Finite Group Databasebeta

← #29686821

Subgroups — #29686821

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455399C211yesmaximal, characteristic
2C2×#16839882C211yesmaximal, characteristic, complemented
3#21454524C211yesmaximal, characteristic, complemented
4#21454520C211yesmaximal, characteristic, complemented
5C2×#16839716C211yesmaximal, characteristic, complemented
6#21446709C211yesmaximal, characteristic, complemented
7#21446571C211yesmaximal, characteristic, complemented
8#21446525C211yesmaximal, characteristic, complemented
9#21446521C211yesmaximal, characteristic, complemented
10#21446266C211yesmaximal, characteristic, complemented
11#21446264C211yesmaximal, characteristic, complemented
12#21445856C211yesmaximal, characteristic, complemented
13#21445704C211yesmaximal, characteristic, complemented
14#21445656C211yesmaximal, characteristic, complemented
15C61×#1764C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18C22×#11758189C2211yescharacteristic
19C22×C8×D61C2211yescharacteristic
20#16840274C2211yescharacteristic
21#16840269C2211yescharacteristic
22C2×#14519734C2211yescharacteristic
23#16840002C412yescomplemented
24#16839997C412yescomplemented
25#16839994C412yescomplemented
26#16839990C412yescomplemented
27#16839882—24no—
28#16839841—22no—
29#16839838—22no—
30#16839835—22no—
31#16839833—22no—
32C2×#14519712C2211yescharacteristic
33C2×#14519712C2211yescharacteristic
34#16839798C2211yescharacteristic
35#16839790C2211yescharacteristic
36#16839789C2211yescharacteristic
37C2×#14519705C2211yescharacteristic
38C2×#14519704C2211yescharacteristic
39C2×#14519703C2211yescharacteristic
40#16839759C2211yescharacteristic
41#16839730C2211yescharacteristic
42#16839729C2211yescharacteristic
43#16839725C2211yescharacteristic
44#16839725C2211yescharacteristic
45#16839717C2211yescharacteristic
46#16839716—24no—
47C2×#14519688C2211yescharacteristic
48C2×#14519687C2211yescharacteristic
49C2×C8×Dic61C2211yescharacteristic
50#16839527C2211yescharacteristic
51#16839523C2211yescharacteristic
52#16839522C2211yescharacteristic
53#16839521C2211yescharacteristic
54C2×#131×C61C2211yescharacteristic
55C2×#104×C61C2211yescharacteristic
56C61×#416C2211yescharacteristic
57C61×#405C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C61×#341C2211yescharacteristic
60C61×#339C2211yescharacteristic
61C2×#11758222C2311yescharacteristic
62C22×C4×D61C2311yescharacteristic
63#14519797—22no—
64#14519797—22no—
65C2×#11758202C2×C411yescharacteristic
66C2×#11758200C2×C411yescharacteristic
67C2×#11758201C2×C411yescharacteristic
68C2×#11758199C2×C411yescharacteristic
69#14519783—22no—
70#14519783—22no—
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C2×#11758189—22no—
74C2×#11758189—22no—
75C2×#11758189—21no—
76C2×#11758189D412yes—
77C2×#11758189D412yes—
78C22×#92418C2311yescharacteristic
79C22×#92418C2311yescharacteristic
80C2×C8×D61—22no—
81C2×C8×D61—22no—
82C2×C8×D61—21no—
83C2×C8×D61C2×C412yes—
84C2×C8×D61C2×C412yes—
85#14519761—21no—
86#14519758—21no—
87#14519757C2×C412yes—
88#14519753C2×C412yes—
89#14519752C2×C412yes—
90#14519748C2×C412yes—
91#14519747—21no—
92#14519745—21no—
93C4×D244—22no—
94C4×C22⋊D61—22no—
95#14519736C2311yescharacteristic
96#14519734—22no—
97#14519734—22no—
98C2×#11758196—21no—
99C2×#11758198—21no—
100C2×C4×Dic61C2311yescharacteristic
101#14519712—22no—
102#14519712—22no—
