Finite Group Databasebeta

← #29716022

Subgroups — #29716022

Order 15616.

1358 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840029C211yesmaximal, characteristic, complemented
2#21455477C211yesmaximal, characteristic, complemented
3#21455123C212yesmaximal, complemented
4C2×#16839882C211yesmaximal, characteristic, complemented
5#21447108C212yesmaximal, complemented
6#21446859C212yesmaximal, complemented
7#21446713C211yesmaximal, characteristic, complemented
8#21446696C211yesmaximal, characteristic, complemented
9#21445878C211yesmaximal, characteristic, complemented
10#21445773C212yesmaximal, complemented
11C2×C61×#576C211yesmaximal, characteristic, complemented
12C2×#14519803C2212yescomplemented
13C2×#14519797C2211yescharacteristic
14C2×#14519783C2211yescharacteristic, complemented
15#16840340C2212yescomplemented
16#16840029—24no—
17#16840017—21no—
18#16840012—21no—
19#16839988—22no—
20#16839924—22no—
21#16839913—22no—
22#16839907—22no—
23#16839896—22no—
24#16839882—24no—
25C2×#14519713C2211yescharacteristic, complemented
26C2×#14519712C2211yescharacteristic, complemented
27C2×#14519709C2211yescharacteristic, complemented
28C2×#14519704C2211yescharacteristic
29C2×#14519704C2212yescomplemented
30C2×#14519703C2211yescharacteristic, complemented
31#16839763C2212yes—
32#16839763C2212yescomplemented
33C2×#14519698C2212yescomplemented
34C2×#14519694C2212yescomplemented
35#16839729C2212yescomplemented
36C2×#14519687C2211yescharacteristic, complemented
37#16839665C2212yescomplemented
38#16839530C2212yescomplemented
39C2×H(5,𝔽2)×C61C2212yes—
40C61×#576—24no—
41C61×#576—24no—
42C2×AGL(1,ℤ/8)×C61—22no—
43C2×AGL(1,ℤ/8)×C61C2212yes—
44C2×#135×C61—21no—
45C2×#135×C61C2211yescharacteristic
46C22×D8×C61—21no—
47C22×D8×C61C2211yescharacteristic
48C2×#131×C61C2211yescharacteristic
49C2×#11758222C2311yescharacteristic
50#14519803—24no—
51#14519803—42no—
52#14519797—22no—
53#14519797—22no—
54C2×#11758202D411yescharacteristic
55C2×#11758200—22no—
56C2×#11758200D411yescharacteristic
57C2×#11758201D411yescharacteristic
58C2×#11758199—22no—
59C2×#11758199D411yescharacteristic
60#14519783—22no—
61#14519783—22no—
62C2×#11758194C2312yes—
63C2×#11758194C2311yescharacteristic
64C2×#11758189—21no—
65C22×#92418C2311yescharacteristic
66C2×C8×D61—21no—
67#14519756—22no—
68#14519753—22no—
69#14519751—22no—
70#14519749—42no—
71#14519748—22no—
72C2×C4⋊Dic61C2312yes—
73C2×C22⋊Dic61—42no—
74C2×C4×Dic61C2311yescharacteristic
75#14519713—22no—
76#14519712—22no—
77#14519709—21no—
78#14519709—21no—
79#14519704—22no—
80#14519704—42no—
81#14519703—22no—
82#14519701—22no—
83#14519701—22no—
84#14519698—22no—
85#14519698—22no—
86#14519698—22no—
87#14519698—22no—
88#14519694—24no—
89C22⋊#92418—22no—
90C22⋊#92418—22no—
91#14519687—22no—
92H(5,𝔽2)×C61—24no—
93H(5,𝔽2)×C61—24no—
94H(5,𝔽2)×C61—24no—
95H(5,𝔽2)×C61—42no—
96C2×#41×C61—22no—
97C2×#41×C61—22no—
98C2×#41×C61C2312yes—
99C2×#41×C61C2311yescharacteristic
100C22×D4×C61—22no—
101C22×D4×C61—22no—
102C22×D4×C61—42no—
103C22×D4×C61C2312yes—
104AGL(1,ℤ/8)×C61—216no—
105AGL(1,ℤ/8)×C61—24no—
106AGL(1,ℤ/8)×C61—24no—
107#135×C61—28no—
108#135×C61—22no—
109#135×C61—22no—
110C2×Q16×C61D411yescharacteristic
111C2×SD16×C61—22no—
112C2×SD16×C61D412yes—
113C2×D8×C61—28no—
114C2×D8×C61—22no—
115C2×D8×C61—21no—
116C2×D8×C61—21no—
117C2×D8×C61—22no—
118C2×D8×C61D411yescharacteristic
119C2×D8×C61D414yescomplemented
120#131×C61—22no—
121#131×C61D414yescomplemented
122C2×M16×C61—21no—
123C2×M16×C61C2311yescharacteristic
124C22×C8×C61—21no—
125C22×C8×C61C2311yescharacteristic
126C24×C61—22no—
127C24×C61—42no—
128#41×C61—21no—
129#41×C61—21no—
130#41×C61—28no—
131#41×C61—22no—
132#41×C61—22no—
133#41×C61—22no—
134#41×C61—22no—
135#41×C61—44no—
136#41×C61—44no—
137#41×C61—44no—
138#41×C61C2×D412yes—
139#41×C61#4112yes—
140C2×Q8×C61C2×D412yes—
141C2×Q8×C61C2×D411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61—28no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—44no—
151C2×D4×C61—42no—
152C2×D4×C61—42no—
153C2×D4×C61—44no—
154C2×D4×C61—48no—
155C2×D4×C61—48no—
156C2×D4×C61—44no—
157C2×D4×C61C2×D412yes—
158C2×D4×C61C2×D414yes—
159C2×D4×C61C2×D411yescharacteristic
160C22×C4×C61—21no—
161C22×C4×C61—22no—
162C22×C4×C61—42no—
163C22×C4×C61C2411yescharacteristic
