Finite Group Databasebeta

← #6635579

Subgroups — #6635579

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37297C211yesmaximal, characteristic, complemented
2#35530C211yesmaximal, characteristic, complemented
3#32913C212yesmaximal, complemented
4#31933C214yesmaximal, complemented
5#31931C214yesmaximal, complemented
6#30988C214yesmaximal, complemented
7#30984C214yesmaximal, complemented
8#29765C212yesmaximal, complemented
9#28307C212yesmaximal, complemented
10#27973C212yesmaximal, complemented
11#24015C212yesmaximal, complemented
12#23924C212yesmaximal, complemented
13#21903C211yesmaximal, characteristic, complemented
14C2×#537C2211yescharacteristic
15C2×#535C2212yes—
16C2×#520C2211yescharacteristic, complemented
17C2×#482C2214yescomplemented
18C2×#460C2212yes—
19C2×#459C2212yes—
20C2×#449C2214yescomplemented
21C2×#450C2212yescomplemented
22C2×#447C2212yescomplemented
23#2856C2212yes—
24#2733—21no—
25#2733—21no—
26#2679—21no—
27#2679—21no—
28#2669—24no—
29#2619C2212yes—
30#2502—21no—
31#2501—21no—
32#2500—21no—
33#2500C2214yes—
34#2499—21no—
35#2443—24no—
36#2433C2214yes—
37#2423C2214yescomplemented
38#2422C2214yescomplemented
39#2317C2211yescharacteristic, complemented
40#2311C2211yescharacteristic
41#2297C2211yescharacteristic
42#2297C2211yescharacteristic, complemented
43C2×#384C2211yescharacteristic, complemented
44#2095C2212yescomplemented
45#2094C2212yescomplemented
46#2085C2214yescomplemented
47#2071C2214yescomplemented
48#2071C2214yescomplemented
49#2067C2214yescomplemented
50#2065C2214yescomplemented
51#2041C2214yes—
52#2034C2214yes—
53#2033C2214yes—
54#2014C2212yes—
55#2011C2212yes—
56#2000C2214yes—
57#1999C2214yes—
58#1970C2214yescomplemented
59#1968C2214yescomplemented
60#1957C2214yes—
61#1923C2214yescomplemented
62#1923C2214yescomplemented
63#1880C2214yescomplemented
64#1879C2214yescomplemented
65#1875C2212yescomplemented
66#1874C2214yescomplemented
67#1874C2214yescomplemented
68#1873C2212yescomplemented
69#1854C2214yescomplemented
70#1834C2214yescomplemented
71#1805C2212yescomplemented
72#1804C2212yescomplemented
73#1798C2212yescomplemented
74#1786C2214yescomplemented
75#1780C2212yes—
76C4×C22⋊C8C2212yescomplemented
77C22×#41C2311yescharacteristic
78C22×#41C2311yescharacteristic
79C23×D4C2312yes—
80C22×SD16C2314yes—
81C22×D8C2314yes—
82#537—42no—
83#537—42no—
84#535—48no—
85C2×#127—22no—
86C2×#127C2312yes—
87C2×#127C2312yes—
88C2×#124—22no—
89C2×#123—24no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8C2312yes—
92C2×#121—22no—
93C2×#121—21no—
94C2×#121—22no—
95C2×#121—22no—
96C2×#121C2314yes—
97C2×#121C2312yes—
98C2×#121C2314yes—
99C2×#120—22no—
100C2×#120—22no—
101C2×#120—24no—
102C2×#120—24no—
103#520—42no—
104#520—42no—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C22×C4⋊C4—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#482—48no—
118#460—44no—
119#459—44no—
120#450—44no—
121#449—48no—
122#447—44no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C4×C8C2314yes—
140#399—24no—
141#398—24no—
142#397—24no—
143#394—22no—
144#394—22no—
145#394—24no—
146#394—24no—
147#394—24no—
148#392—22no—
149#392—22no—
150#392—24no—
151#392—24no—
152#391—21no—
153#391D411yescharacteristic
154#391D411yescharacteristic
155#391C2312yes—
156#390—21no—
157#390—22no—
158#390—22no—
159#390D411yescharacteristic
160#390D411yescharacteristic
161#390C2314yes—
162#390C2314yes—
163#390C2312yes—
164#389C2314yes—
165#388—24no—
166#388—24no—
167#388—24no—
168#387—24no—
169#386—22no—
170#386—22no—
171#386D414yes—
172#385—22no—
173#385—24no—
174#384—21no—
175#384—21no—
176#384—21no—
177#384C2314yes—
178#384C2312yes—
179#382—21no—
180#382C2312yes—
181#380C2314yes—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2312yes—
191#336C2312yes—
192C25—22no—
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—41no—
198C2×#41—41no—
199C2×#41—42no—
200C2×#41—42no—
201C22×Q8C2×D411yescharacteristic
202C22×Q8C2411yescharacteristic
203C22×D4—21no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4C2×D411yescharacteristic
217C22×D4C2×D412yes—
218C22×D4C2414yes—
219C22×D4C2411yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4C2×D412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C2×SD16—28no—
228C2×SD16—44no—
229C2×SD16—44no—
230C2×D8—28no—
231C2×D8—44no—
232C2×D8—44no—
233C22×C8C2414yes—
234C22×C8C2414yes—
235#127—44no—
236#127—42no—
237#127—42no—
238#127—48no—
239#124—48no—
240#123—88no—
241C22⋊Q8—24no—
242C22⋊Q8—42no—
243C22⋊Q8—44no—
244#121—24no—
