Finite Group Databasebeta

← #29647856

Subgroups — #29647856

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459671C211yesmaximal, characteristic
2#21454848C211yesmaximal, characteristic, complemented
3#21444373C212yesmaximal, complemented
4#21444369C211yesmaximal, characteristic, complemented
5#21444365C211yesmaximal, characteristic, complemented
6#21444360C212yesmaximal, complemented
7#21444330C211yesmaximal, characteristic, complemented
8#21444326C211yesmaximal, characteristic, complemented
9#21444321C212yesmaximal, complemented
10#21444308C212yesmaximal, complemented
11C61×#2353C211yesmaximal, characteristic, complemented
12#16840495—22no—
13#16840482C412yescomplemented
14#16840477C412yescomplemented
15#16840462—22no—
16#16840433C412yescomplemented
17#16840417C412yescomplemented
18#16840350—21no—
19#16840350—21no—
20#16840329—21no—
21#16840319C2211yescharacteristic
22#16840312C2211yescharacteristic
23#16840273C2212yes—
24#16840273C2212yes—
25#16840243—21no—
26#16839834—22no—
27#16839834—22no—
28#16839819C2211yescharacteristic
29#16839819C2211yescharacteristic
30#16839726C2212yes—
31C2×C22⋊#92418C2212yes—
32#16839498C2212yes—
33#16839498C2212yes—
34#16839498C2212yes—
35#16839498C2212yes—
36#16839495C2212yes—
37#16839495C2212yes—
38#16839495C2212yes—
39#16839495C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#388—21no—
42C61×#387—21no—
43C61×#386—21no—
44C61×#383C2211yescharacteristic
45C61×#381C2211yescharacteristic
46C61×#380C2211yescharacteristic
47C4×C22⋊C4×C61—21no—
48C4×C22⋊C4×C61C2211yescharacteristic
49C61×#376C2211yescharacteristic
50C2×#95×C61C2211yescharacteristic
51C2×#11758222—21no—
52C2×#11758222C2×C412yes—
53C2×#11758222C2×C412yescomplemented
54C2×#11758194—22no—
55C22×#92418—22no—
56#14519777C2×C411yescharacteristic, complemented
57#14519777C2×C411yescharacteristic, complemented
58#14519777C2×C411yescharacteristic, complemented
59#14519774C2×C411yescharacteristic, complemented
60#14519773—22no—
61#14519770—22no—
62#14519770—22no—
63#14519769—21no—
64#14519765—21no—
65#14519764—22no—
66#14519763—22no—
67#14519763—22no—
68#14519760—22no—
69#14519760—22no—
70#14519759—22no—
71#14519759—22no—
72#14519756—21no—
73#14519756C2×C412yes—
74#14519756C2×C412yescomplemented
75#14519756C2312yes—
76#14519751—21no—
77#14519751C2×C412yes—
78#14519751C2×C412yescomplemented
79#14519751C2312yes—
80#14519750—22no—
81#14519750—22no—
82#14519748—22no—
83#14519748—22no—
84#14519746—22no—
85#14519746—22no—
86C4×Dic122—21no—
87C4×Dic122C2×C412yescomplemented
88#14519742—22no—
89C4×D244—21no—
90C4×D244C2×C412yescomplemented
91#14519739—22no—
92C4×C22⋊D61—22no—
93C4×C22⋊D61—22no—
94C4×C22⋊D61—22no—
95C4×C22⋊D61—22no—
96#14519736—21no—
97#14519736C2×C412yes—
98#14519736C2×C412yescomplemented
99#14519735—22no—
100C4⋊C4×D61—22no—
101C4⋊#92418—21no—
102C4⋊#92418—21no—
103C22⋊#92418—22no—
104C22⋊#92418—22no—
105C22⋊#92418C2312yes—
106C22⋊#92418C2312yes—
107#14519688—21no—
108#14519688—21no—
109C22×D4×C61D411yescharacteristic
110C23×C4×C61—21no—
111C23×C4×C61C2311yescharacteristic
112C23×C4×C61C2311yescharacteristic
113C4×D4×C61C2×C412yes—
114C4×D4×C61C2×C412yes—
115C4×D4×C61D412yes—
116C4×D4×C61D412yes—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61D411yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61D411yescharacteristic
123C2×C22⋊C4×C61D411yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132#95×C61C2311yescharacteristic
133#95×C61C2311yescharacteristic
134#95×C61C2311yescharacteristic
135#95×C61C2311yescharacteristic
136C24×C61C2×D411yescharacteristic
137C24×C61C2×D411yescharacteristic
138C2×D4×C61—41no—
139C2×D4×C61—41no—
140C2×D4×C61C22⋊C412yes—
141C2×D4×C61C22⋊C412yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—81no—
149C22×C4×C61—81no—
150C22×C4×C61—81no—
151C22×C4×C61C22×C412yes—
152C22×C4×C61C22×C412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D412yes—
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—22no—
161C2×C22⋊D61—22no—
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—22no—
165#11758222—42no—
166#11758222—42no—