103#14519710C2×C412yes—
104#14519708C2×C412yes—
105C4⋊#92418—22no—
106#14519706—22no—
107#14519705—22no—
108#14519704—22no—
109#14519703—22no—
110#14519702C2×C412yes—
111#14519699C2×C412yes—
112C22⋊#92418—22no—
113#14519689—22no—
114#14519689—22no—
115#14519689—22no—
116#14519688—22no—
117#14519687—21no—
118#14519687D414yescomplemented
119C8×Dic61—21no—
120C8×Dic61—22no—
121C2×#41×C61C2311yescharacteristic
122#131×C61—22no—
123#131×C61—22no—
124C2×M16×C61—21no—
125C2×M16×C61C2311yescharacteristic
126C2×M16×C61C2311yescharacteristic
127C22×C8×C61—21no—
128C22×C8×C61C2311yescharacteristic
129C22×C8×C61C2311yescharacteristic
130#117×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132#104×C61—22no—
133#103×C61C2×C412yes—
134#102×C61C2×C412yes—
135#97×C61—21no—
136C4×C8×C61—22no—
137C4×C8×C61—21no—
138C23×D61#4111yescharacteristic
139#41×C61—21no—
140#41×C61—21no—
141#41×C61—21no—
142#41×C61—21no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—21no—
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—21no—
150C2×C22⋊D61—22no—
151C22×Dic61—21no—
152C22×Dic61#4111yescharacteristic
153#11758222—21no—
154#11758222—21no—
155#11758222—21no—
156#11758222—21no—
157C2×D244—22no—
158C2×D244C22×C411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61C22×C412yes—
165C2×C4×D61C2×D412yes—
166C2×Dic122C22×C411yescharacteristic
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61—21no—
171M16×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—42no—
177C2×C8×C61C22×C412yes—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D412yes—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61#4112yes—
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61#4112yes—
187C22⋊Dic61—21no—
188C22⋊Dic61—22no—
189#11758202—22no—
190#11758201—22no—
191#11758200—22no—
192#11758199—22no—
193#11758198—24no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61C22×C412yes—
199#11758196—24no—
200#11758196—22no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61C2×D412yes—
204#11758194—21no—
205#11758194—21no—
206#11758194—21no—
207#11758194—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C22×C411yescharacteristic
213C2×#92418C22×C411yescharacteristic
214C2×#92418C2×D412yes—
215C2×#92418C2×D412yes—
216C2×#92418#4112yes—
217#11758189—24no—
218#11758189—24no—
219#11758189—42no—
220C8×D61—24no—
221C8×D61—24no—
222C8×D61—42no—
223C23×C61—21no—
224C23×C61C2×#4111yescharacteristic
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—42no—
229C22×D61C4×D412yes—
230Q8×C61—21no—
231Q8×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—42no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—41no—
249C22⋊D61—41no—
250C22⋊D61—82no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—42no—
257C2×Dic61C4×D412yes—
258D244—21no—
259D244—21no—
260D244—82no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—41no—
266C4×D61—41no—
267Dic122—21no—
268Dic122—21no—
269C8×C61—22no—
270C8×C61—41no—
271C8×C61—41no—
272C8×C61#12112yes—
273C8×C61#12112yes—
274#92418—22no—
275#92418—41no—
276#92418—41no—
277#92418C4×D412yes—
278#92418C4×D412yes—
279#21547—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—42no—
282C22×C61—41no—
283C22×C61C2×C4×D411yescharacteristic
284C22×C61#52511yescharacteristic
285C22×C61#52911yescharacteristic
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—41no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61C2×C4×D411yescharacteristic
