61#54711yescharacteristic
290C4×C61#54811yescharacteristic
291C4×C61#55211yescharacteristic
292C4×C61#55611yescharacteristic
293Dic61—42no—
294Dic61—41no—
295Dic61—41no—
296Dic61—41no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300C2×#477D6111yescharacteristic, complemented
301C2×#433—611no—
302#2628—611no—
303#2323—611no—
304#2076—611no—
305#2075—611no—
306#2036—611no—
307#2035—611no—
308#1988—611no—
309#1949—611no—
310#1944—611no—
311#1942—611no—
312#1935—611no—
313#1933—611no—
314#1805—611no—
315C2×C61#241511yescharacteristic
316C2×C61#305911yescharacteristic
317C2×C61#306211yescharacteristic
318C2×C61#312111yescharacteristic
319C2×C61#312411yescharacteristic
320C2×C61#324211yescharacteristic
321C2×C61#324311yescharacteristic
322D61—81no—
323C2×C4⋊Q8C2×D6111yescharacteristic
324C2×C22⋊Q8—611no—
325C2×C4×Q8C2×D6111yescharacteristic
326C2×#117—611no—
327#477—24no—
328#433—1224no—
329C2×#107—611no—
330C2×#107C2×D6111yescharacteristic
331C2×#107C2×D6111yescharacteristic
332C2×#106—611no—
333C2×C4⋊C8—611no—
334C2×C4⋊C8C2×D6111yescharacteristic
335C2×#103—611no—
336C2×#103—611no—
337C2×#103C2×D6111yescharacteristic
338C2×#103C2×D6111yescharacteristic
339C2×C22⋊C8—611no—
340C2×C22⋊C8—611no—
341C2×#97—611no—
342C2×C4×C8—611no—
343#400—1221no—
344#399—1221no—
345#395—1221no—
346#393—1221no—
347#391—611no—
348#389—611no—
349#389—611no—
350#389—611no—
351#389—1221no—
352#388—1221no—
353#388—1221no—
354#387—1221no—
355#382—611no—
356#380—611no—
357#380—611no—
358#380—611no—
359C4×C4⋊C4—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368C61#2471411yescharacteristic, complemented, Sylow
369C22×Q8—21no—
370C22×Q8C22×D6111yescharacteristic
371C23×C4—611no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C4⋊Q8—22no—
377C4⋊Q8—22no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C4×Q8—22no—
381C4×Q8—22no—
382#117—1222no—
383#117—1222no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C22⋊C4—611no—
400C2×C22⋊C4—611no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C42—21no—
406C2×C42—611no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409#107—22no—
410#107—22no—
411#107—1222no—
412#106—1222no—
413C4⋊C8—1222no—
414C4⋊C8C22⋊D6114yes—
415#103—22no—
416#103—22no—
417#103—1222no—
418#103—1222no—
419C22⋊C8—1222no—
420C22⋊C8—1222no—
421#97—1222no—
422C4×C8—1222no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434C24—611no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—81no—
442C2×Q8—81no—
443C2×Q8#1175822612yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8C2×C22⋊D6112yes—
475C2×C8C2×C22⋊D6112yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—612no—
485C4⋊C4—1224no—
486C4⋊C4—1221no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—4881no—
496C4⋊C4—4881no—
497C4⋊C4—4881no—
498C22⋊C4—1224no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—4881no—
502C22⋊C4—4881no—
503C22⋊C4—4881no—
504C42—42no—
505C42—1221no—
506C42—1221no—
507C42—4881no—
508C42#1175822412yes—
509C42C2×C22⋊D6112yes—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—1221no—
514C23—1221no—
515C23—1221no—
516C23—1221no—
517C23C2×D4×D6111yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—84no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
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—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4#1451974812yes—
565C2×C4#1451974812yes—
566C2×C4C22×C22⋊D6111yescharacteristic
567C2×C4#41×D6111yescharacteristic
568C2×C4C2×#1175822411yescharacteristic
569C2×C4C2×#1175822611yescharacteristic
570C2×C4#1451982411yescharacteristic
571C2×C4#1451982511yescharacteristic
572C8—22no—
573C8—22no—
574C8—2441no—
575C8—2441no—
576C22—2441no—
577C22—2441no—
578C22—2441no—
579C22—2441no—
580C22—2441no—
581C22—2441no—
582C22—2441no—
583C22C2×#1451974811yescharacteristic
584C22#1684027511yescharacteristic
585C22#1684033511yescharacteristic
586C22#1684056611yescharacteristic
587C22#1684057211yescharacteristic
588C22#1684057511yescharacteristic
589C22#1684057511yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—4881no—
597C4—4881no—
598C4—4881no—
599C4—2441no—
600C4—2441no—
601C4—2441no—
602C4—2442no—
603C4#1684041111yescharacteristic
604C4#1684042311yescharacteristic
605C4#1684045411yescharacteristic
606C4#1684048311yescharacteristic
607C2—4881no—
608C2#2144924411yescharacteristic
609C2#2145547011yescharacteristic
610C2#2145549311yescharacteristic
611C2#2145605311yescharacteristic
612C2#2145608811yescharacteristic
613C2#2145708611yescharacteristic
614C2#2145710711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#477C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#477D61
hypocenterC61#24714
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
8. #21446772C2semidirect
9. #21446589C2semidirect
10. #21446560C2semidirect
11. #21446550C2semidirect
12. #21446506C2semidirect
13. #21446492C2semidirect
14. #21445834C2semidirect
15. C2×C61×#477C2semidirect
300. C2×#477D61semidirect
368. C61#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#168398222
#214516412
#214484952
#214472482
#214472432
#214470212
#214470142
#214467722
#214465892
#214465602
#214465502
#214465062
#214464922
#214458342
C2×C61×#4772
#2471461