Finite Group Databasebeta

← #29707192

Subgroups — #29707192

Order 15616.

1916 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
1C2×#16840560C212yesmaximal, complemented
2#21454264C212yesmaximal, complemented
3#21454263C211yesmaximal, characteristic, complemented
4#21446922C218yesmaximal, complemented
5C2×#16839520C211yesmaximal, characteristic, complemented
6C2×C61×#575C211yesmaximal, characteristic, complemented
7C22×#11758222C2211yescharacteristic
8#16840560—24no—
9#16840560—24no—
10#16840560—24no—
11#16840560C2218yescomplemented
12C2×#14519808—22no—
13C2×#14519807—22no—
14C2×#14519803—22no—
15C2×#14519802—22no—
16C2×#14519797—24no—
17C22×#11758194C2212yes—
18C22×#11758194C2211yescharacteristic
19#16839730C2214yescomplemented
20#16839729—24no—
21#16839729C2218yescomplemented
22#16839728—22no—
23#16839726—24no—
24#16839725—22no—
25#16839520C2214yescomplemented
26C22×#41×C61C2212yes—
27C61×#575—24no—
28C61×#575—22no—
29C61×#575C2214yescomplemented
30C2×#137×C61—22no—
31C2×AGL(1,ℤ/8)×C61—22no—
32C2×#135×C61—22no—
33C22×M16×C61C2211yescharacteristic
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—22no—
37C2×#11758222—22no—
38C2×#11758222—22no—
39C2×#11758222—22no—
40C2×#11758222D412yes—
41C2×#11758222C2312yes—
42C22×Dic122D411yescharacteristic
43C22×D244D411yescharacteristic
44C22×C22⋊D61—21no—
45C22×C4×D61—21no—
46#14519808—28no—
47#14519808—28no—
48#14519807—28no—
49#14519807—28no—
50#14519803—28no—
51#14519803—28no—
52#14519802—28no—
53#14519802—28no—
54#14519797—216no—
55#14519797—216no—
56C2×#11758202—24no—
57C2×#11758200—24no—
58C2×#11758201—24no—
59C2×#11758199—24no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194—21no—
65C2×#11758194—22no—
66C2×#11758194—21no—
67C2×#11758194—22no—
68C2×#11758194D414yes—
69C2×#11758194C2314yes—
70C22×#92418D414yes—
71C22×#92418C2312yes—
72C22⋊#92418—216no—
73#14519689—28no—
74C2×#41×C61—22no—
75C2×#41×C61—22no—
76C2×#41×C61—22no—
77C2×#41×C61—22no—
78C2×#41×C61—22no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61D414yes—
85C2×#41×C61C2314yes—
86C22×Q8×C61D412yes—
87C22×D4×C61—22no—
88C22×D4×C61D412yes—
89C23×C4×C61—22no—
90C23×C4×C61C2311yescharacteristic
91#137×C61—28no—
92#137×C61—28no—
93AGL(1,ℤ/8)×C61—28no—
94AGL(1,ℤ/8)×C61—28no—
95#135×C61—28no—
96#135×C61—28no—
97C2×Q16×C61—22no—
98C2×SD16×C61—24no—
99C2×D8×C61—22no—
100C2×M16×C61—22no—
101C2×M16×C61—22no—
102C2×M16×C61D412yes—
103C2×M16×C61C2312yes—
104C22×C8×C61D412yes—
105C24×C61—22no—
106C24×C61#4111yescharacteristic
107C23×D61—21no—
108#41×C61—24no—
109#41×C61—24no—
110#41×C61—24no—
111#41×C61—24no—
112#41×C61—24no—
113#41×C61—24no—
114#41×C61—24no—
115#41×C61—24no—
116#41×C61—42no—
117#41×C61—42no—
118#41×C61—42no—
119#41×C61—42no—
120#41×C61—42no—
121#41×C61—42no—
122#41×C61—42no—
123#41×C61—42no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61C2×D414yes—
130C2×D4×C61—24no—
131C2×D4×C61—24no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—42no—
138C2×D4×C61—42no—
139C2×D4×C61—42no—
140C2×D4×C61—42no—
141C2×D4×C61C2×D414yes—
142C22×C4×C61—22no—
143C22×C4×C61—24no—
