Finite Group Databasebeta

← #29686820

Subgroups — #29686820

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#21455397C211yesmaximal, characteristic
2C2×#16839881C211yesmaximal, characteristic, complemented
3#21454523C211yesmaximal, characteristic, complemented
4#21454518C211yesmaximal, characteristic, complemented
5C2×#16839718C211yesmaximal, characteristic, complemented
6#21446709C211yesmaximal, characteristic, complemented
7#21446571C211yesmaximal, characteristic, complemented
8#21446524C211yesmaximal, characteristic, complemented
9#21446520C211yesmaximal, characteristic, complemented
10#21446267C211yesmaximal, characteristic, complemented
11#21445855C211yesmaximal, characteristic, complemented
12#21445702C211yesmaximal, characteristic, complemented
13#21445657C211yesmaximal, characteristic, complemented
14#21445653C211yesmaximal, characteristic, complemented
15C61×#1897C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519785C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19C22×#11758194C2211yescharacteristic
20C23×#92418C2211yescharacteristic
21#16840273C2211yescharacteristic
22#16840001C412yescomplemented
23#16839996C412yescomplemented
24#16839993C412yescomplemented
25#16839989C412yescomplemented
26#16839881—24no—
27#16839840—22no—
28#16839837—22no—
29#16839836—22no—
30#16839834—22no—
31C2×#14519712C2211yescharacteristic
32C2×#14519712C2211yescharacteristic
33#16839798C2211yescharacteristic
34#16839790C2211yescharacteristic
35#16839789C2211yescharacteristic
36C2×#14519705C2211yescharacteristic
37C2×#14519705C2211yescharacteristic
38#16839758C2211yescharacteristic
39#16839756C2211yescharacteristic
40#16839730C2211yescharacteristic
41#16839729C2211yescharacteristic
42#16839726C2211yescharacteristic
43#16839720C2211yescharacteristic
44#16839718—24no—
45#16839718C2211yescharacteristic
46C2×#14519688C2211yescharacteristic
47C2×#14519688C2211yescharacteristic
48C2×C4×#92418C2211yescharacteristic
49C2×C4×#92418C2211yescharacteristic
50#16839526C2211yescharacteristic
51#16839526C2211yescharacteristic
52#16839522C2211yescharacteristic
53#16839522C2211yescharacteristic
54C61×#525C2211yescharacteristic
55C2×#117×C61C2211yescharacteristic
56C2×#104×C61C2211yescharacteristic
57C2×#104×C61C2211yescharacteristic
58C61×#407C2211yescharacteristic
59C61×#339C2211yescharacteristic
60C61×#339C2211yescharacteristic
61C2×#11758222C2311yescharacteristic
62#14519797—22no—
63#14519797—22no—
64C2×#11758202C2×C411yescharacteristic
65C2×#11758200C2×C411yescharacteristic
66C2×#11758201C2×C411yescharacteristic
67C2×#11758199C2×C411yescharacteristic
68#14519785—22no—
69#14519785—22no—
70#14519784—22no—
71#14519784—22no—
72C2×#11758194—22no—
73C2×#11758194—21no—
74C2×#11758194—22no—
75C2×#11758194D412yes—
76C2×#11758194D412yes—
77C2×#11758194C2311yescharacteristic
78C2×#11758194C2311yescharacteristic
79C2×#11758189—21no—
80C22×#92418—22no—
81C22×#92418—22no—
82C22×#92418—21no—
83C22×#92418C2×C412yes—
84C22×#92418C2×C412yes—
85C22×#92418C2311yescharacteristic
86C22×#92418C2311yescharacteristic
87C22×#92418C2311yescharacteristic
88C22×#92418C2311yescharacteristic
89C2×C8×D61—21no—
90#14519760—21no—
91#14519756C2×C412yes—
92#14519751C2×C412yes—
93#14519746—21no—
94C4×C22⋊D61—22no—
95#14519736C2311yescharacteristic
96#14519712—22no—
97#14519712—22no—
98#14519710C2×C412yes—
99#14519708C2×C412yes—
100C4⋊#92418—22no—
101#14519706—22no—
102#14519705—22no—
103#14519705—22no—
104#14519701C2×C412yes—
105#14519700C2×C412yes—
106#14519698C2×C412yes—
107#14519697C2×C412yes—
108C22⋊#92418—22no—
109C22⋊#92418—22no—
110#14519689—22no—
111#14519688—22no—
112#14519688—21no—
113#14519688D414yescomplemented
114#14519687—21no—
115C4×#92418—22no—
116C4×#92418—22no—
117C4×#92418—21no—
118C8×Dic61—21no—
119C2×#41×C61C2311yescharacteristic
120C23×C4×C61C2311yescharacteristic
121C2×M16×C61C2311yescharacteristic
122C2×M16×C61C2311yescharacteristic
123#123×C61—21no—
124C22⋊Q8×C61C2×C412yes—
125#121×C61C2×C412yes—
126#120×C61—21no—
127C4×D4×C61—22no—
128#117×C61—22no—
129#117×C61—22no—
130#117×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132C2×C22⋊C4×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C2×C42×C61C2311yescharacteristic
