Finite Group Databasebeta

← #6650284

Subgroups — #6650284

Order 512.

2762 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35000C211yesmaximal, characteristic, complemented
2C2×#2541C211yesmaximal, characteristic, complemented
3#33094C212yesmaximal, complemented
4#32026C214yesmaximal, complemented
5#30701C218yesmaximal, complemented
6#30499C214yesmaximal, complemented
7#29622C212yesmaximal, complemented
8#24378C214yesmaximal, complemented
9#24002C212yesmaximal, complemented
10#23918C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic, complemented
12C2×#534—21no—
13C22×#127—21no—
14C22×#127C2211yescharacteristic, complemented
15C22×#121C2212yes—
16C22×#121C2211yescharacteristic
17C22×#121C2212yescomplemented
18C22×C4×D4C2211yescharacteristic
19C2×#480C2214yescomplemented
20C2×#466C2218yescomplemented
21C2×#447C2214yes—
22C2×#447C2214yescomplemented
23C22×C22⋊C8C2212yescomplemented
24#2854C2212yes—
25#2847C2212yes—
26#2618—21no—
27#2617—21no—
28#2543—22no—
29#2541—22no—
30#2541—22no—
31#2541C2214yes—
32#2541C2214yescomplemented
33#2534—22no—
34#2486C2212yescomplemented
35#2416C2214yescomplemented
36#2408C2212yescomplemented
37C2×#392—21no—
38C2×#392—21no—
39#2308—21no—
40C2×#390—21no—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic, complemented
43#2041C2218yescomplemented
44#2033C2218yescomplemented
45#2033C2218yescomplemented
46#2011C2218yescomplemented
47#1999C2218yescomplemented
48#1999C2218yescomplemented
49#1925C2218yescomplemented
50#1918C2218yescomplemented
51#1893C2218yescomplemented
52#1874C2214yes—
53#1874C2214yescomplemented
54#1874C2214yescomplemented
55#1869C2214yescomplemented
56#1869C2214yescomplemented
57#1852C2214yescomplemented
58#1817C2212yescomplemented
59#1786C2214yescomplemented
60#1779C2212yes—
61C23×D4—21no—
62C23×D4—22no—
63C23×D4—22no—
64C23×D4—21no—
65C23×D4—21no—
66C23×D4D411yescharacteristic
67C23×D4C2312yes—
68C23×D4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C22×D8C2318yes—
72C23×C8C2314yes—
73#534—48no—
74C2×#127—24no—
75C2×#127—24no—
76C2×#127—22no—
77C2×#127—44no—
78C2×#127—48no—
79C2×#127C2314yes—
80C2×#123—22no—
81C2×#123—22no—
82C2×#121—24no—
83C2×#121—21no—
84C2×#121—22no—
85C2×#121—22no—
86C2×#121—21no—
87C2×#121—24no—
88C2×#121—48no—
89C2×#121—48no—
90C2×#121—44no—
91C2×#121C2312yes—
92C2×#121C2314yes—
93C2×#121C2314yes—
94C2×#121C2318yes—
95C2×#121C2314yes—
96C2×#121C2312yes—
97C2×#120—22no—
98C2×#120—22no—
99C2×C4×D4—44no—
100C2×C4×D4D412yes—
101C2×C4×D4D412yes—
102C2×C4×D4C2312yes—
103C2×C4×D4C2312yes—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4D411yescharacteristic
113C22×C22⋊C4D411yescharacteristic
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116#480—48no—
117#466—416no—
118#447—48no—
119#447—48no—
120C2×#107C2318yes—
121C2×#102C2318yes—
122C2×#102C2318yes—
123C2×#102C2318yes—
124C2×#102C2318yes—
125C2×C22⋊C8—44no—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130#398—22no—
131#397—22no—
132#396—24no—
133#394—22no—
134#394—24no—
135#392—22no—
136#392—24no—
137#392—24no—
138#392—24no—
139#392—24no—
140#390—24no—
141#390—24no—
142#390C2314yes—
143#389C2314yes—
144#388—24no—
145#386—22no—
146#386—22no—
147#386—24no—
148#386—24no—
149#381—22no—
150#380—21no—
151#380—21no—
152#380C2312yes—
153#380C2314yes—
154#380C2312yes—
155#379—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4C2314yes—
158C2×#95—21no—
159C2×#95—21no—
160#340C2318yes—
161#340C2318yes—
162#336C2318yes—
163C25—22no—
164C25—21no—
165C25C2×D411yescharacteristic
166C25C2×D411yescharacteristic
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—22no—
174C22×D4—24no—
175C22×D4—21no—
176C22×D4—24no—
177C22×D4—28no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—42no—
181C22×D4—44no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—48no—
185C22×D4—48no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—48no—
192C22×D4—48no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—84no—
196C22×D4—84no—
197C22×D4—84no—
198C22×D4D818yescomplemented
199C22×D4C2×D412yes—
200C22×D4C2×D412yes—
201C22×D4C2414yes—
202C22×D4C2412yes—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—21no—
207C23×C4—22no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C2×D8—48no—
222C2×D8—48no—
223C2×D8—48no—
224C22×C8—44no—
225C22×C8—48no—
226C22×C8C2418yes—
227#127—44no—
228#127—44no—
229#127—48no—
230#127—416no—
231#127—416no—
232#123—48no—
233#123—48no—
234#121—28no—
235#121—28no—
236#121—216no—
237#121—28no—
238#121—44no—
239#121—44no—
240#121—48no—
241#121—44no—
242#121—48no—
243#121—48no—
244#121—416no—
245#121—44no—
246#121—44no—
247#121—416no—
248#121—48no—
249#120—48no—
250#120—48no—
251C4×D4—44no—
