Finite Group Databasebeta

← #6632008

Subgroups — #6632008

Order 512.

3414 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic, complemented
2C2×#2408C211yesmaximal, characteristic, complemented
3#32878C212yesmaximal, complemented
4#31260C214yesmaximal, complemented
5#30499C214yesmaximal, complemented
6#30396C218yesmaximal, complemented
7#29890C212yesmaximal, complemented
8#27582C212yesmaximal, complemented
9#24378C214yesmaximal, complemented
10#23930C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C2×#535—21no—
14C2×#532C2211yescharacteristic, complemented
15C22×#121C2212yes—
16C22×#120—21no—
17C22×C4×D4C2211yescharacteristic
18C2×#466C2218yescomplemented
19C2×#447C2214yes—
20C2×#447C2218yescomplemented
21C22×C22⋊C8C2212yescomplemented
22#2847C2212yes—
23#2847C2212yescomplemented
24#2617—21no—
25#2617—21no—
26#2547C2212yescomplemented
27#2544—22no—
28#2541C2212yescomplemented
29#2537C2214yescomplemented
30#2535—22no—
31#2408—22no—
32#2408—22no—
33#2408C2214yes—
34#2408C2214yes—
35C2×#394—21no—
36C2×#392—21no—
37#2311—21no—
38C2×#390C2212yes—
39#2285C2211yescharacteristic, complemented
40C2×#380C2211yescharacteristic
41C2×#379—21no—
42#2093C2214yescomplemented
43#2075C2218yescomplemented
44#2071C2214yescomplemented
45#2033C2218yescomplemented
46#2033C2218yescomplemented
47#2023C2214yescomplemented
48#2011C2214yescomplemented
49#1999C2218yescomplemented
50#1925C2218yescomplemented
51#1893C2218yescomplemented
52#1879C2218yescomplemented
53#1874C2214yescomplemented
54#1869C2214yescomplemented
55#1869C2214yescomplemented
56#1852C2214yescomplemented
57#1817C2212yescomplemented
58#1786C2214yes—
59#1786C2218yescomplemented
60#1779C2212yes—
61C26—21no—
62C22×#41—21no—
63C23×D4—22no—
64C23×D4—21no—
65C23×D4—22no—
66C23×D4—44no—
67C23×D4—48no—
68C23×D4D411yescharacteristic
69C23×D4C2312yes—
70C23×D4C2314yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C22×D8C2318yes—
74C23×C8C2314yescomplemented
75#535—48no—
76#532—44no—
77C2×C4⋊Q8C2311yescharacteristic, complemented
78C2×#127—22no—
79C2×#127C2311yescharacteristic, complemented
80C2×#124—22no—
81C2×#124—21no—
82C2×#121—24no—
83C2×#121—21no—
84C2×#121—24no—
85C2×#121—48no—
86C2×#121C2312yes—
87C2×#121C2318yes—
88C2×#121C2314yes—
89C2×#121C2314yescomplemented
90C2×#120—22no—
91C2×#120—22no—
92C2×#120—24no—
93C2×#120—24no—
94C2×#120—48no—
95C2×C4×D4—21no—
96C2×C4×D4—44no—
97C2×C4×D4D412yes—
98C2×C4×D4D412yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yescomplemented
102C2×#117—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—22no—
108C22×C22⋊C4—22no—
109C22×C22⋊C4D411yescharacteristic
110C22×C22⋊C4D411yescharacteristic
111C22×C42C2311yescharacteristic
112#466—416no—
113#447—48no—
114#447—416no—
115C2×#107C2318yescomplemented
116C2×#102C2318yes—
117C2×#102C2318yes—
118C2×#102C2318yes—
119C2×#102C2318yescomplemented
120C2×C22⋊C8—44no—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2318yes—
123C2×C22⋊C8C2314yescomplemented
124#397—22no—
125#397—22no—
126#394—24no—
127#394—24no—
128#394—24no—
129#394—24no—
130#392—22no—
131#392—22no—
132#392—24no—
133#392—24no—
134#391—22no—
135#390—22no—
136#390—22no—
137#390—22no—
138#390C2314yes—
139#390C2314yes—
140#386—22no—
141#386—22no—
142#386—24no—
143#386—24no—
144#384C2312yescomplemented
145#382—21no—
146#380—21no—
147#380—22no—
148#380—24no—