245#121—28no—
246#121—28no—
247#121—44no—
248#121—44no—
249#121—42no—
250#121—44no—
251#121—48no—
252#121—84no—
253#121—84no—
254#120—48no—
255#120—48no—
256#120—88no—
257#120—88no—
258C4×Q8—24no—
259C4×Q8—42no—
260C4×D4—24no—
261C4×D4—42no—
262C4×D4—48no—
263#117—24no—
264#117—41no—
265#117—41no—
266#117—44no—
267#117—44no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—84no—
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4C2×D414yes—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D412yes—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—42no—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2412yes—
313C2×C42C2412yes—
314#107—44no—
315#106—44no—
316C4⋊C8—42no—
317C4⋊C8—42no—
318#103—44no—
319#102—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323#102—44no—
324#102—44no—
325#102—44no—
326C22⋊C8—28no—
327C22⋊C8—44no—
328C22⋊C8—42no—
329C22⋊C8—42no—
330C4×C8—44no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95C2×D414yes—
338#95C2×D414yes—
339C24—21no—
340C24—24no—
341C24—44no—
342C24—82no—
343C24—82no—
344C24—84no—
345C24—84no—
346C24C22×D412yes—
347C24C22×D411yescharacteristic
348#41—48no—
349#41—48no—
350#41—84no—
351#41—84no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8H(5,𝔽2)12yes—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—82no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4H(5,𝔽2)12yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—84no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C4C2511yescharacteristic
433SD16—88no—
434D8—88no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—48no—
455C4⋊C4—84no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—84no—
459C22⋊C4—28no—
460C22⋊C4—28no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C42—22no—
477C42—22no—
478C42—24no—
479C42—41no—
480C42—41no—
481C42—44no—
482C42—44no—
483C42—42no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—88no—
506C23—168no—
507C23—168no—
508C23—164no—
509C23—164no—
510C23#53512yes—
511C23C23×D411yescharacteristic
512Q8—24no—
513Q8—42no—
514Q8—82no—
515D4—24no—
516D4—48no—
517D4—42no—
518D4—44no—
519D4—44no—
520D4—44no—
521D4—44no—
522D4—82no—
523D4—84no—
524D4—88no—
525D4—164no—
526D4—164no—
527D4—164no—
528D4—164no—
529D4—168no—
530D4—164no—
531D4—168no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—164no—
565C2×C4—162no—
566C2×C4—162no—
567C2×C4#53512yes—
568C2×C4#57614yes—
569C2×C4#57714yes—
570C2×C4C2×H(5,𝔽2)12yes—
571C2×C4C2×H(5,𝔽2)11yescharacteristic
572C2×C4C2×H(5,𝔽2)11yescharacteristic
573C2×C4#58512yes—
574C8—44no—
575C8—44no—
576C22—22no—
577C22—24no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—82no—
590C22—82no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—1616no—
599C22C2×#53511yescharacteristic
600C22#348411yescharacteristic
601C22#348511yescharacteristic
602C22C2×#57612yes—
603C22C2×#57712yes—
604C4—24no—
605C4—24no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—82no—
612C4—84no—
613C4—82no—
614C4—82no—
615C4#353412yes—
616C4#353412yes—
617C2—42no—
618C2—82no—
619C2—82no—
620C2—164no—
621C2#3672111yescharacteristic
622C2#6082212yes—
623C2#6166111yescharacteristic
624C2#6166411yescharacteristic
625C2#6176612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37297C2semidirect
2. #35530C2semidirect
3. #32913C2semidirect
4. #31933C2semidirect
5. #31931C2semidirect
6. #30988C2semidirect
7. #30984C2semidirect
8. #29765C2semidirect
9. #28307C2semidirect
10. #27973C2semidirect
11. #24015C2semidirect
12. #23924C2semidirect
13. #21903C2semidirect
16. C2×#520C22semidirect
17. C2×#482C22semidirect
20. C2×#449C22semidirect
21. C2×#450C22semidirect
22. C2×#447C22semidirect
37. #2423C22semidirect
38. #2422C22semidirect
39. #2317C22semidirect
42. #2297C22semidirect
43. C2×#384C22semidirect
44. #2095C22semidirect
45. #2094C22semidirect
46. #2085C22semidirect
47. #2071C22semidirect
48. #2071C22semidirect
49. #2067C22semidirect
50. #2065C22semidirect
58. #1970C22semidirect
59. #1968C22semidirect
61. #1923C22semidirect
62. #1923C22semidirect
63. #1880C22semidirect
64. #1879C22semidirect
65. #1875C22semidirect
66. #1874C22semidirect
67. #1874C22semidirect
68. #1873C22semidirect
69. #1854C22semidirect
70. #1834C22semidirect
71. #1805C22semidirect
72. #1804C22semidirect
73. #1798C22semidirect
74. #1786C22semidirect
76. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#372972
#355302
#329132
#319332
#319312
#309882
#309842
#297652
#283072
#279732
#240152
#239242
#219032