135#104×C61—22no—
136#104×C61—22no—
137C22⋊C8×C61—22no—
138C24×C61#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×C61—22no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C22×C412yes—
155C22×C4×C61C2×D412yes—
156C22×C4×C61#4111yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—21no—
159C2×C22⋊D61—22no—
160#11758222—21no—
161#11758222—21no—
162#11758222—21no—
163#11758222—21no—
164C2×D244C22×C411yescharacteristic
165C2×C4×D61—21no—
166C2×Dic122C22×C411yescharacteristic
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C22×C412yes—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61C2×D412yes—
184C42×C61#4112yes—
185C22⋊Dic61—22no—
186C22⋊Dic61—21no—
187#11758202—22no—
188#11758201—22no—
189#11758200—22no—
190#11758199—22no—
191#11758198—22no—
192C4⋊Dic61C22×C412yes—
193#11758196—22no—
194C4×Dic61—21no—
195#11758194—24no—
196#11758194—24no—
197#11758194—21no—
198#11758194—21no—
199#11758194—21no—
200#11758194—21no—
201#11758194—42no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—24no—
207C2×#92418—24no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—42no—
211C2×#92418C22×C412yes—
212C2×#92418C22×C411yescharacteristic
213C2×#92418C22×C411yescharacteristic
214C2×#92418C2×D412yes—
215C2×#92418C2×D412yes—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D412yes—
219C2×#92418#4112yes—
220C2×#92418#4112yes—
221#11758189—42no—
222C8×D61—42no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—42no—
227C23×C61C4×D412yes—
228C23×C61C2×#4111yescharacteristic
229C22×D61—21no—
230Q8×C61—21no—
231Q8×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—82no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61C4×D412yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—41no—
259C22⋊D61—41no—
260C22⋊D61—82no—
261C2×Dic61—22no—
262C2×Dic61—21no—
263D244—21no—
264D244—21no—
265C4×D61—41no—
266C4×D61—41no—
267Dic122—21no—
268Dic122—21no—
269C8×C61—41no—
270C8×C61—41no—
271#92418—22no—
272#92418—22no—
273#92418—41no—
274#92418—41no—
275#92418C4×D412yes—
276#92418C4×D412yes—
277#92418#12112yes—
278#92418#12112yes—
279#21547—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—41no—
288C22×C61C2×C4×D411yescharacteristic
289C22×C61#52511yescharacteristic
290C22×C61#52911yescharacteristic
291C2×D61—42no—
292C2×D61—41no—
293C4×C61—22no—
294C4×C61—42no—
295C4×C61—42no—
296C4×C61—42no—
297C4×C61—41no—
298C4×C61C2×C4×D411yescharacteristic
299C4×C61C4×#4111yescharacteristic
300C4×C61C2×#12111yescharacteristic
301C4×C61#53211yescharacteristic
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#1897D6111yescharacteristic, complemented
315#1810—611no—
316#1773—611no—
317#1764—611no—
318#1764—611no—
319C2×C61—21no—
320C2×C61—21no—
321C2×C61—42no—
322C2×C61—81no—
323C2×C61C4×#12111yescharacteristic
324C2×C61#295712yes—
325D61—81no—
326C2×#135—611no—
327C2×#131—611no—
328C2×#131—611no—
329C22×M16—611no—
330C23×C8—611no—
331#525—611no—
332#525C2×D6111yescharacteristic
333C2×#117C2×D6111yescharacteristic
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×#104C2×D6111yescharacteristic
351C2×#104C2×D6111yescharacteristic
352#416—611no—
353#416—611no—
354#408—611no—
355#408—611no—
356#407—611no—
357#407C2×D6111yescharacteristic
358#405—611no—
359#405—611no—
360#405—1224no—
361C2×#97—611no—
362C2×#97—611no—
363C2×C4×C8—611no—
364C2×C4×C8—611no—
365#341—611no—
366#341—611no—
367#339—611no—
368#339—611no—
369#339C2×D6111yescharacteristic
370#339C2×D6111yescharacteristic
371C61#2154711yescharacteristic, complemented, Sylow
372C2×#41—611no—
373C2×#41C22×D6111yescharacteristic
374C23×C4C22×D6111yescharacteristic
375#135—1222no—
376#135—1222no—
377C2×Q16—611no—
378C2×SD16—611no—
379C2×SD16—611no—
380C2×D8—611no—
381#131—1222no—
382#131—1222no—
383#131—1222no—
384#131—1222no—
385C2×M16—612no—
386C2×M16—612no—
387C2×M16—611no—
388C2×M16—611no—
389C2×M16—1221no—
390C2×M16—1222no—
391C2×M16—1222no—
392C2×M16—1221no—
393C2×M16C22×D6111yescharacteristic
394C2×M16C22×D6111yescharacteristic
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—612no—
398C22×C8—612no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—1221no—
402C22×C8—1222no—
