Finite Group Databasebeta

← #6519443

Subgroups — #6519443

Order 512.

2095 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35565C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#26246C214yesmaximal, complemented
4#26244C214yesmaximal, complemented
5#26238C214yesmaximal, complemented
6#26230C214yesmaximal, complemented
7#24015C212yesmaximal, complemented
8#24011C212yesmaximal, complemented
9#24007C212yesmaximal, complemented
10#24002C212yesmaximal, complemented
11#21813C212yesmaximal, complemented
12#21792C211yesmaximal, characteristic, complemented
13#21368C212yesmaximal, complemented
14C2×#537C2211yescharacteristic
15C2×#535C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#525C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C2×#484C2212yescomplemented
21C2×#484C2212yescomplemented
22C2×#481C2212yescomplemented
23C2×#482C2214yescomplemented
24C2×#480C2214yes—
25C2×#480C2212yescomplemented
26C2×#433C2212yes—
27#2541—22no—
28#2541C2214yes—
29#2540—22no—
30#2540C2214yes—
31#2539—22no—
32#2433—22no—
33#2376—22no—
34#2358—22no—
35#2352C2214yescomplemented
36D4×C4⋊C4C2214yescomplemented
37#2329—22no—
38#2325—22no—
39#2314C2212yes—
40#2311C2212yes—
41C2×#380C2212yes—
42#2274C2211yescharacteristic
43#2036C2214yescomplemented
44#2035C2214yescomplemented
45#2034C2214yescomplemented
46#2033C2214yes—
47#2033C2214yes—
48#2033C2214yescomplemented
49#2033C2214yescomplemented
50#2033C2214yescomplemented
51#1983C2214yescomplemented
52#1982C2214yescomplemented
53#1961C2212yescomplemented
54#1959C2212yescomplemented
55#1958C2212yescomplemented
56#1957C2212yescomplemented
57#1955C2214yescomplemented
58#1954C2214yescomplemented
59#1949C2214yescomplemented
60#1947C2214yescomplemented
61#1875C2212yescomplemented
62#1874C2214yes—
63#1874C2212yescomplemented
64#1874C2214yescomplemented
65#1873C2212yescomplemented
66#1872C2212yescomplemented
67#1854C2214yes—
68#1852C2214yes—
69#1788C2214yes—
70#1786C2214yes—
71#1780C2212yes—
72#1779C2212yes—
73#1776C2214yescomplemented
74C4×#107C2214yescomplemented
75C22×#41C2311yescharacteristic
76C22×#41C2311yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80#537—44no—
81#535—28no—
82#535—44no—
83C2×#127—21no—
84C2×#127—24no—
85C2×#127C2312yes—
86C2×#124—21no—
87C2×#124C2312yes—
88#525—28no—
89#525—44no—
90C2×#123—21no—
91C2×#123D412yes—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8D412yes—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×#121—22no—
97C2×#121—21no—
98C2×#121—22no—
99C2×#121—22no—
100C2×#121—44no—
101C2×#121D412yes—
102C2×#121C2314yes—
103C2×#121C2312yes—
104C2×#121C2314yes—
105C2×#121C2312yes—
106C2×#120—21no—
107C2×#120—22no—
108C2×#120—22no—
109C2×#120D412yes—
110C2×C4×D4—22no—
111C2×C4×D4—24no—
112C2×C4×D4C2314yes—
113C2×#117C2311yescharacteristic
114C2×#117C2312yes—
115C2×#117C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—22no—
122#484—44no—
123#484—44no—
124#482—48no—
125#481—44no—
126#480—48no—
127#480—44no—
128#433—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151#399—24no—
152#392—24no—
153#392—24no—
154#392—24no—
155#391—22no—
156#390—22no—
157#390—24no—
158#390—24no—
159#390—24no—
160#389—24no—
161#388—24no—
162#388—24no—
163#387—24no—
164#387—24no—
165#386—22no—
166#386—22no—
167#386—24no—
168#385—24no—
169#385—24no—
170#384C2312yes—
171#382C2312yes—
172#380—22no—
173#380—22no—
174#380C2314yes—
175#380C2314yes—
176#380C2314yes—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—24no—
181C4×C22⋊C4—22no—
182C2×#95—22no—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187C25—21no—
188C25C2×D411yescharacteristic
189C2×#41—24no—
190C2×#41—24no—
191C2×#41—42no—
192C2×#41—42no—
193C2×#41—42no—
194C2×#41—42no—
195C22×Q8C2×D411yescharacteristic
196C22×Q8C2411yescharacteristic
197C22×D4—21no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—84no—
209C22×D4C2×D414yes—
210C22×D4C2×D412yes—
211C22×D4C2×D411yescharacteristic
212C22×D4C2412yes—
213C22×D4C2411yescharacteristic
214C23×C4—21no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C22×C8C2414yes—
228C22×C8C2414yes—
229#127—24no—
230#127—24no—
231#127—44no—
232#127—88no—
233#124—24no—
234#124—24no—
235#124—44no—
236#123—28no—
237#123—44no—
238C22⋊Q8—28no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—42no—
242C22⋊Q8—42no—
243C22⋊Q8—44no—
244#121—28no—
245#121—28no—
246#121—24no—
247#121—28no—
248#121—24no—
249#121—44no—
250#121—42no—
251#121—44no—
252#121—42no—
253#121—48no—
254#121—44no—
255#121—48no—
256#121—48no—
257#120—28no—
258#120—44no—
259#120—48no—
260#120—48no—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4—48no—
264C4×D4—88no—
265#117—24no—
266#117—44no—