167#11758222—82no—
168#11758222—82no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244C22×C412yes—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122C22×C412yes—
178C4⋊C4×C61—81no—
179C4⋊C4×C61C22⋊C412yes—
180C4⋊C4×C61C22⋊C412yes—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—81no—
184C22⋊C4×C61C22⋊C412yes—
185C22⋊C4×C61C22⋊C412yes—
186C22⋊C4×C61C22⋊C412yes—
187C22⋊C4×C61C22⋊C412yes—
188C42×C61—42no—
189C42×C61—81no—
190C42×C61C22×C411yescharacteristic
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197C22⋊Dic61—22no—
198C22⋊Dic61—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C22×C412yes—
208C4⋊Dic61C22×C412yes—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216#11758194—82no—
217C2×#92418—22no—
218C2×#92418—22no—
219C2×#92418—82no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C2×C22⋊C412yes—
229C23×C61C2×C22⋊C412yes—
230C23×C61C22×D411yescharacteristic
231C22×D61—22no—
232D4×C61—42no—
233D4×C61—42no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—84no—
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C412yes—
259C2×C4×C61C2×C22⋊C412yes—
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C22⋊D61—42no—
266C22⋊D61—42no—
267C22⋊D61—82no—
268C22⋊D61—82no—
269C22⋊D61—82no—
270C22⋊D61—82no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274D244—42no—
275D244—42no—
276D244—81no—
277C4×D61—44no—
278C4×D61—82no—
279C4×D61—82no—
280Dic122—42no—
281Dic122—42no—
282Dic122—81no—
283#92418—42no—
284#92418—42no—
285#17629—611nomaximal, Sylow
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61C22×C22⋊C411yescharacteristic
299C22×C61#57511yescharacteristic
300C22×C61#57511yescharacteristic
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61—21no—
312C4×C61—42no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61—81no—
316Dic61—42no—
317Dic61—42no—
318Dic61—42no—
319#3506—611no—
320#2973—611no—
321#2353D6111yescharacteristic, complemented
322#1516—612no—
323#1515—611no—
324#1514—611no—
325#1513—612no—
326#1506—611no—
327#1505—611no—
328#1504—612no—
329#1501—612no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61—42no—
333C2×C61—42no—
334C2×C61#288311yescharacteristic
335C2×C61#289611yescharacteristic
336C2×C61#290111yescharacteristic
337D61—82no—
338#559—1222no—
339#556—612no—
340#555—612no—
341#553—1222no—
342#550—612no—
343#547—612no—
344#538—1221no—
345#538—1221no—
346#535—1221no—
347#533—611no—
348#532—611no—
349#525—612no—
350#525—612no—
351#520—1221no—
352C2×C4×D4C2×D6111yescharacteristic
353#435—1222no—
354#435—1222no—
355#432—611no—
356#432—611no—
357#407—612no—
358C2×C22⋊C8—612no—
359#388—21no—
360#387—21no—
361#386—21no—
362#383C2×D6111yescharacteristic
363#381C2×D6111yescharacteristic
364#380C2×D6111yescharacteristic
365C4×C22⋊C4—21no—
366C4×C22⋊C4C2×D6111yescharacteristic
367#376C2×D6111yescharacteristic
368C2×#95C2×D6111yescharacteristic
369#328—612no—
370#328—612no—
371#328—612no—
372#328—612no—
373#327—612no—
374#327—612no—
375#327—612no—
376#327—612no—
377C61#1762911yescharacteristic, complemented, Sylow
378C2×#41—612no—
379C2×#41—612no—
380C2×#41—1221no—
381C22×D4C22⋊D6111yescharacteristic
382C23×C4—21no—
383C23×C4C22×D6111yescharacteristic
384C23×C4C22×D6111yescharacteristic
385C2×M16—1222no—
386C22×C8—1222no—
387C4⋊Q8—611no—
388C4⋊Q8—611no—
389C4⋊Q8—611no—
390#127—611no—
391#127—1222no—
392#126—1222no—
393#126—1222no—
394#125—1221no—
395#124—1222no—
396#124—1222no—
397#124—1222no—
398#124—1221no—
399#123—1222no—
400#123—1222no—
401#123—1222no—
402#123—1222no—
403C22⋊Q8—612no—
404C22⋊Q8—612no—
405C22⋊Q8—612no—
406C22⋊Q8—1221no—
407#121—612no—
408#121—612no—
409#121—612no—
410#121—1222no—
411#121—1222no—
412#121—1221no—
413#121—1222no—
414#121—1222no—
415#120—1222no—
416#120—1222no—
417C4×Q8—612no—
418C4×Q8—1222no—
419C4×Q8—1221no—
420C4×D4—612no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1221no—
427C4×D4C4×D6112yes—
428C4×D4C4×D6112yes—
429C4×D4D24412yes—
430C4×D4D24412yes—
431#117—612no—
432#117—612no—
433#117—1221no—
434#117—1222no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—1222no—
