Finite Group Databasebeta

← #29658482

Subgroups — #29658482

Order 15616.

1688 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456737C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic, complemented
4#21444998C214yesmaximal, complemented
5#21444873C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×D42×C61C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic, complemented
12#16840138C2212yescomplemented
13#16840108C2212yescomplemented
14#16840075—21no—
15#16840042C2212yescomplemented
16#16839977—22no—
17#16839967C2212yescomplemented
18#16839957C2212yescomplemented
19#16839957C2212yescomplemented
20#16839954—22no—
21#16839896—22no—
22#16839896—21no—
23C4×#11758199—22no—
24#16839815C2211yescharacteristic, complemented
25#16839815C2211yescharacteristic, complemented
26#16839786C2211yescharacteristic, complemented
27#16839761—21no—
28#16839506C2214yescomplemented
29D4⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C23×D4×C61—21no—
32C23×D4×C61—21no—
33D42×C61—22no—
34D42×C61—22no—
35D42×C61—22no—
36D42×C61C2214yes—
37C2×#127×C61C2211yescharacteristic
38C2×#121×C61—21no—
39C2×#121×C61—21no—
40C2×#120×C61—22no—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758222—21no—
44#14519803—44no—
45C2×#11758200—22no—
46C2×#11758199—22no—
47C2×#11758199—22no—
48C2×#11758199—24no—
49C2×#11758194—21no—
50#14519777C2311yescharacteristic, complemented
51#14519774C2311yescharacteristic, complemented
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57#14519710—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yescomplemented
61C4⋊#92418C2312yescomplemented
62#14519700—22no—
63#14519698—24no—
64#14519698—22no—
65#14519697—22no—
66#14519697—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yescomplemented
71C25×C61—22no—
72C22×D4×C61—24no—
73C22×D4×C61—24no—
74C22×D4×C61—22no—
75C22×D4×C61—22no—
76C22×D4×C61—24no—
77C22×D4×C61—24no—
78C22×D4×C61—21no—
79C22×D4×C61—21no—
80C22×D4×C61—48no—
81C22×D4×C61—44no—
82C22×D4×C61—44no—
83C22×D4×C61D411yescharacteristic
84C22×D4×C61D411yescharacteristic
85C22×D4×C61D412yes—
86C23×C4×C61—21no—
87C23×C4×C61—21no—
88#127×C61—21no—
89#127×C61—21no—
90#127×C61D412yes—
91#127×C61C2312yes—
92#121×C61—24no—
93#121×C61—24no—
94#121×C61—24no—
95#121×C61—24no—
96#120×C61—28no—
97#120×C61—28no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61D412yes—
103C4×D4×C61D412yes—
104C4×D4×C61C2312yes—
105C4×D4×C61C2312yes—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C22⋊C4×C61—22no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C42×C61C2311yescharacteristic
112C24×C61—21no—
113C24×C61—21no—
114C24×C61—28no—
115C24×C61—22no—
116C24×C61—48no—
117C24×C61—48no—
118C24×C61—84no—
119C2×D4×C61—24no—
120C2×D4×C61—28no—
121C2×D4×C61—28no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—44no—
131C2×D4×C61—416no—
132C2×D4×C61—48no—
133C2×D4×C61—48no—
134C2×D4×C61—416no—
135C2×D4×C61—48no—
136C2×D4×C61—48no—
137C2×D4×C61—48no—
138C2×D4×C61—48no—
139C2×D4×C61—42no—
140C2×D4×C61—42no—
141C2×D4×C61—82no—
142C2×D4×C61D814yescomplemented
143C2×D4×C61C2×D412yes—
144C2×D4×C61C2×D412yes—
145C2×D4×C61C2×D414yes—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—44no—
153C22×C4×C61—42no—
154C22×C4×C61—42no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C2×C22⋊D61—41no—
159#11758222—42no—
160#11758222—42no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—28no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61C2411yescharacteristic
179#11758200—44no—
180#11758199—44no—
181#11758199—44no—
182#11758199—84no—
183#11758198—41no—
184#11758198—41no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187#11758194—42no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C23×C61—24no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—24no—
195C23×C61—24no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—44no—
