Finite Group Databasebeta

← #29647705

Subgroups — #29647705

Order 15616.

930 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459573C211yesmaximal, characteristic
2#21454473C211yesmaximal, characteristic, complemented
3#21448710C211yesmaximal, characteristic, complemented
4#21444393C212yesmaximal, complemented
5#21444392C211yesmaximal, characteristic, complemented
6#21444368C212yesmaximal, complemented
7#21444362C211yesmaximal, characteristic
8#21444341C212yesmaximal, complemented
9#21444338C211yesmaximal, characteristic, complemented
10#21444316C212yesmaximal, complemented
11C61×#1501C211yesmaximal, characteristic
12#16840482—21no—
13#16840469—21no—
14#16840452—21no—
15#16840452—21no—
16#16840449—22no—
17#16840443—22no—
18#16840390C411yescharacteristic, complemented
19#16840379C412yescomplemented
20#16840372C412yescomplemented
21#16840333C411yescharacteristic, complemented
22#16840314C2212yes—
23#16840293C2212yes—
24C2×#14519756C2211yescharacteristic
25#16840245C411yescharacteristic, complemented
26#16840239C411yescharacteristic, complemented
27C2×#14519736C2211yescharacteristic
28#16839820—22no—
29#16839816C2212yes—
30#16839812C2212yes—
31#16839788—21no—
32#16839726C2211yescharacteristic
33#16839717—21no—
34#16839660—21no—
35#16839656—22no—
36#16839629C2212yes—
37#16839622C2211yescharacteristic
38C4×C4⋊Dic61—21no—
39#16839613C2212yes—
40#16839500C2212yes—
41#16839500C2212yes—
42#16839499C2212yes—
43#16839498C2212yes—
44#16839498C2212yes—
45#16839496C2212yes—
46C61×#525C2211yescharacteristic
47C2×C22⋊C8×C61C2211yescharacteristic
48C4×C22⋊C4×C61C2211yescharacteristic
49C61×#328C2211yescharacteristic
50C61×#328C2211yescharacteristic
51C61×#327C2211yescharacteristic
52C61×#327C2211yescharacteristic
53C22×Dic122C2×C411yescharacteristic
54C2×#11758194—21no—
55#14519777—21no—
56#14519776C2×C412yes—
57#14519775C2×C412yes—
58#14519770—22no—
59#14519770—22no—
60#14519770—22no—
61#14519770—22no—
62#14519769—21no—
63#14519768—22no—
64#14519768C2×C412yes—
65#14519767—22no—
66#14519767—22no—
67#14519764—22no—
68#14519764C2×C412yes—
69#14519758—22no—
70#14519758—22no—
71#14519758—22no—
72#14519756—21no—
73#14519756—21no—
74#14519756—21no—
75#14519756—21no—
76#14519756C2×C412yes—
77#14519756C2312yes—
78#14519754—22no—
79#14519754—22no—
80#14519754—22no—
81Q8×Dic61C2×C412yes—
82C4×Dic122—21no—
83C4×Dic122—21no—
84#14519742—22no—
85#14519742C2×C412yes—
86#14519742C2×C412yes—
87D4×Dic61—22no—
88D4×Dic61—22no—
89D4×Dic61C2×C412yes—
90#14519736—21no—
91#14519736C2×C411yescharacteristic
92#14519736C2×C411yescharacteristic
93#14519736C2×C412yes—
94#14519736D412yes—
95#14519736D411yescharacteristic
96#14519736D411yescharacteristic
97#14519733—22no—
98#14519733—22no—
99#14519733—22no—
100C2×C4⋊Dic61—21no—
101C2×C4⋊Dic61—21no—
102C2×C4⋊Dic61C2×C411yescharacteristic
103C2×C4⋊Dic61D411yescharacteristic
104C2×C4⋊Dic61D411yescharacteristic
105C2×#11758196—21no—
106C2×#11758196—22no—
107C2×C22⋊Dic61—21no—
108C2×C22⋊Dic61—21no—
109C2×C22⋊Dic61D412yes—
110C2×C4×Dic61—22no—
111C2×C4×Dic61C2312yes—
112#14519706—22no—
113#14519706—22no—
114C22⋊#92418—21no—
115C22⋊#92418C2312yes—
116#14519687—22no—
117C8×Dic61—22no—
118#14519684—22no—
119#14519684—22no—
120#14519684C2312yes—
121C2×#41×C61C2×C411yescharacteristic
122C23×C4×C61C2311yescharacteristic
123C22×C8×C61—21no—
124#123×C61—21no—
125C22⋊Q8×C61C2×C411yescharacteristic
126C22⋊Q8×C61C2311yescharacteristic
127#121×C61C2×C411yescharacteristic
128#121×C61C2311yescharacteristic
129#120×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132#117×C61C2×C411yescharacteristic
133C2×C22⋊C4×C61—21no—
134C2×C42×C61—21no—
135C22⋊C8×C61—21no—
136C22⋊C8×C61C2311yescharacteristic
137C22⋊C8×C61C2311yescharacteristic
138#95×C61C2311yescharacteristic
139#95×C61C2311yescharacteristic
140C24×C61C2×D411yescharacteristic
141#41×C61—42no—
142C2×Q8×C61C22×C411yescharacteristic
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—42no—
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61C2×D412yes—