144C22×C4×C61—24no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61#4111yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C2×C22⋊D61—22no—
160C2×C22⋊D61—22no—
161C2×C22⋊D61—42no—
162C2×C22⋊D61—42no—
163C22×Dic61—21no—
164#11758222—24no—
165#11758222—24no—
166#11758222—24no—
167#11758222—24no—
168#11758222—41no—
169#11758222—41no—
170#11758222—42no—
171#11758222—42no—
172#11758222—41no—
173#11758222—41no—
174C2×D244—22no—
175C2×D244—22no—
176C2×D244—21no—
177C2×D244C2×D412yes—
178C2×C4×D61—22no—
179C2×C4×D61—22no—
180C2×C4×D61—42no—
181C2×C4×D61—42no—
182C2×Dic122—22no—
183C2×Dic122—22no—
184C2×Dic122—21no—
185C2×Dic122C2×D412yes—
186Q16×C61—28no—
187SD16×C61—216no—
188D8×C61—28no—
189M16×C61—24no—
190M16×C61—24no—
191C2×C8×C61—22no—
192C2×C8×C61—22no—
193C2×C8×C61C2×D414yes—
194#11758202—216no—
195#11758201—216no—
196#11758200—216no—
197#11758199—216no—
198#11758194—24no—
199#11758194—24no—
200#11758194—24no—
201#11758194—24no—
202#11758194—42no—
203#11758194—41no—
204#11758194—41no—
205C2×#92418—24no—
206C2×#92418—24no—
207C2×#92418—22no—
208C2×#92418—24no—
209C2×#92418C2×D418yes—
210C23×C61—24no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—81no—
214C23×C61—84no—
215C23×C61#12111yescharacteristic
216C23×C61#12111yescharacteristic
217C23×C61#12112yes—
218C23×C61#12112yes—
219C23×C61C2×#4111yescharacteristic
220C22×D61—22no—
221C22×D61—42no—
222C22×D61—82no—
223Q8×C61—24no—
224Q8×C61—24no—
225Q8×C61—42no—
226Q8×C61—42no—
227D4×C61—24no—
228D4×C61—24no—
229D4×C61—44no—
230D4×C61—44no—
231D4×C61—44no—
232D4×C61—44no—
233D4×C61—42no—
234D4×C61—42no—
235C2×C4×C61—24no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—44no—
239C2×C4×C61—44no—
240C2×C4×C61—44no—
241C2×C4×C61—44no—
242C2×C4×C61#12012yes—
243C2×C4×C61#12012yes—
244C2×C4×C61#12012yes—
245C2×C4×C61#12012yes—
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61#12012yes—
249C2×C4×C61#12111yescharacteristic
250C2×C4×C61#12111yescharacteristic
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61#12712yes—
254C2×C4×C61#12711yescharacteristic
255C2×C4×C61#12711yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D412yes—
258C2×C4×C61C22×D412yes—
259C2×C4×C61C22×D411yescharacteristic
260C22⋊D61—44no—
261C22⋊D61—42no—
262C22⋊D61—42no—
263C2×Dic61—22no—
264C2×Dic61—42no—
265D244—24no—
266D244—41no—
267D244—41no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61—44no—
271Dic122—24no—
272Dic122—41no—
273Dic122—41no—
274C8×C61—24no—
275#92418—28no—
276#92418—42no—
277#23434—611nomaximal, Sylow
278C22×C61—41no—
279C22×C61—42no—
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—41no—
283C22×C61—48no—
284C22×C61—44no—
285C22×C61—84no—
286C22×C61—84no—
287C22×C61#39212yes—
288C22×C61#39211yescharacteristic
289C22×C61#39211yescharacteristic
290C22×C61C2×#12111yescharacteristic
291C22×C61C2×#12112yes—
292C2×D61—44no—
293C2×D61—42no—
294C2×D61—84no—
295C4×C61—44no—
296C4×C61#39211yescharacteristic
297C4×C61#39211yescharacteristic
298C4×C61#39212yes—
299C4×C61C2×#12012yes—
300C4×C61C2×#12011yescharacteristic
301C4×C61C2×#12711yescharacteristic
302Dic61—42no—
303C2×#575—612no—
304C2×#575D6111yescharacteristic, complemented