297C4×C61C4×#4111yescharacteristic
298C4×C61C2×#12111yescharacteristic
299C4×C61#53211yescharacteristic
300Dic61—22no—
301Dic61—42no—
302Dic61—42no—
303Dic61—41no—
304#3050—611no—
305C2×#444—611no—
306#2930—611no—
307#2930—611no—
308C2×#405—611no—
309#1975—611no—
310#1946—611no—
311#1937—611no—
312#1937—611no—
313#1897—611no—
314#1897—611no—
315#1810—611no—
316#1773—611no—
317#1764—611no—
318#1764D6111yescharacteristic, complemented
319C2×C61—21no—
320C2×C61—21no—
321C2×C61—81no—
322C2×C61C4×#12111yescharacteristic
323C2×C61#295712yes—
324D61—42no—
325D61—81no—
326C2×#135—611no—
327C2×#131—611no—
328C2×#131C2×D6111yescharacteristic
329C22×M16—611no—
330C23×C8—611no—
331#525—611no—
332#525—611no—
333C2×#117—611no—
334#467—612no—
335#466—612no—
336#465—612no—
337#465—612no—
338#444—1224no—
339#436—1222no—
340#436—1222no—
341#435—1222no—
342#435—1222no—
343C2×#108—611no—
344C2×#108—611no—
345#427—611no—
346#424—611no—
347#424—611no—
348C2×#104—611no—
349C2×#104—611no—
350C2×#104—611no—
351C2×#104C2×D6111yescharacteristic
352#416—611no—
353#416C2×D6111yescharacteristic
354#408—611no—
355#408—611no—
356#407—611no—
357#407—611no—
358#405—611no—
359#405—1224no—
360#405C2×D6111yescharacteristic
361C2×#97—611no—
362C2×#97—611no—
363C2×C4×C8—611no—
364C2×C4×C8C2×D6111yescharacteristic
365#341—611no—
366#341C2×D6111yescharacteristic
367#339—611no—
368#339—611no—
369#339—611no—
370#339C2×D6111yescharacteristic
371C61#2154711yescharacteristic, complemented, Sylow
372C2×#41—611no—
373C2×#41C22×D6111yescharacteristic
374C23×C4—611no—
375#135—1222no—
376#135—1222no—
377C2×Q16—611no—
378C2×SD16—611no—
379C2×SD16—611no—
380C2×D8—611no—
381#131—22no—
382#131—22no—
383#131—1222no—
384#131—1222no—
385C2×M16—21no—
386C2×M16—612no—
387C2×M16—612no—
388C2×M16—611no—
389C2×M16—611no—
390C2×M16—1222no—
391C2×M16—1222no—
392C2×M16—1221no—
393C2×M16C22×D6111yescharacteristic
394C2×M16C22×D6111yescharacteristic
395C22×C8—21no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—612no—
399C22×C8—612no—
400C22×C8—1222no—
401C22×C8—1222no—
402C22×C8—1221no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405#123—1221no—
406#123—1221no—
407C22⋊Q8—612no—
408C22⋊Q8—612no—
409#121—612no—
410#121—612no—
411#120—1221no—
412#120—1221no—
413C4×D4—1222no—
414C4×D4—1222no—
415#117—611no—
416#117—1222no—
417#117—1222no—
418#117C22×D6111yescharacteristic
419C2×C4⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—611no—
422C2×C42C22×D6111yescharacteristic
423#108—1222no—
424#108—1222no—
425#107—612no—
426#106—612no—
427C4⋊C8—1222no—
428C4⋊C8—1222no—
429#104—22no—
430#104—1222no—
431#104—1222no—
432#104—1222no—
433#103—612no—
434#103C4×D6112yes—
435#102—612no—
436#102C4×D6112yes—
437C22⋊C8—1222no—
438C22⋊C8—1222no—
439C22⋊C8—1222no—
440C22⋊C8—1222no—
441#97—21no—
442#97—614no—
443#97—1222no—
444#97—1221no—
445C4×C8—22no—
446C4×C8—21no—
447C4×C8—1222no—
448C4×C8—1221no—
449C24—611no—
450#41—21no—
451#41—21no—
452#41—21no—
453#41—21no—
454#41—1221no—
455#41—1221no—
456#41—1221no—
457#41—1221no—
458C2×Q8—611no—
459C2×Q8C2×C4×D6111yescharacteristic
460C2×D4—21no—
461C2×D4—611no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4C2×C4×D6111yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—611no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4C23×D6111yescharacteristic