162C22×Dic61C2×D412yes—
163C2×Dic122—21no—
164C2×Dic122—22no—
165C2×Dic122—42no—
166C2×Dic122C22×C412yes—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—42no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C22×C411yescharacteristic
173C4⋊C4×C61C22×C411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—42no—
180C42×C61—21no—
181C42×C61—42no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—81no—
188C22⋊Dic61C22⋊C412yes—
189C22⋊Dic61C22⋊C412yes—
190C22⋊Dic61C22⋊C412yes—
191C22⋊Dic61C22⋊C412yes—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—42no—
196C4⋊Dic61C22⋊C411yescharacteristic
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22⋊C412yes—
199C4⋊Dic61C22⋊C411yescharacteristic
200C4⋊Dic61C22⋊C411yescharacteristic
201C4⋊Dic61C22⋊C412yes—
202C4⋊Dic61C22×C412yes—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—44no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—82no—
216C4×Dic61C22×C412yes—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220#11758194—42no—
221C2×#92418—22no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61C2×C22⋊C411yescharacteristic
228C23×C61C2×C22⋊C411yescharacteristic
229C23×C61C22×D411yescharacteristic
230Q8×C61—41no—
231D4×C61—41no—
232D4×C61—42no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—81no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—82no—
266C2×Dic61—81no—
267C2×Dic61C2×C22⋊C412yes—
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×C22⋊C412yes—
270C2×Dic61C2×C22⋊C412yes—
271Dic122—42no—
272Dic122—42no—
273Dic122—82no—
274C8×C61—41no—
275C8×C61—41no—
276#92418—42no—
277#17612—611nomaximal, Sylow
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61#57711yescharacteristic
293C22×C61#57811yescharacteristic
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—81no—
302C4×C61#41012yes—
303Dic61—22no—
304Dic61—22no—
305Dic61—22no—
306Dic61—22no—
307Dic61—42no—
308Dic61—42no—
309Dic61—82no—
310#3495—611no—
311#2924—611no—
312#2347—611no—
313#1521—612no—
314#1520—611no—
315#1515—612no—
316#1513—611no—
317#1509—612no—
318#1508—611no—
319#1503—612no—
320#1501D6111yescharacteristic
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—41no—
324C2×C61—41no—
325C2×C61—81no—
326C2×C61C2×#41011yescharacteristic
327C2×C61#289811yescharacteristic
328C2×C61#290111yescharacteristic
329#556—1221no—
330#554—1221no—
331#552—1221no—
332#552—1221no—
333#552—1222no—
334#551—1222no—
335#544—611no—
336#542—612no—
337#541—612no—
338#536—611no—
339#533—612no—
340#529—612no—
341#525C2×D6111yescharacteristic
342C2×C22⋊Q8—611no—
343#520—611no—
344#519—611no—
345C2×#117—611no—
346#433—1222no—
347#432—612no—
348#431—612no—
349#424—1221no—
350#407—611no—
351C2×C22⋊C8C2×D6111yescharacteristic
352#405—1221no—
353#389—1221no—
354#388—1222no—
355#383—612no—
356#380—611no—
357C4×C4⋊C4—612no—
358C4×C4⋊C4—1221no—
359C4×C22⋊C4C2×D6111yescharacteristic
360#328—612no—
361#328—612no—
362#328—612no—
363#328—612no—
364#328—612no—
365#328C2×D6111yescharacteristic
366#328C2×D6111yescharacteristic
367#327—612no—
368#327C2×D6111yescharacteristic
369#327C2×D6111yescharacteristic
370C61#1761211yescharacteristic, complemented, Sylow
371C2×#41C4×D6111yescharacteristic
372C22×Q8—611no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—1221no—
375C22×C8—21no—
376C4⋊Q8—612no—
377C4⋊Q8—612no—
378C4⋊Q8—1221no—
379#126—1222no—
380#126—1222no—
381#126—1222no—
382#126—1222no—
383#125—612no—
384#125—1221no—
385#125—1222no—
386#125—1222no—
387#125—1222no—
388#124—612no—
389#124—1222no—
390#123—21no—
391#123—1222no—
392#123—1222no—
393#123—1222no—
394C22⋊Q8—612no—
395C22⋊Q8—612no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C22⋊Q8—1222no—
399C22⋊Q8—1221no—
400C22⋊Q8—1221no—
401C22⋊Q8—1221no—
402C22⋊Q8—1221no—
403C22⋊Q8C4×D6111yescharacteristic
404C22⋊Q8C22×D6111yescharacteristic
405#121C4×D6111yescharacteristic
406#121C22×D6111yescharacteristic
407#120—21no—
408C4×Q8—612no—
409C4×Q8—612no—
410C4×Q8—612no—
411C4×Q8—1221no—