200C23×C61—416no—
201C23×C61—416no—
202C23×C61—44no—
203C23×C61—42no—
204C23×C61—44no—
205C23×C61—88no—
206C23×C61—832no—
207C23×C61#12011yescharacteristic
208C22×D61—41no—
209D4×C61—24no—
210D4×C61—22no—
211D4×C61—22no—
212D4×C61—24no—
213D4×C61—44no—
214D4×C61—44no—
215D4×C61—416no—
216D4×C61—416no—
217D4×C61—42no—
218D4×C61—42no—
219D4×C61—44no—
220D4×C61—44no—
221D4×C61—82no—
222D4×C61C2×D814yes—
223D4×C61AGL(1,ℤ/8)14yescomplemented
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—44no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—44no—
234C2×C4×C61—44no—
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61C2×D812yes—
239C2×C4×C61AGL(1,ℤ/8)12yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C22⋊D61—82no—
244C2×Dic61—41no—
245D244—42no—
246D244—41no—
247D244—42no—
248D244—41no—
249C4×D61—82no—
250Dic122—41no—
251Dic122—41no—
252#92418—42no—
253#92418—42no—
254#92418—42no—
255#18746—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—24no—
259C22×C61—24no—
260C22×C61—48no—
261C22×C61—48no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—44no—
267C22×C61—44no—
268C22×C61—44no—
269C22×C61—88no—
270C22×C61—832no—
271C22×C61#44712yes—
272C22×C61C2×#12011yescharacteristic
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61#44712yes—
282C4×C61C22×D811yescharacteristic
283C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C61C2×AGL(1,ℤ/8)12yes—
285Dic61—81no—
286C2×D42D6111yescharacteristic, complemented
287#3205—611no—
288#3100—611no—
289#3036—611no—
290#1639—614no—
291#1619—614no—
292#1545—611no—
293#1486—611no—
294#1478—611no—
295C2×C61—22no—
296C2×C61—44no—
297C2×C61—44no—
298C2×C61—84no—
299C2×C61C2×#44711yescharacteristic
300C2×C61#300311yescharacteristic
301C2×C61#301911yescharacteristic
302D61—161no—
303C23×D4—21no—
304C23×D4—21no—
305C2×AGL(1,ℤ/8)—1221no—
306D42—22no—
307D42—22no—
308D42—22no—
309D42C2×D6114yes—
310#532—611no—
311C2×#127C2×D6111yescharacteristic
312C2×#121—21no—
313C2×#121—21no—
314C2×#120—22no—
315C2×C4×D4C2×D6111yescharacteristic
316C2×C4×D4C2×D6111yescharacteristic
317#493—612no—
318#486—612no—
319#480—1221no—
320#474—612no—
321#463—1222no—
322#461—612no—
323#459—612no—
324#459—612no—
325#459—1222no—
326#447—1221no—
327#447—1222no—
328C4×D8—1222no—
329#431—611no—
330#431—611no—
331#423—611no—
332#417—1221no—
333#331—614no—
334D4⋊C8—614no—
335D4⋊C8—614no—
336C61#1874611yescharacteristic, complemented, Sylow
337C25—22no—
338C2×#41—1221no—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4—24no—
342C22×D4—24no—
343C22×D4—24no—
344C22×D4—24no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4—48no—
348C22×D4—44no—
349C22×D4—44no—
350C22×D4C22⋊D6111yescharacteristic
351C22×D4C22⋊D6111yescharacteristic
352C22×D4C22⋊D6112yes—
353C23×C4—21no—
354C23×C4—21no—
355AGL(1,ℤ/8)—2444no—
356C2×SD16—1222no—
357C2×D8—1224no—
358C2×D8—1222no—
359C2×D8—1222no—
360C2×M16—1221no—
361C4⋊Q8—611no—
362#127—21no—
363#127—21no—
364#127—611no—
365#127C22⋊D6112yes—
366#127C22×D6112yes—
367#124—1221no—
368#121—24no—
369#121—24no—
370#121—24no—
371#121—24no—
372#121—1221no—
373#121—1221no—
374#120—28no—
375#120—28no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—1221no—
381C4×D4—1221no—
382C4×D4C22⋊D6112yes—
383C4×D4C22⋊D6112yes—
384C4×D4C22×D6112yes—
385C4×D4C22×D6112yes—
386C2×C4⋊C4C22⋊D6111yescharacteristic
387C2×C4⋊C4C22⋊D6111yescharacteristic
388C2×C22⋊C4—22no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42C22×D6111yescharacteristic
392#107—1222no—
393C4⋊C8—612no—
394C4⋊C8—612no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397#103—1222no—
398#102—1222no—
399#102—1224no—
400#102—1222no—
401#102—1222no—
402C22⋊C8—1221no—
403C22⋊C8—1221no—
404C4×C8—612no—
405C4×C8—1221no—
406C24—21no—
407C24—21no—
408C24—28no—
409C24—22no—
410C24—48no—
411C24—48no—