437C2×C4⋊C4C22⋊D6111yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4C22⋊D6111yescharacteristic
442C2×C22⋊C4C22⋊D6111yescharacteristic
443C2×C42—21no—
444C2×C42—21no—
445C2×C42C22×D6111yescharacteristic
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C22⋊C8—612no—
449C22⋊C8—612no—
450C22⋊C8—1222no—
451C22⋊C8—1222no—
452#97—1221no—
453#97—1221no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95C22×D6111yescharacteristic
460#95C22×D6111yescharacteristic
461#95C22×D6111yescharacteristic
462#95C22×D6111yescharacteristic
463C24C2×C22⋊D6111yescharacteristic
464C24C2×C22⋊D6111yescharacteristic
465#41—4882no—
466#41—4882no—
467#41—2442no—
468#41—2442no—
469C2×Q8—612no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—612no—
475C2×D4—1222no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4#1175819812yes—
483C2×D4#1175819812yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4C2×C4×D6112yes—
497C22×C4C2×C4×D6112yes—
498C22×C4C2×D24412yes—
499C22×C4C2×D24412yes—
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C2×C22⋊D6111yescharacteristic
504C22×C4C23×D6111yescharacteristic
505M16—4882no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—4882no—
509C4⋊C4—81no—
510C4⋊C4—612no—
511C4⋊C4—612no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4#1175819812yes—
521C4⋊C4#1175819812yes—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4#1175819812yes—
535C22⋊C4#1175819812yes—
536C22⋊C4#1175819812yes—
537C22⋊C4#1175819812yes—
538C42—42no—
539C42—81no—
540C42—1222no—
541C42—1222no—
542C42—1222no—
543C42C2×C4×D6111yescharacteristic
544C42C2×C4×D6111yescharacteristic
545C42C2×D24411yescharacteristic
546C42C2×D24411yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—1222no—
556C23C2×#1175819812yes—
557C23C2×#1175819812yes—
558C23C22×C22⋊D6111yescharacteristic
559Q8—4881no—
560Q8—2442no—
561Q8—2442no—
562D4—42no—
563D4—42no—
564D4—4881no—
565D4—4882no—
566D4—4882no—
567D4—4882no—
568D4—4882no—
569D4—2442no—
570D4—2442no—
571D4—2442no—
572D4—2442no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—4882no—
600C2×C4—4882no—
601C2×C4—2444no—
602C2×C4C2×#1175819811yescharacteristic
603C2×C4C2×#1175819812yes—
604C2×C4C2×#1175819812yes—
605C2×C4C2×#1175819811yescharacteristic
606C2×C4C2×#1175819811yescharacteristic
607C2×C4C2×#1175819811yescharacteristic
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C22×D24411yescharacteristic
610C8—2442no—
611C8—2442no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—2442no—
625C22—2442no—
626C22—2442no—
627C22C22×#1175819811yescharacteristic
628C22#1684056011yescharacteristic
629C22#1684056011yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—2442no—
643C4—2442no—
644C4—2442no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—4882no—
650C2#2145427911yescharacteristic
651C2#2145433611yescharacteristic
652C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2353C2
socleC22×C61C22×C22⋊C4
prime radical#21459671C2
cosocleC22×C4×C61C24
hypercenter#2353D61
hypocenterC61#17629
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2353

Complements

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

NH ≅ G/Ntype
2. #21454848C2semidirect
3. #21444373C2semidirect
4. #21444369C2semidirect
5. #21444365C2semidirect
6. #21444360C2semidirect
7. #21444330C2semidirect
8. #21444326C2semidirect
9. #21444321C2semidirect
10. #21444308C2semidirect
11. C61×#2353C2semidirect
13. #16840482C4semidirect
14. #16840477C4semidirect
16. #16840433C4semidirect
17. #16840417C4semidirect
53. C2×#11758222C2×C4semidirect
56. #14519777C2×C4semidirect
57. #14519777C2×C4semidirect
58. #14519777C2×C4semidirect
59. #14519774C2×C4semidirect
74. #14519756C2×C4semidirect
78. #14519751C2×C4semidirect
87. C4×Dic122C2×C4semidirect
90. C4×D244C2×C4semidirect
98. #14519736C2×C4semidirect
321. #2353D61semidirect
377. C61#17629semidirect

Maximal subgroups

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

subgroupindex
#214596712
#214548482
#214443732
#214443692
#214443652
#214443602
#214443302
#214443262
#214443212
#214443082
C61×#23532
#1762961