305#2879—612no—
306#2879—611no—
307#2017—618no—
308C2×#338—611no—
309C2×C61—21no—
310C2×C61—22no—
311C2×C61—22no—
312C2×C61—21no—
313C2×C61—21no—
314C2×C61—84no—
315C2×C61C2×#39211yescharacteristic
316D61—82no—
317C22×#41—611no—
318C22×#41C2×D6112yes—
319#575—24no—
320#575—22no—
321#575—618no—
322#575—1224no—
323#575—1224no—
324#575—1224no—
325#575C2×D6114yescomplemented
326C2×#137—22no—
327C2×#137—1222no—
328C2×#137—1222no—
329C2×AGL(1,ℤ/8)—22no—
330C2×AGL(1,ℤ/8)—1222no—
331C2×AGL(1,ℤ/8)—1222no—
332C2×#135—22no—
333C2×#135—1224no—
334C22×M16—612no—
335C22×M16—611no—
336C22×M16C2×D6111yescharacteristic
337#408—618no—
338#408—614no—
339#408—1224no—
340#408—1222no—
341#407—1224no—
342#407—1222no—
343#338—614no—
344C61#2343411yescharacteristic, complemented, Sylow
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41—22no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41—612no—
356C2×#41—612no—
357C2×#41—1221no—
358C2×#41—1221no—
359C2×#41—1222no—
360C2×#41—1222no—
361C2×#41—1222no—
362C2×#41—1222no—
363C2×#41C22⋊D6114yes—
364C2×#41C22×D6114yes—
365C22×Q8—611no—
366C22×Q8C22⋊D6112yes—
367C22×D4—22no—
368C22×D4—611no—
369C22×D4—1221no—
370C22×D4C22⋊D6112yes—
371C23×C4—22no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374#137—28no—
375#137—28no—
376#137—1228no—
377#137—1228no—
378#137—1228no—
379#137—1228no—
380AGL(1,ℤ/8)—28no—
381AGL(1,ℤ/8)—28no—
382AGL(1,ℤ/8)—1228no—
383AGL(1,ℤ/8)—1228no—
384AGL(1,ℤ/8)—1228no—
385AGL(1,ℤ/8)—1228no—
386#135—28no—
387#135—28no—
388#135—12216no—
389#135—12216no—
390C2×Q16—22no—
391C2×Q16—1224no—
392C2×SD16—24no—
393C2×SD16—1224no—
394C2×SD16—1224no—
395C2×D8—22no—
396C2×D8—1224no—
397C2×M16—22no—
398C2×M16—22no—
399C2×M16—614no—
400C2×M16—614no—
401C2×M16—1222no—
402C2×M16—1222no—
403C2×M16—1222no—
404C2×M16—1222no—
405C2×M16—1222no—
406C2×M16—1222no—
407C2×M16—1221no—
408C2×M16—1221no—
409C2×M16C22⋊D6112yes—
410C2×M16C22×D6112yes—
411C22×C8—614no—
412C22×C8—612no—
413C22×C8C22⋊D6112yes—
414C22⋊C8—12216no—
415C22⋊C8—1228no—
416C24—22no—
417C24—1221no—
418C24#1175822411yescharacteristic
419#41—24no—
420#41—24no—
421#41—24no—
422#41—24no—
423#41—24no—
424#41—24no—
425#41—24no—
426#41—24no—
427#41—42no—
428#41—42no—
429#41—42no—
430#41—42no—
431#41—42no—
432#41—42no—
433#41—42no—
434#41—42no—
435#41—1224no—
436#41—1224no—
437#41—1224no—
438#41—1224no—
439#41—2441no—
440#41—2441no—
441#41—2442no—
442#41—2442no—
443#41—2441no—
444#41—2441no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—612no—
451C2×Q8—1221no—
452C2×Q8—1222no—
453C2×Q8—1222no—
454C2×Q8C2×C22⋊D6114yes—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—612no—
467C2×D4—1222no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1221no—
472C2×D4—2442no—
473C2×D4—2442no—
474C2×D4C2×C22⋊D6114yes—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—1221no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—2442no—
486C22×C4—2442no—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6112yes—
490C22×C4D4×D6112yes—
491C22×C4#1175822411yescharacteristic
492C22×C4C2×C22⋊D6112yes—
493C22×C4C2×C22⋊D6112yes—
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497Q16—28no—