412C24—84no—
413#41—2442no—
414#41—2442no—
415C2×Q8—1221no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—44no—
428C2×D4—416no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—416no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—48no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—82no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—2441no—
443C2×D4#1175819914yescomplemented
444C2×D4C2×C22⋊D6112yes—
445C2×D4C2×C22⋊D6112yes—
446C2×D4C2×C22⋊D6114yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—2441no—
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C2×C22⋊D6111yescharacteristic
460SD16—2444no—
461D8—4884no—
462D8—2444no—
463D8—2444no—
464M16—2442no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1222no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4C2×C22⋊D6112yes—
473C4⋊C4C2×C22⋊D6112yes—
474C22⋊C4—28no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—2441no—
480C22⋊C4—2441no—
481C42C2×C22⋊D6111yescharacteristic
482C42C2×C22⋊D6111yescharacteristic
483C42C2×C22⋊D6111yescharacteristic
484C42C23×D6111yescharacteristic
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—416no—
495C23—416no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—88no—
500C23—832no—
501C23—2441no—
502C23#1451974611yescharacteristic
503Q8—2441no—
504Q8—2441no—
505D4—24no—
506D4—22no—
507D4—22no—
508D4—24no—
509D4—44no—
510D4—44no—
511D4—416no—
512D4—416no—
513D4—42no—
514D4—42no—
515D4—44no—
516D4—44no—
517D4—82no—
518D4—4882no—
519D4—2442no—
520D4—2441no—
521D4—2442no—
522D4—2441no—
523D4C2×#1175819914yes—
524D4#1451980314yescomplemented
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—4882no—
537C2×C4—2441no—
538C2×C4#1451974611yescharacteristic
539C2×C4#1451974611yescharacteristic
540C2×C4#1451974611yescharacteristic
541C2×C4C2×#1175819912yes—
542C2×C4#1451980312yes—
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C22×C22⋊D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C8—2442no—
547C8—2442no—
548C8—2442no—
549C22—21no—
550C22—21no—
551C22—24no—
552C22—24no—
553C22—48no—
554C22—48no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—88no—
563C22—832no—
564C22—4881no—
565C22—4881no—
566C22—4881no—
567C22#1683989612yes—
568C22C2×#1451974611yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—42no—
574C4—4881no—
575C4#1683989612yes—
576C4C22×#1175819911yescharacteristic
577C4C2×#1451980311yescharacteristic
578C4C2×#1451980312yes—
579C2—22no—
580C2—44no—
581C2—44no—
582C2—84no—
583C2—9761no—
584C2C2×#1683989611yescharacteristic
585C2#2145504611yescharacteristic
586C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×D42×C61C2
socleC22×C61C2×#120
hypercenterC2×D42D61
hypocenterC61#18746
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. #21456737C2semidirect
2. #21455819C2semidirect
3. #21455283C2semidirect
4. #21444998C2semidirect
5. #21444873C2semidirect
6. #21444487C2semidirect
7. #21444244C2semidirect
8. #21444210C2semidirect
9. C2×D42×C61C2semidirect
11. #16840312C22semidirect
12. #16840138C22semidirect
13. #16840108C22semidirect
15. #16840042C22semidirect
17. #16839967C22semidirect
18. #16839957C22semidirect
19. #16839957C22semidirect
24. #16839815C22semidirect
25. #16839815C22semidirect
26. #16839786C22semidirect
28. #16839506C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
50. #14519777C23semidirect
51. #14519774C23semidirect
60. C4⋊#92418C23semidirect
61. C4⋊#92418C23semidirect
70. C4×#92418C23semidirect
142. C2×D4×C61D8semidirect
223. D4×C61AGL(1,ℤ/8)semidirect
286. C2×D42D61semidirect
336. C61#18746semidirect
443. C2×D4#11758199semidirect
524. D4#14519803semidirect

Maximal subgroups

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

subgroupindex
#214567372
#214558192
#214552832
#214449982
#214448732
#214444872
#214442442
#214442102
C2×D42×C612
#1874661