478Q16—1222no—
479SD16—1222no—
480SD16—1222no—
481D8—1222no—
482M16—21no—
483M16—21no—
484M16—21no—
485M16—21no—
486M16—42no—
487M16—1224no—
488M16—1224no—
489M16—1221no—
490M16—1221no—
491M16—1221no—
492M16—1221no—
493M16—2442no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—42no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—612no—
502C2×C8—612no—
503C2×C8—612no—
504C2×C8—1224no—
505C2×C8—1224no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—2442no—
511C2×C8C2×C4×D6112yes—
512C2×C8#1175822212yes—
513C2×C8D4×D6112yes—
514C2×C8D4×D6111yescharacteristic
515C2×C8D4×D6111yescharacteristic
516C4⋊C4—612no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1224no—
520C4⋊C4C2×C4×D6112yes—
521C22⋊C4—21no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1224no—
526C22⋊C4—1222no—
527C22⋊C4—1221no—
528C42—21no—
529C42—21no—
530C42—612no—
531C42—1221no—
532C42—1221no—
533C42#1175822212yes—
534C23—21no—
535C23—612no—
536C23—1221no—
537C23—1221no—
538C23—1221no—
539C23—2442no—
540C23#41×D6111yescharacteristic
541Q8—21no—
542Q8—21no—
543Q8—1221no—
544Q8—1221no—
545D4—21no—
546D4—21no—
547D4—41no—
548D4—41no—
549D4—1221no—
550D4—1221no—
551D4—4882no—
552D4—4882no—
553D4—2441no—
554D4—2441no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—612no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1221no—
570C2×C4—1222no—
571C2×C4—2442no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C2×C4C2×#1175822211yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C8—22no—
581C8—41no—
582C8—41no—
583C8—612no—
584C8—612no—
585C8—1222no—
586C8—2441no—
587C8—2441no—
588C8#1451975012yes—
589C8#1451975012yes—
590C22—22no—
591C22—41no—
592C22—42no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—2442no—
597C22—2442no—
598C22—2441no—
599C22—2442no—
600C22C4×D4×D6111yescharacteristic
601C22#1684027111yescharacteristic
602C22#1684029011yescharacteristic
603C4—42no—
604C4—41no—
605C4—42no—
606C4—1222no—
607C4—2442no—
608C4—2441no—
609C4—2442no—
610C4C4×D4×D6111yescharacteristic
611C4C4×#1175822211yescharacteristic
612C4C2×#1451975011yescharacteristic
613C4#1684030911yescharacteristic
614C2—21no—
615C2—21no—
616C2—81no—
617C2—4881no—
618C2—2442no—
619C2C4×#1451975011yescharacteristic
620C2#2145473112yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1764C2
socleC22×C61C2×C4×D4
prime radical#21455399C2
cosocleC22×C4×C61C24
hypercenter#1764D61
hypocenterC61#21547
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1764

Complements

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

NH ≅ G/Ntype
2. C2×#16839882C2semidirect
3. #21454524C2semidirect
4. #21454520C2semidirect
5. C2×#16839716C2semidirect
6. #21446709C2semidirect
7. #21446571C2semidirect
8. #21446525C2semidirect
9. #21446521C2semidirect
10. #21446266C2semidirect
11. #21446264C2semidirect
12. #21445856C2semidirect
13. #21445704C2semidirect
14. #21445656C2semidirect
15. C61×#1764C2semidirect
23. #16840002C4semidirect
24. #16839997C4semidirect
25. #16839994C4semidirect
26. #16839990C4semidirect
118. #14519687D4semidirect
318. #1764D61semidirect
371. C61#21547semidirect

Maximal subgroups

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

subgroupindex
#214553992
C2×#168398822
#214545242
#214545202
C2×#168397162
#214467092
#214465712
#214465252
#214465212
#214462662
#214462642
#214458562
#214457042
#214456562
C61×#17642
#2154761