412C4×Q8—1221no—
413C4×Q8—1222no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—612no—
417C4×D4—1222no—
418C4×D4—1222no—
419#117—611no—
420#117—611no—
421#117—612no—
422#117—611no—
423#117—612no—
424#117—611no—
425#117—1222no—
426#117—1222no—
427#117—1222no—
428#117—1221no—
429#117C4×D6111yescharacteristic
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1221no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—612no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C42—21no—
442C2×C42—612no—
443C2×C42—1222no—
444C4⋊C8—1222no—
445C4⋊C8—1222no—
446C22⋊C8—21no—
447C22⋊C8—612no—
448C22⋊C8—1221no—
449C22⋊C8C22×D6111yescharacteristic
450C22⋊C8C22×D6111yescharacteristic
451#97—1222no—
452C4×C8—1222no—
453#95—612no—
454#95—1222no—
455#95—1222no—
456#95C22×D6111yescharacteristic
457#95C22×D6111yescharacteristic
458C24D4×D6111yescharacteristic
459#41—42no—
460C2×Q8—612no—
461C2×Q8—1222no—
462C2×Q8—1221no—
463C2×Q8—2442no—
464C2×Q8C2×C4×D6111yescharacteristic
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×C4×D6111yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—1222no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1222no—
479C22×C4C2×C4×D6111yescharacteristic
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4C23×D6111yescharacteristic
485M16—2442no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—42no—
489C2×C8—1222no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—611no—
493C4⋊C4—611no—
494C4⋊C4—612no—
495C4⋊C4—611no—
496C4⋊C4—611no—
497C4⋊C4—612no—
498C4⋊C4—612no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1221no—
509C4⋊C4—1222no—
510C4⋊C4—2444no—
511C4⋊C4—2442no—
512C4⋊C4C2×C4×D6111yescharacteristic
513C4⋊C4C2×C4×D6111yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—42no—
520C22⋊C4—612no—
521C22⋊C4—612no—
522C22⋊C4—612no—
523C22⋊C4—612no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—4881no—
530C42—21no—
531C42—42no—
532C42—612no—
533C42—612no—
534C42—612no—
535C42—612no—
536C42—1222no—
537C42—1222no—
538C42—1222no—
539C42—4882no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23C22⋊C4×D6111yescharacteristic
546C23C22⋊C4×D6111yescharacteristic
547C23C2×D4×D6111yescharacteristic
548Q8—41no—
549Q8—4882no—
550Q8—2442no—
551Q8—2442no—
552D4—41no—
553D4—42no—
554D4—41no—
555D4—41no—
556D4—81no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—4882no—
582C2×C4—4881no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4C22⋊C4×D6111yescharacteristic
589C2×C4C22⋊C4×D6111yescharacteristic
590C2×C4C22×C4×D6111yescharacteristic
591C2×C4C2×D4×D6111yescharacteristic
592C8—41no—
593C8—41no—
594C8—2442no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—81no—
608C22C2×C22⋊C4×D6111yescharacteristic
609C22#1684056911yescharacteristic
610C22#1684057611yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—1222no—
620C4—1222no—
621C4—1222no—
622C4—1222no—
623C4—4882no—
624C4—2442no—
625C4—2442no—
626C4#1683973412yes—
627C2—21no—
628C2—21no—
629C2—41no—
630C2—41no—
631C2—81no—
632C2C2×#1683973411yescharacteristic
633C2#2145434811yescharacteristic
634C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1501C2
socleC22×C61C22×C22⋊C4
prime radicalC61×#525C22
cosocleC22×C4×C61C24
hypercenter#1501D61
hypocenterC61#17612
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1501

Complements

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

NH ≅ G/Ntype
2. #21454473C2semidirect
3. #21448710C2semidirect
4. #21444393C2semidirect
5. #21444392C2semidirect
6. #21444368C2semidirect
8. #21444341C2semidirect
9. #21444338C2semidirect
10. #21444316C2semidirect
18. #16840390C4semidirect
19. #16840379C4semidirect
20. #16840372C4semidirect
21. #16840333C4semidirect
25. #16840245C4semidirect
26. #16840239C4semidirect
370. C61#17612semidirect

Maximal subgroups

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

subgroupindex
#214595732
#214544732
#214487102
#214443932
#214443922
#214443682
#214443622
#214443412
#214443382
#214443162
C61×#15012
#1761261