164C2×C22⋊D61—21no—
165C2×C22⋊D61—42no—
166C22×Dic61—21no—
167C22×Dic61—22no—
168#11758222—21no—
169#11758222—21no—
170#11758222—21no—
171#11758222—21no—
172C2×D244C2×D411yescharacteristic
173C2×C4×D61—21no—
174C2×Dic122C2×D411yescharacteristic
175Q16×C61—22no—
176SD16×C61—28no—
177SD16×C61—24no—
178D8×C61—28no—
179D8×C61—22no—
180D8×C61—24no—
181D8×C61—24no—
182M16×C61—21no—
183M16×C61—24no—
184M16×C61C2×D412yes—
185C2×C8×C61—21no—
186C2×C8×C61—24no—
187C2×C8×C61C2×D411yescharacteristic
188C2×C8×C61C2×D411yescharacteristic
189C2×C8×C61C2×D412yes—
190C2×C8×C61C2×D412yes—
191C22⋊Dic61—42no—
192C22⋊Dic61—84no—
193#11758202—22no—
194#11758201—22no—
195#11758200—22no—
196#11758200—44no—
197#11758199—22no—
198#11758199—44no—
199#11758198—42no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202#11758196—42no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205#11758194—21no—
206#11758194—22no—
207#11758194—21no—
208#11758194—22no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—22no—
212C2×#92418—21no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215#11758189—42no—
216C8×D61—42no—
217C23×C61—21no—
218C23×C61—22no—
219C23×C61—24no—
220C23×C61—48no—
221C23×C61—42no—
222C23×C61—42no—
223C23×C61—82no—
224C23×C61—82no—
225C23×C61—162no—
226C23×C61—162no—
227C23×C61H(5,𝔽2)11yescharacteristic
228C22×D61—21no—
229Q8×C61—21no—
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61#12112yes—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—22no—
236D4×C61—24no—
237D4×C61—21no—
238D4×C61—24no—
239D4×C61—22no—
240D4×C61—48no—
241D4×C61—48no—
242D4×C61—42no—
243D4×C61—42no—
244D4×C61—42no—
245D4×C61—42no—
246D4×C61#12112yes—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—48no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C22⋊D61—41no—
261C22⋊D61—41no—
262C22⋊D61—162no—
263C2×Dic61—21no—
264C2×Dic61—42no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—82no—
268D244—21no—
269D244—21no—
270C4×D61—41no—
271C4×D61—41no—
272Dic122—21no—
273Dic122—21no—
274C8×C61—22no—
275C8×C61#12012yes—
276C8×C61#12012yes—
277#92418—21no—
278#92418—21no—
279#92418—41no—
280#92418—42no—
281#24294—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—44no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—84no—
290C22×C61—82no—
291C22×C61—82no—
292C22×C61—162no—
293C22×C61—168no—
294C22×C61#53511yescharacteristic
295C22×C61D4211yescharacteristic
296C22×C61#54611yescharacteristic
297C2×D61—41no—
298C2×D61—42no—
299C4×C61—22no—
300C4×C61—42no—
301C4×C61C2×#12011yescharacteristic
302C4×C61C2×#12111yescharacteristic
303C4×C61D4211yescharacteristic
304C4×C61#54611yescharacteristic
305Dic61—41no—
306Dic61—81no—
307Dic61—82no—
308C2×#576D6111yescharacteristic, complemented
309C2×#472—611no—
310#3060—611no—
311#3012—612no—
312C2×#444—611no—
313#2051—612no—
314#2006—612no—
315#1975—611no—
316#1972—611no—
317#1815—611no—
318#1792—612no—
319C2×C61—21no—
320C2×C61—21no—
321C2×C61—42no—
322C2×C61—82no—
323C2×C61—84no—
324C2×C61#240811yescharacteristic
325C2×C61#328812yes—
326D61—81no—
327C2×H(5,𝔽2)C2×D6112yes—
328#576—24no—
329#576—24no—
330C2×AGL(1,ℤ/8)—22no—
331C2×AGL(1,ℤ/8)—612no—
332C2×AGL(1,ℤ/8)C2×D6112yes—
333C2×#135—21no—
334C2×#135—611no—
335C2×#135C2×D6111yescharacteristic
336C22×D8—21no—
337C22×D8C2×D6111yescharacteristic
338C2×#131—611no—
339C2×#131C2×D6111yescharacteristic
340#537—612no—
341#472—1224no—
342#470—1221no—
343#469—1221no—
344#465—1222no—
345#452—1222no—
346#450—1222no—
347#449—1222no—
348#447—1222no—
349#444—1224no—
350C2×#108—611no—
351C2×#108—611no—
352C2×#106—611no—
353C2×#104—611no—
354C2×#104—611no—
355C2×#104—612no—
356#417—612no—
357#417—612no—
358C2×#102—612no—
359C2×#100—612no—
360#408—612no—
361C2×#97—611no—
362#390—612no—
363#343—612no—
364C61#2429411yescharacteristic, complemented, Sylow
365H(5,𝔽2)—24no—
366H(5,𝔽2)—24no—
367H(5,𝔽2)—24no—
368H(5,𝔽2)—42no—
369C2×#41—22no—
370C2×#41—22no—
371C2×#41—611no—
372C2×#41C22×D6111yescharacteristic
373C2×#41C22×D6112yes—
374C22×D4—22no—
375C22×D4—22no—
376C22×D4—42no—
377C22×D4C22×D6112yes—
378AGL(1,ℤ/8)—216no—
379AGL(1,ℤ/8)—24no—
380AGL(1,ℤ/8)—24no—
381AGL(1,ℤ/8)—1224no—
382AGL(1,ℤ/8)—2442no—
383#135—28no—
384#135—22no—
385#135—22no—
386#135—1222no—
387#135—1222no—
388C2×Q16—611no—
389C2×Q16C22⋊D6111yescharacteristic