252C4×D4—44no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2414yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—44no—
294C2×C42C2×D412yes—
295C2×C42C2412yes—
296#107—216no—
297#102—216no—
298#102—48no—
299#102—48no—
300#102—48no—
301C22⋊C8—28no—
302C22⋊C8—44no—
303C22⋊C8—44no—
304C22⋊C8—44no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—44no—
321C24—44no—
322C24—42no—
323C24—44no—
324C24—44no—
325C24—42no—
326C24—44no—
327C24—44no—
328C24—44no—
329C24—42no—
330C24—82no—
331C24—82no—
332C24C2×D814yes—
333C24C2×D814yes—
334C24C22×D411yescharacteristic
335C24C22×D412yes—
336C24C22×D411yescharacteristic
337C24C22×D411yescharacteristic
338C2×D4—28no—
339C2×D4—24no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—416no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—48no—
350C2×D4—416no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—42no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—416no—
365C2×D4—48no—
366C2×D4—44no—
367C2×D4—48no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—816no—
371C2×D4—82no—
372C2×D4—88no—
373C2×D4—816no—
374C2×D4—88no—
375C2×D4—84no—
376C2×D4—88no—
377C2×D4—84no—
378C2×D4—832no—
379C2×D4—832no—
380C2×D4—816no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—816no—
384C2×D4C2×D818yes—
385C2×D4C2×D818yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4C2×D814yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C4C2511yescharacteristic
431D8—816no—
432C2×C8—28no—
433C2×C8—28no—
434C2×C8—28no—
435C2×C8—88no—
436C4⋊C4—28no—
437C4⋊C4—24no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—48no—
444C4⋊C4—82no—
445C4⋊C4H(5,𝔽2)18yes—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—416no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—24no—
462C42—22no—
463C42—22no—
464C42—48no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—28no—
480C23—28no—
481C23—48no—
482C23—48no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—84no—
500C23—84no—
501C23—88no—
502C23—88no—
503C23—88no—
504C23—88no—
505C23—84no—
506C23—88no—
507C23—88no—
508C23—84no—
509C23—84no—
510C23—88no—
511C23—164no—
512C23—164no—
513C23#53412yes—
514C23C22×D812yes—
515C23C22×D814yes—
516C23C22×D812yes—
517C23C22×D812yes—
518C23#57512yes—
519C23C23×D411yescharacteristic
520D4—48no—
521D4—48no—
522D4—48no—
523D4—48no—
524D4—48no—
525D4—816no—
526D4—88no—
527D4—816no—
528D4—832no—
529D4—816no—
530D4—816no—
531D4—84no—
532D4—88no—
533D4—88no—
534D4—168no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—44no—
550C2×C4—416no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—84no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—84no—
569C2×C4—164no—
570C2×C4#53412yes—
571C2×C4C22×D814yes—
572C2×C4C2×H(5,𝔽2)12yes—
573C2×C4C2×H(5,𝔽2)14yes—
574C8—48no—
575C22—22no—
576C22—22no—
577C22—24no—
578C22—24no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—28no—
584C22—24no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—41no—
591C22—42no—
592C22—44no—
593C22—42no—
594C22—88no—
595C22—88no—
596C22—84no—
597C22—82no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—1616no—
609C22—168no—
610C22#318614yes—
611C22C2×#53411yescharacteristic
612C22#348412yes—
613C22C23×D811yescharacteristic
614C22C2×#57511yescharacteristic
615C22#358512yes—
616C4—24no—
617C4—22no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—48no—
623C4—82no—
624C4—84no—
625C4#354314yes—
626C2—24no—
627C2—42no—
628C2—44no—
629C2—42no—
630C2—164no—
631C2—162no—
632C2#3723211yescharacteristic
633C2C2×#318611yescharacteristic
634C2#6075311yescharacteristic
635C2#6155912yes—
636C2#6157712yes—
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. #35000C2semidirect
2. C2×#2541C2semidirect
3. #33094C2semidirect
4. #32026C2semidirect
5. #30701C2semidirect
6. #30499C2semidirect
7. #29622C2semidirect
8. #24378C2semidirect
9. #24002C2semidirect
10. #23918C2semidirect
11. #21738C2semidirect
14. C22×#127C22semidirect
17. C22×#121C22semidirect
19. C2×#480C22semidirect
20. C2×#466C22semidirect
22. C2×#447C22semidirect
23. C22×C22⋊C8C22semidirect
32. #2541C22semidirect
34. #2486C22semidirect
35. #2416C22semidirect
36. #2408C22semidirect
42. C2×C4×C4⋊C4C22semidirect
43. #2041C22semidirect
44. #2033C22semidirect
45. #2033C22semidirect
46. #2011C22semidirect
47. #1999C22semidirect
48. #1999C22semidirect
49. #1925C22semidirect
50. #1918C22semidirect
51. #1893C22semidirect
53. #1874C22semidirect
54. #1874C22semidirect
55. #1869C22semidirect
56. #1869C22semidirect
57. #1852C22semidirect
58. #1817C22semidirect
59. #1786C22semidirect
198. C22×D4D8semidirect

Maximal subgroups

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

subgroupindex
#350002
C2×#25412
#330942
#320262
#307012
#304992
#296222
#243782
#240022
#239182
#217382