267#117#4118yes—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—84no—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D412yes—
304C2×C42—21no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42C2412yes—
309#107—28no—
310#107—28no—
311#107—28no—
312#106—28no—
313C4⋊C8—28no—
314#103—44no—
315#103—44no—
316#102—44no—
317#102—44no—
318#102—44no—
319#102—44no—
320#102—44no—
321#102—44no—
322C22⋊C8—28no—
323C22⋊C8—42no—
324C22⋊C8—44no—
325C22⋊C8—42no—
326C22⋊C8—44no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329#97—24no—
330C4×C8—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—21no—
341C24—41no—
342C24—42no—
343C24—41no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—82no—
348C24—82no—
349C24C22×D412yes—
350C24C22×D411yescharacteristic
351C24C22×D412yes—
352#41—88no—
353#41—88no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—28no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—44no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—44no—
377C2×D4—48no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—42no—
388C2×D4—48no—
389C2×D4—88no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4—82no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—816no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—88no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D414yes—
437C22×C4C2×#4114yes—
438C22×C4C2511yescharacteristic
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C4⋊C4—24no—
446C4⋊C4—28no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—84no—
459C4⋊C4#12318yes—
460C4⋊C4H(5,𝔽2)14yes—
461C4⋊C4H(5,𝔽2)14yes—
462C22⋊C4—24no—
463C22⋊C4—28no—
464C22⋊C4—28no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—84no—
474C22⋊C4—82no—
475C22⋊C4—84no—
476C22⋊C4#12318yes—
477C42—24no—
478C42—82no—
479C42—84no—
480C42—84no—
481C42C2×#4114yes—
482C42C2×#4114yes—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—21no—
488C23—22no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—44no—
499C23—48no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—82no—
511C23—84no—
512C23—84no—
513C23—82no—
514C23—84no—
515C23—84no—
516C23—164no—
517C23—164no—
518C23C2×#12312yes—
519C23#57614yes—
520C23C23×D411yescharacteristic
521Q8—44no—
522Q8—82no—
523D4—416no—
524D4—44no—
525D4—44no—
526D4—44no—
527D4—82no—
528D4—84no—
529D4—84no—
530D4—88no—
531D4—816no—
532D4—84no—
533D4—168no—
534D4—168no—
535D4—164no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—82no—
568C2×C4—84no—
569C2×C4—164no—
570C2×C4—164no—
571C2×C4—164no—
572C2×C4C2×#12318yes—
573C2×C4C2×#12312yes—
574C2×C4C22×#4112yes—
575C2×C4C2×H(5,𝔽2)12yes—
576C2×C4C2×H(5,𝔽2)12yes—
577C8—44no—
578C8—44no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—22no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—88no—
597C22—82no—
598C22—84no—
599C22—82no—
600C22—84no—
601C22—82no—
602C22—81no—
603C22—81no—
604C22—82no—
605C22—81no—
606C22—82no—
607C22—168no—
608C22C22×#12311yescharacteristic
609C22#348411yescharacteristic
610C22#348411yescharacteristic
611C22C2×#57612yes—
612C22#358612yes—
613C4—28no—
614C4—22no—
615C4—44no—
616C4—44no—
617C4—42no—
618C4—42no—
619C4—84no—
620C4—84no—
621C4—81no—
622C4#350814yes—
623C2—44no—
624C2—42no—
625C2—82no—
626C2—161no—
627C2—162no—
628C2#3555411yescharacteristic
629C2#6052112yes—
630C2#6162412yes—
631C2#6163011yescharacteristic
632C2#6165211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35565C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34360C2semidirect
3. #26246C2semidirect
4. #26244C2semidirect
5. #26238C2semidirect
6. #26230C2semidirect
7. #24015C2semidirect
8. #24011C2semidirect
9. #24007C2semidirect
10. #24002C2semidirect
11. #21813C2semidirect
12. #21792C2semidirect
13. #21368C2semidirect
20. C2×#484C22semidirect
21. C2×#484C22semidirect
22. C2×#481C22semidirect
23. C2×#482C22semidirect
25. C2×#480C22semidirect
35. #2352C22semidirect
36. D4×C4⋊C4C22semidirect
43. #2036C22semidirect
44. #2035C22semidirect
45. #2034C22semidirect
48. #2033C22semidirect
49. #2033C22semidirect
50. #2033C22semidirect
51. #1983C22semidirect
52. #1982C22semidirect
53. #1961C22semidirect
54. #1959C22semidirect
55. #1958C22semidirect
56. #1957C22semidirect
57. #1955C22semidirect
58. #1954C22semidirect
59. #1949C22semidirect
60. #1947C22semidirect
61. #1875C22semidirect
63. #1874C22semidirect
64. #1874C22semidirect
65. #1873C22semidirect
66. #1872C22semidirect
73. #1776C22semidirect
74. C4×#107C22semidirect

Maximal subgroups

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

subgroupindex
#355652
#343602
#262462
#262442
#262382
#262302
#240152
#240112
#240072
#240022
#218132
#217922
#213682