403C22×C8—1222no—
404C22×C8—1221no—
405#123—21no—
406#123—1221no—
407C22⋊Q8—612no—
408C22⋊Q8C4×D6112yes—
409#121—612no—
410#121C4×D6112yes—
411#120—21no—
412#120—1221no—
413C4×D4—22no—
414C4×D4—1222no—
415#117—22no—
416#117—22no—
417#117—611no—
418#117C22×D6111yescharacteristic
419C2×C4⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C42C22×D6111yescharacteristic
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—22no—
431#104—1222no—
432#104—1222no—
433#103—612no—
434#103—612no—
435#102—612no—
436#102—612no—
437C22⋊C8—22no—
438C22⋊C8—1222no—
439C22⋊C8—1222no—
440C22⋊C8—1222no—
441#97—614no—
442#97—1222no—
443#97—1221no—
444#97—1221no—
445C4×C8—1222no—
446C4×C8—1221no—
447C4×C8—1221no—
448C4×C8—1222no—
449C24#1175822411yescharacteristic
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—22no—
462C2×D4—611no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4C2×C4×D6111yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1221no—
474C22×C4C2×C4×D6112yes—
475C22×C4D4×D6112yes—
476C22×C4#1175822411yescharacteristic
477C22×C4C23×D6111yescharacteristic
478Q16—1222no—
479SD16—1222no—
480SD16—1222no—
481D8—1222no—
482M16—21no—
483M16—21no—
484M16—21no—
485M16—21no—
486M16—1224no—
487M16—1224no—
488M16—1221no—
489M16—1221no—
490M16—1221no—
491M16—1221no—
492M16—2442no—
493M16—2442no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—612no—
499C2×C8—612no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—612no—
503C2×C8—612no—
504C2×C8—612no—
505C2×C8—612no—
506C2×C8—1224no—
507C2×C8—1224no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—2442no—
515C2×C8—2442no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—612no—
519C4⋊C4—1222no—
520C4⋊C4C2×C4×D6112yes—
521C22⋊C4—24no—
522C22⋊C4—21no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—1221no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—1221no—
532C42D4×D6112yes—
533C42#1175822412yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—42no—
538C23—1221no—
539C23#1451973712yes—
540C23C2×#1175822411yescharacteristic
541Q8—21no—
542Q8—21no—
543Q8—1221no—
544Q8—1221no—
545D4—21no—
546D4—21no—
547D4—41no—
548D4—41no—
549D4—82no—
550D4—1221no—
551D4—1221no—
552D4—4882no—
553D4—2441no—
554D4—2441no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4#1451973712yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C2×C4C2×#1175822411yescharacteristic
580C8—41no—
581C8—41no—
582C8—612no—
583C8—612no—
584C8—612no—
585C8—612no—
586C8—1222no—
587C8—1222no—
588C8—2441no—
589C8—2441no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—42no—
598C22—2441no—
599C22—2442no—
600C22C2×#1451973711yescharacteristic
601C22#1684026811yescharacteristic
602C22#1684028911yescharacteristic
603C4—22no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—42no—
608C4—2441no—
609C4—2442no—
610C4C4×D4×D6111yescharacteristic
611C4C4×#1175822411yescharacteristic
612C4#121×D6111yescharacteristic
613C4#1684030711yescharacteristic
614C2—21no—
615C2—21no—
616C2—42no—
617C2—81no—
618C2—4881no—
619C2#2144832911yescharacteristic
620C2#2145473212yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1897C2
socleC22×C61C2×C4×D4
prime radical#21455397C2
cosocleC22×C4×C61C24
hypercenter#1897D61
hypocenterC61#21547
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1897

Complements

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

NH ≅ G/Ntype
2. C2×#16839881C2semidirect
3. #21454523C2semidirect
4. #21454518C2semidirect
5. C2×#16839718C2semidirect
6. #21446709C2semidirect
7. #21446571C2semidirect
8. #21446524C2semidirect
9. #21446520C2semidirect
10. #21446267C2semidirect
11. #21445855C2semidirect
12. #21445702C2semidirect
13. #21445657C2semidirect
14. #21445653C2semidirect
15. C61×#1897C2semidirect
22. #16840001C4semidirect
23. #16839996C4semidirect
24. #16839993C4semidirect
25. #16839989C4semidirect
113. #14519688D4semidirect
314. #1897D61semidirect
371. C61#21547semidirect

Maximal subgroups

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

subgroupindex
#214553972
C2×#168398812
#214545232
#214545182
C2×#168397182
#214467092
#214465712
#214465242
#214465202
#214462672
#214458552
#214457022
#214456572
#214456532
C61×#18972
#2154761