149#380C2314yes—
150#380C2314yes—
151#380C2312yes—
152#380C2314yescomplemented
153#379—24no—
154#379—24no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2312yescomplemented
157C2×#95—21no—
158#340C2318yescomplemented
159#340C2318yescomplemented
160#336C2314yescomplemented
161#336C2314yescomplemented
162C25—22no—
163C25—24no—
164C25—44no—
165C25—48no—
166C25—44no—
167C25—44no—
168C25—84no—
169C25C2×D411yescharacteristic
170C25C2×D411yescharacteristic
171C2×#41—44no—
172C2×#41—44no—
173C2×#41—44no—
174C22×Q8—21no—
175C22×D4—28no—
176C22×D4—216no—
177C22×D4—22no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—24no—
184C22×D4—21no—
185C22×D4—28no—
186C22×D4—44no—
187C22×D4—48no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—44no—
193C22×D4—48no—
194C22×D4—416no—
195C22×D4—48no—
196C22×D4—464no—
197C22×D4—48no—
198C22×D4—48no—
199C22×D4—84no—
200C22×D4D818yescomplemented
201C22×D4C2×D412yes—
202C22×D4C2×D412yes—
203C22×D4C2412yes—
204C22×D4C2414yes—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C2×D8—216no—
221C2×D8—48no—
222C2×D8—48no—
223C22×C8—44no—
224C22×C8—48no—
225C22×C8C2418yes—
226C4⋊Q8—42no—
227#127—42no—
228#127—48no—
229#124—44no—
230#124—48no—
231#121—216no—
232#121—28no—
233#121—48no—
234#121—44no—
235#121—416no—
236#121—416no—
237#121—48no—
238#121—48no—
239#120—48no—
240#120—48no—
241#120—416no—
242#120—416no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—44no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—44no—
251#117—44no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—84no—
282C2×C22⋊C4C2×D412yes—
283C2×C22⋊C4C2×D412yes—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D412yes—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292#107—48no—
293#102—216no—
294#102—48no—
295#102—48no—
296#102—48no—
297C22⋊C8—28no—
298C22⋊C8—44no—
299C22⋊C8—48no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—28no—
310C24—24no—
311C24—22no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—416no—
317C24—44no—
318C24—44no—
319C24—48no—
320C24—44no—
321C24—48no—
322C24—432no—
323C24—44no—
324C24—44no—
325C24—44no—
326C24—48no—
327C24—88no—
328C24—816no—
329C24—832no—
330C24—832no—
331C24—816no—
332C24C2×D814yes—
333C24C2×D814yes—
334C24C22×D412yes—
335C24C22×D412yes—
336C24C22×D411yescharacteristic
337#41—816no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—28no—
346C2×D4—24no—
347C2×D4—432no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—416no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—416no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—416no—
360C2×D4—432no—
361C2×D4—464no—
362C2×D4—432no—
363C2×D4—416no—
364C2×D4—416no—
365C2×D4—44no—
366C2×D4—416no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—42no—
373C2×D4—82no—
374C2×D4—88no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—832no—
380C2×D4C2×D818yes—
381C2×D4C2×D818yes—
382C2×D4H(5,𝔽2)18yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—416no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—88no—
413C22×C4C2×D814yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4H(5,𝔽2)14yes—
422C22×C4C2511yescharacteristic
423D8—816no—
424C2×C8—28no—
425C2×C8—28no—
426C2×C8—28no—
427C2×C8—88no—
428C4⋊C4—28no—
429C4⋊C4—24no—
430C4⋊C4—28no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—82no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—416no—
