Finite Group Databasebeta

← #29721373

Subgroups — #29721373

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×#16840055C211yesmaximal, characteristic, complemented
2C2×#16839820C211yesmaximal, characteristic
3#21451644C211yesmaximal, characteristic
4#21448503C211yesmaximal, characteristic
5#21447247C211yesmaximal, characteristic
6#21447245C211yesmaximal, characteristic
7#21447012C211yesmaximal, characteristic
8#21446769C211yesmaximal, characteristic, complemented
9#21446584C211yesmaximal, characteristic, complemented
10#21446561C211yesmaximal, characteristic, complemented
11#21446553C211yesmaximal, characteristic, complemented
12#21446511C211yesmaximal, characteristic, complemented
13#21446492C211yesmaximal, characteristic, complemented
14#21445832C211yesmaximal, characteristic, complemented
15C61×#2036C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#16840055—24no—
20#16839820—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839704—21no—
32#16839698—21no—
33#16839683—21no—
34#16839674—21no—
35#16839669C2211yescharacteristic
36#16839661—21no—
37#16839661C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839655—21no—
41#16839655—21no—
42#16839648—21no—
43#16839627C2211yescharacteristic
44#16839622C2211yescharacteristic
45#16839622C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C22⋊Q8×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519776—22no—
65#14519776—22no—
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68#14519736—22no—
69#14519736—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61D411yescharacteristic
84C2×C22⋊Dic61D411yescharacteristic
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711—22no—
90#14519710—22no—
91#14519708—22no—
92#14519706—22no—
93#14519706D414yes—
94#14519701—22no—
95#14519701—22no—
96C22⋊#92418—22no—
97#14519687—22no—
98C8×Dic61—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109C22×Q8×C61C2311yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22⋊Q8×C61—22no—
114C22⋊Q8×C61—22no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121#107×C61—22no—
122#103×C61—22no—
123#103×C61—22no—
124C22⋊C8×C61—22no—
125#95×C61—21no—
126C24×C61C2×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×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×Dic122—42no—
154C2×Dic122—81no—
155C2×Dic122—81no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61C2×D412yes—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61#4112yes—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—81no—
171C22⋊Dic61—22no—
172C22⋊Dic61—22no—
173C22⋊Dic61—81no—
174C22⋊Dic61—81no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—42no—
186C4⋊Dic61—81no—
187C4⋊Dic61—81no—
188#11758196—42no—
189#11758196—81no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—42no—
193C4×Dic61—81no—
194C4×Dic61C2×D412yes—
195C4×Dic61#4112yes—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C2×D412yes—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61C22×D411yescharacteristic
210Q8×C61—22no—
211Q8×C61—22no—
212Q8×C61—41no—
213Q8×C61—41no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
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—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
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—21no—
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—
260C2×Dic61—41no—
261C2×Dic61#12112yes—
262C2×Dic61#12112yes—
263Dic122—84no—
264C8×C61—22no—
265C8×C61—41no—
266#92418—22no—
267#92418—41no—
268#24714—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61C2×#12111yescharacteristic
277C22×C61#52511yescharacteristic
278C22×C61#53611yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57811yescharacteristic
282C22×C61#57811yescharacteristic
283C4×C61—42no—
284C4×C61—41no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61#54711yescharacteristic
289C4×C61#54811yescharacteristic
290C4×C61#55211yescharacteristic
291C4×C61#55611yescharacteristic
292Dic61—22no—
293Dic61—22no—
294Dic61—41no—
295Dic61—41no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300C2×#477—611no—
301C2×#433—611no—
302#2628—611no—
303#2323—611no—
304#2076—611no—
305#2075—611no—
306#2036D6111yescharacteristic
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—81no—
316C2×C61#241511yescharacteristic
317C2×C61#305911yescharacteristic
318C2×C61#306211yescharacteristic
319C2×C61#312111yescharacteristic
320C2×C61#312411yescharacteristic
321C2×C61#324211yescharacteristic