390C2×SD16—22no—
391C2×SD16—611no—
392C2×SD16—611no—
393C2×SD16—1222no—
394C2×SD16C22⋊D6112yes—
395C2×D8—28no—
396C2×D8—22no—
397C2×D8—21no—
398C2×D8—21no—
399C2×D8—22no—
400C2×D8—611no—
401C2×D8—1222no—
402C2×D8C22⋊D6111yescharacteristic
403C2×D8C22⋊D6114yescomplemented
404#131—22no—
405#131—1222no—
406#131—1222no—
407#131C22⋊D6114yescomplemented
408C2×M16—21no—
409C2×M16—611no—
410C2×M16—612no—
411C2×M16—1221no—
412C2×M16C22×D6111yescharacteristic
413C22×C8—21no—
414C22×C8—611no—
415C22×C8—1221no—
416C22×C8C22×D6111yescharacteristic
417C22⋊Q8—1222no—
418C22⋊Q8—1222no—
419#121—1222no—
420#121—1222no—
421#121—2442no—
422C2×C4⋊C4—612no—
423C2×C22⋊C4—2442no—
424C2×C42—611no—
425#108—1222no—
426#108—1222no—
427#106—1221no—
428#106—1221no—
429#104—1222no—
430#104—1222no—
431#104—2442no—
432#103—1222no—
433#103—1222no—
434#102—1222no—
435#102—1222no—
436#102—1222no—
437#102—1222no—
438#100—1224no—
439C22⋊C8—1222no—
440C22⋊C8—1222no—
441#97—1222no—
442C24—22no—
443C24—42no—
444#41—21no—
445#41—21no—
446#41—28no—
447#41—22no—
448#41—22no—
449#41—22no—
450#41—22no—
451#41—44no—
452#41—44no—
453#41—44no—
454#41—1221no—
455#41—1221no—
456#41—1221no—
457#41—1221no—
458#41#1175822412yes—
459#41C2×C22⋊D6112yes—
460C2×Q8—611no—
461C2×Q8D4×D6111yescharacteristic
462C2×Q8C2×C22⋊D6112yes—
463C2×D4—21no—
464C2×D4—28no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—44no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—44no—
475C2×D4—48no—
476C2×D4—48no—
477C2×D4—44no—
478C2×D4—611no—
479C2×D4—1221no—
480C2×D4—2442no—
481C2×D4D4×D6111yescharacteristic
482C2×D4C2×C22⋊D6114yes—
483C2×D4C2×C22⋊D6112yes—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—1221no—
488C22×C4—1222no—
489C22×C4—1221no—
490C22×C4C23×D6111yescharacteristic
491Q16—22no—
492Q16—1222no—
493SD16—28no—
494SD16—24no—
495SD16—1222no—
496SD16—1222no—
497SD16—2444no—
498D8—28no—
499D8—22no—
500D8—24no—
501D8—24no—
502D8—1222no—
503D8—2444no—
504M16—21no—
505M16—24no—
506M16—1221no—
507M16—1222no—
508M16—1221no—
509M16—1222no—
510M16—2442no—
511M16C2×C22⋊D6112yes—
512C2×C8—21no—
513C2×C8—24no—
514C2×C8—611no—
515C2×C8—611no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1222no—
519C2×C8—1221no—
520C2×C8—2442no—
521C2×C8C2×C22⋊D6112yes—
522C2×C8C2×C22⋊D6111yescharacteristic
523C2×C8C2×C22⋊D6111yescharacteristic
524C2×C8C2×C22⋊D6112yes—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—2442no—
528C22⋊C4—4884no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—1221no—
532C42—1221no—
533C23—21no—
534C23—22no—
535C23—24no—
536C23—48no—
537C23—42no—
538C23—42no—
539C23—82no—
540C23—82no—
541C23—162no—
542C23—162no—
543C23—1221no—
544C23#1451982211yescharacteristic
545Q8—21no—
546Q8—22no—
547Q8—22no—
548Q8—1221no—
549Q8—1221no—
550Q8#1451974812yes—
551D4—22no—
552D4—22no—
553D4—22no—
554D4—24no—
555D4—21no—
556D4—24no—
557D4—22no—
558D4—48no—
559D4—48no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—1221no—
565D4—1221no—
566D4—9762no—
567D4—2441no—
568D4—2441no—
569D4#1451974812yes—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—1221no—
578C2×C4—4882no—
579C2×C4—2442no—
580C2×C4—2441no—
581C2×C4—2442no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4C22×C22⋊D6111yescharacteristic
586C2×C4C22×C22⋊D6111yescharacteristic
587C2×C4C22×C22⋊D6111yescharacteristic
588C2×C4C2×#1175822411yescharacteristic
589C2×C4#1451982211yescharacteristic
590C8—22no—
591C8—1221no—
592C8—1221no—
593C8—2441no—
594C8—2442no—
595C8#1451974612yes—
596C8#1451974612yes—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—84no—
605C22—82no—
606C22—82no—
607C22—162no—
608C22—168no—
609C22—2442no—
610C22—2441no—
611C22#1684032911yescharacteristic
612C22D4×C22⋊D6111yescharacteristic
613C22#1684040611yescharacteristic
614C4—22no—
615C4—42no—
616C4—4881no—
617C4—4882no—
618C4—2441no—
619C4C2×#1451974611yescharacteristic
620C4C2×#1451974811yescharacteristic
621C4D4×C22⋊D6111yescharacteristic
622C4#1684040611yescharacteristic
623C2—21no—
624C2—21no—
625C2—42no—
626C2—82no—
627C2—84no—
628C2—4881no—
629C2#2144916911yescharacteristic
630C2#2145758312yes—
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#576C2
socleC22×C61D42
hypercenterC2×#576D61
hypocenterC61#24294
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#576