498Q16—12216no—
499SD16—216no—
500SD16—12216no—
501SD16—12216no—
502D8—28no—
503D8—12216no—
504M16—24no—
505M16—24no—
506M16—1224no—
507M16—1224no—
508M16—1224no—
509M16—1224no—
510M16—2441no—
511M16—2442no—
512M16—2441no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—618no—
516C2×C8—1224no—
517C2×C8—1224no—
518C2×C8—1224no—
519C2×C8—1222no—
520C2×C8C2×C22⋊D6114yes—
521C23—24no—
522C23—42no—
523C23—42no—
524C23—81no—
525C23—84no—
526C23—1222no—
527C23—4882no—
528C23—2442no—
529C23#1451974811yescharacteristic
530C23#1451974811yescharacteristic
531C23#1451974912yes—
532C23#1451974912yes—
533C23C2×#1175822411yescharacteristic
534Q8—24no—
535Q8—24no—
536Q8—42no—
537Q8—42no—
538Q8—1224no—
539Q8—2441no—
540Q8—2441no—
541D4—24no—
542D4—24no—
543D4—44no—
544D4—44no—
545D4—44no—
546D4—44no—
547D4—42no—
548D4—42no—
549D4—1224no—
550D4—2444no—
551D4—2442no—
552D4—2442no—
553D4—2441no—
554D4—2441no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—1222no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2444no—
567C2×C4#1451974512yes—
568C2×C4#1451974512yes—
569C2×C4#1451974512yes—
570C2×C4#1451974512yes—
571C2×C4#1451974611yescharacteristic
572C2×C4#1451974611yescharacteristic
573C2×C4#1451974612yes—
574C2×C4#1451974811yescharacteristic
575C2×C4#1451974811yescharacteristic
576C2×C4#1451974912yes—
577C2×C4#1451974912yes—
578C2×C4#1451977311yescharacteristic
579C2×C4#1451977312yes—
580C2×C4#1451977311yescharacteristic
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×D4×D6112yes—
583C2×C4C22×C22⋊D6112yes—
584C2×C4C22×C22⋊D6111yescharacteristic
585C8—24no—
586C8—1228no—
587C8—2442no—
588C22—42no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—48no—
594C22—44no—
595C22—84no—
596C22—84no—
597C22—4884no—
598C22—2444no—
599C22—2442no—
600C22#1683967011yescharacteristic
601C22#1683967012yes—
602C22#1683967011yescharacteristic
603C22C2×#1451974811yescharacteristic
604C22C2×#1451974912yes—
605C4—44no—
606C4—2442no—
607C4#1683967011yescharacteristic
608C4#1683967012yes—
609C4#1683967011yescharacteristic
610C4C2×#1451974512yes—
611C4C2×#1451974611yescharacteristic
612C4C2×#1451977311yescharacteristic
613C2—22no—
614C2—21no—
615C2—22no—
616C2—21no—
617C2—21no—
618C2—84no—
619C2—4882no—
620C2C2×#1683967011yescharacteristic
subgroupG/H
centerC4C2×#14519773
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#575C2
socleC2×C61C2×#392
hypercenterC2×#575D61
hypocenterC61#23434
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#575

Complements

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

NH ≅ G/Ntype
1. C2×#16840560C2semidirect
2. #21454264C2semidirect
3. #21454263C2semidirect
4. #21446922C2semidirect
5. C2×#16839520C2semidirect
6. C2×C61×#575C2semidirect
11. #16840560C22semidirect
19. #16839730C22semidirect
21. #16839729C22semidirect
25. #16839520C22semidirect
29. C61×#575C22semidirect
304. C2×#575D61semidirect
325. #575C2×D61semidirect
344. C61#23434semidirect

Maximal subgroups

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

subgroupindex
C2×#168405602
#214542642
#214542632
#214469222
C2×#168395202
C2×C61×#5752
#2343461