442C22⋊C4—44no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—44no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—28no—
471C23—28no—
472C23—48no—
473C23—48no—
474C23—44no—
475C23—44no—
476C23—48no—
477C23—416no—
478C23—416no—
479C23—416no—
480C23—432no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—82no—
493C23—82no—
494C23—84no—
495C23—832no—
496C23—88no—
497C23—88no—
498C23—88no—
499C23—816no—
500C23—832no—
501C23—8128no—
502C23—816no—
503C23—88no—
504C23#53412yes—
505C23C22×D814yes—
506C23C22×D814yes—
507C23C22×D812yes—
508C23#57512yes—
509C23C23×D411yescharacteristic
510Q8—84no—
511D4—216no—
512D4—48no—
513D4—48no—
514D4—416no—
515D4—432no—
516D4—48no—
517D4—48no—
518D4—84no—
519D4—84no—
520D4—832no—
521D4—88no—
522D4—168no—
523D4—168no—
524C2×C4—28no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—84no—
555C2×C4—88no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—88no—
559C2×C4#53412yes—
560C2×C4C22×D812yes—
561C2×C4C22×D812yes—
562C2×C4C2×H(5,𝔽2)12yes—
563C2×C4C2×H(5,𝔽2)14yes—
564C8—48no—
565C22—22no—
566C22—22no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—28no—
572C22—28no—
573C22—416no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—48no—
581C22—48no—
582C22—48no—
583C22—88no—
584C22—832no—
585C22—864no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—84no—
594C22—84no—
595C22#318614yes—
596C22C2×#53411yescharacteristic
597C22#348412yes—
598C22C23×D811yescharacteristic
599C22C2×#57511yescharacteristic
600C22#358512yes—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—42no—
608C4—82no—
609C4—84no—
610C4—84no—
611C4#353214yes—
612C2—24no—
613C2—44no—
614C2—44no—
615C2—88no—
616C2—162no—
617C2#3670211yescharacteristic
618C2C2×#318611yescharacteristic
619C2#6075311yescharacteristic
620C2#6155912yes—
621C2#6157612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35165C2semidirect
2. C2×#2408C2semidirect
3. #32878C2semidirect
4. #31260C2semidirect
5. #30499C2semidirect
6. #30396C2semidirect
7. #29890C2semidirect
8. #27582C2semidirect
9. #24378C2semidirect
10. #23930C2semidirect
11. #21738C2semidirect
14. C2×#532C22semidirect
18. C2×#466C22semidirect
20. C2×#447C22semidirect
21. C22×C22⋊C8C22semidirect
23. #2847C22semidirect
26. #2547C22semidirect
28. #2541C22semidirect
29. #2537C22semidirect
39. #2285C22semidirect
42. #2093C22semidirect
43. #2075C22semidirect
44. #2071C22semidirect
45. #2033C22semidirect
46. #2033C22semidirect
47. #2023C22semidirect
48. #2011C22semidirect
49. #1999C22semidirect
50. #1925C22semidirect
51. #1893C22semidirect
52. #1879C22semidirect
53. #1874C22semidirect
54. #1869C22semidirect
55. #1869C22semidirect
56. #1852C22semidirect
57. #1817C22semidirect
59. #1786C22semidirect
74. C23×C8C23semidirect
77. C2×C4⋊Q8C23semidirect
79. C2×#127C23semidirect
89. C2×#121C23semidirect
101. C2×C4×D4C23semidirect
115. C2×#107C23semidirect
119. C2×#102C23semidirect
123. C2×C22⋊C8C23semidirect
144. #384C23semidirect
152. #380C23semidirect
156. C4×C4⋊C4C23semidirect
158. #340C23semidirect
159. #340C23semidirect
160. #336C23semidirect
161. #336C23semidirect
200. C22×D4D8semidirect

Maximal subgroups

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

subgroupindex
#351652
C2×#24082
#328782
#312602
#304992
#303962
#298902
#275822
#243782
#239302
#217382