Complements

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

NH ≅ G/Ntype
1. C2×#16840029C2semidirect
2. #21455477C2semidirect
3. #21455123C2semidirect
4. C2×#16839882C2semidirect
5. #21447108C2semidirect
6. #21446859C2semidirect
7. #21446713C2semidirect
8. #21446696C2semidirect
9. #21445878C2semidirect
10. #21445773C2semidirect
11. C2×C61×#576C2semidirect
12. C2×#14519803C22semidirect
14. C2×#14519783C22semidirect
15. #16840340C22semidirect
25. C2×#14519713C22semidirect
26. C2×#14519712C22semidirect
27. C2×#14519709C22semidirect
29. C2×#14519704C22semidirect
30. C2×#14519703C22semidirect
32. #16839763C22semidirect
33. C2×#14519698C22semidirect
34. C2×#14519694C22semidirect
35. #16839729C22semidirect
36. C2×#14519687C22semidirect
37. #16839665C22semidirect
38. #16839530C22semidirect
119. C2×D8×C61D4semidirect
121. #131×C61D4semidirect
308. C2×#576D61semidirect
364. C61#24294semidirect
403. C2×D8C22⋊D61semidirect
407. #131C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400292
#214554772
#214551232
C2×#168398822
#214471082
#214468592
#214467132
#214466962
#214458782
#214457732
C2×C61×#5762
#2429461