322C2×C61#324311yescharacteristic
323C2×C4⋊Q8—611no—
324C2×C22⋊Q8C2×D6111yescharacteristic
325C2×C4×Q8—611no—
326C2×#117—611no—
327#477—1224no—
328#433—1224no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#107C2×D6111yescharacteristic
332C2×#106—611no—
333C2×C4⋊C8—611no—
334C2×C4⋊C8—611no—
335C2×#103—611no—
336C2×#103—611no—
337C2×#103C2×D6111yescharacteristic
338C2×#103C2×D6111yescharacteristic
339C2×C22⋊C8—611no—
340C2×C22⋊C8C2×D6111yescharacteristic
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#380C2×D6111yescharacteristic
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#340C2×D6111yescharacteristic
368C61#2471411yescharacteristic, complemented, Sylow
369C22×Q8—1221no—
370C22×Q8C22×D6111yescharacteristic
371C23×C4C22×D6111yescharacteristic
372C22×C8—611no—
373C22×C8—611no—
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C4⋊Q8—1222no—
377C4⋊Q8—1222no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C4×Q8—1222no—
381C4×Q8—1222no—
382#117—1222no—
383#117—1222no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
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⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—611no—
402C2×C22⋊C4—611no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—1221no—
408C2×C42—1221no—
409#107—22no—
410#107—1222no—
411#107—1222no—
412#106—1222no—
413C4⋊C8—614no—
414C4⋊C8—1222no—
415#103—22no—
416#103—22no—
417#103—1222no—
418#103—1222no—
419C22⋊C8—22no—
420C22⋊C8—1222no—
421#97—1222no—
422C4×C8—1222no—
423#95—21no—
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—
434C24D4×D6111yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—4881no—
441C2×Q8—4881no—
442C2×Q8—2442no—
443C2×Q8#1175822412yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4D4×D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—612no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8D4×D6112yes—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—4881no—
493C4⋊C4—4881no—
494C4⋊C4—4881no—
495C4⋊C4—2442no—
496C4⋊C4—2442no—
497C4⋊C4#1175822412yes—
498C22⋊C4—24no—
499C22⋊C4—81no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—4881no—
503C22⋊C4—4881no—
504C42—612no—
505C42—612no—
506C42—1221no—
507C42—1221no—
508C42—4881no—
509C42—2442no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C2×D4×D6111yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—4884no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
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—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
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—2441no—
565C2×C4—2441no—
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4#41×D6111yescharacteristic
568C2×C4C2×#1175822411yescharacteristic
569C2×C4C2×#1175822411yescharacteristic
570C2×C4#1451982411yescharacteristic
571C2×C4#1451982411yescharacteristic
572C8—22no—
573C8—41no—
574C8—1222no—
575C8—2441no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22#121×D6111yescharacteristic
584C22#1684026811yescharacteristic
585C22#1684033411yescharacteristic
586C22#1684056611yescharacteristic
587C22#1684057111yescharacteristic
588C22#1684057511yescharacteristic
589C22#1684057611yescharacteristic
590C4—41no—
591C4—42no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—1222no—
596C4—1222no—
597C4—4881no—
598C4—4881no—
599C4—4881no—
600C4—4881no—
601C4—2441no—
602C4—2441no—
603C4#1684041211yescharacteristic
604C4#1684041811yescharacteristic
605C4#1684044911yescharacteristic
606C4#1684048111yescharacteristic
607C2—81no—
608C2#2144924111yescharacteristic
609C2#2145546911yescharacteristic
610C2#2145549411yescharacteristic
611C2#2145604911yescharacteristic
612C2#2145607411yescharacteristic
613C2#2145708511yescharacteristic
614C2#2145710611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2036C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2036D61
hypocenterC61#24714
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2036

Complements

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

NH ≅ G/Ntype
1. C2×#16840055C2semidirect
8. #21446769C2semidirect
9. #21446584C2semidirect
10. #21446561C2semidirect
11. #21446553C2semidirect
12. #21446511C2semidirect
13. #21446492C2semidirect
14. #21445832C2semidirect
368. C61#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#168400552
C2×#168398202
#214516442
#214485032
#214472472
#214472452
#214470122
#214467692
#214465842
#214465612
#214465532
#214465112
#214464922
#214458322
C61×#20362
#2471461