Finite Group Databasebeta

← #6660370

Subgroups — #6660370

Order 512.

3626 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C212yesmaximal, complemented
2#32707C214yesmaximal, complemented
3#29891C214yesmaximal, complemented
4#29732C214yesmaximal, complemented
5#29594C214yesmaximal, complemented
6#27603C211yesmaximal, characteristic, complemented
7#24553C214yesmaximal, complemented
8#24378C214yesmaximal, complemented
9#24005C212yesmaximal, complemented
10#23918C211yesmaximal, characteristic, complemented
11C2×#1779C211yesmaximal, characteristic, complemented
12C25×C4C2211yescharacteristic
13C22×#121C2212yes—
14C22×#121C2212yescomplemented
15C22×#121C2212yescomplemented
16C22×#120—22no—
17C2×#481C2214yescomplemented
18C2×#466C2214yescomplemented
19C2×#447C2214yescomplemented
20C22×C22⋊C8C2212yescomplemented
21C22×C22⋊C8C2214yescomplemented
22#2847—22no—
23#2847—22no—
24#2847—22no—
25#2847—22no—
26#2847C2214yes—
27#2847C2214yes—
28#2847C2214yescomplemented
29#2847C2214yescomplemented
30C2×#386—22no—
31C2×#386—22no—
32C2×#386—22no—
33C2×#380C2212yes—
34C2×#380C2212yescomplemented
35C2×#380C2212yescomplemented
36#2077C2218yescomplemented
37#2076C2214yescomplemented
38#2076C2214yescomplemented
39#2034C2218yescomplemented
40#2034C2218yescomplemented
41#2034C2218yescomplemented
42#2033C2218yescomplemented
43#1999C2214yescomplemented
44#1999C2214yescomplemented
45#1999C2218yescomplemented
46#1893C2218yescomplemented
47#1874C2214yescomplemented
48#1874C2214yescomplemented
49#1853C2214yescomplemented
50#1852C2214yescomplemented
51#1824C2214yescomplemented
52#1807C2214yescomplemented
53#1786C2214yescomplemented
54#1779C2211yescharacteristic
55#1779C2211yescharacteristic
56#1779C2214yescomplemented
57#1779C2212yescomplemented
58C26D411yescharacteristic
59C23×D4—22no—
60C23×D4—22no—
61C23×D4—22no—
62C23×D4C2312yes—
63C24×C4—48no—
64C24×C4—44no—
65C24×C4D411yescharacteristic
66C24×C4D411yescharacteristic
67C24×C4D412yes—
68C24×C4D412yes—
69C24×C4D411yescharacteristic
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C22×D8C2314yescomplemented
76C23×C8C2314yes—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—48no—
84C2×#121—48no—
85C2×#121—48no—
86C2×#121C2314yes—
87C2×#121C2314yes—
88C2×#121C2314yes—
89C2×#121C2314yes—
90C2×#121C2314yes—
91C2×#121C2314yes—
92C2×#120—24no—
93C2×#120—24no—
94C2×#120—24no—
95C2×#120—24no—
96C2×#120—416no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4—22no—
101C22×C22⋊C4—22no—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—22no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106#481—48no—
107#466—48no—
108#447—48no—
109C2×#107C2314yescomplemented
110C2×#106C2318yescomplemented
111C2×#102C2318yescomplemented
112C2×#102C2318yescomplemented
113C2×#102C2318yescomplemented
114C2×C22⋊C8—48no—
115C2×C22⋊C8—44no—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2314yes—
118C2×C22⋊C8C2312yes—
119C2×C22⋊C8C2314yes—
120C2×C22⋊C8C2318yes—
121C2×C22⋊C8C2312yes—
122#386—24no—
123#386—24no—
124#386—24no—
125#386—24no—
126#386—24no—
127#386—24no—
128#386—24no—
129#386—24no—
130#386—24no—
131#386—24no—
132#386—24no—
133#386—24no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380C2314yes—
141#380C2314yes—
142#380C2314yes—
143#380C2314yes—
144#380C2314yes—
145#380C2314yes—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95—22no—
149#340C2314yescomplemented
150#340C2314yescomplemented
151#340C2318yescomplemented
152#340C2318yescomplemented
153C25—24no—
154C25—28no—
155C25—22no—
156C25—44no—
157C25—42no—
158C25—42no—
159C25C2×D412yes—
160C25C2×D411yescharacteristic
161C25C2×D411yescharacteristic
162C25C2×D412yes—
163C25C2×D411yescharacteristic
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—28no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—48no—
175C22×D4—48no—
176C22×D4—48no—
177C22×D4—48no—
178C22×D4—48no—
179C22×D4—48no—
180C22×D4—48no—
181C22×D4—48no—
182C22×D4—48no—
183C22×D4C2414yes—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—28no—
187C23×C4—28no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—464no—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C2×D8—48no—
231C2×D8—44no—
232C22×C8—28no—
233C22×C8—48no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236#121—28no—
237#121—216no—
238#121—28no—
239#121—44no—
240#121—48no—
241#121—48no—
242#121—48no—
243#121—44no—
244#121—48no—
245#121—48no—
246#121—48no—
247#121—48no—
248#121—48no—
249#121—48no—
250#120—416no—
251#120—416no—
252#120—416no—
253#120—416no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2414yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281#107—44no—
282#106—48no—
283#102—48no—
284#102—48no—
285#102—48no—
286C22⋊C8—28no—
287C22⋊C8—28no—
288C22⋊C8—48no—
289C22⋊C8—42no—
290C22⋊C8—44no—
291C22⋊C8—42no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—41no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—41no—
323C24—41no—
324C24—42no—
325C24—41no—
326C24—432no—
327C24—416no—
328C24—432no—
329C24—416no—
330C24—44no—
331C24—44no—
332C24—44no—
333C24—44no—
334C24—416no—
335C24—44no—
336C24—44no—
337C24—44no—
338C24#13514yes—
339C24AGL(1,ℤ/8)18yes—
340C24C22×D411yescharacteristic
341C24C22×D412yes—
342C24C22×D411yescharacteristic
343C24C22×D412yes—
344C24C22×D411yescharacteristic
345C2×D4—24no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—24no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—816no—
371C2×D4—832no—
372C2×D4—832no—
373C2×D4—832no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—432no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—432no—
425C22×C4—416no—
426C22×C4—432no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—432no—
430C22×C4—416no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4H(5,𝔽2)18yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C4C2511yescharacteristic
448D8—88no—
449C2×C8—28no—
450C2×C8—28no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—88no—
454C4⋊C4—24no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—21no—
477C23—22no—
478C23—22no—
479C23—21no—
480C23—216no—
481C23—28no—
482C23—216no—
483C23—216no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—216no—
492C23—22no—
493C23—21no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—21no—
499C23—44no—
500C23—48no—
501C23—44no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—44no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—44no—
516C23—48no—
517C23—44no—
518C23—44no—
519C23—48no—
520C23—44no—
521C23—4128no—
522C23—48no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—88no—
532C23—88no—
533C23—88no—
534C23—88no—
535C23—88no—
536C23—88no—
537C23—88no—
538C23#53414yes—
539C23#53412yes—
540C23C2×#13512yes—
541C23C2×#13514yes—
542C23C2×AGL(1,ℤ/8)14yes—
543C23C2×AGL(1,ℤ/8)14yes—
544C23C2×AGL(1,ℤ/8)14yes—
545C23#57512yes—
546C23C23×D411yescharacteristic
547D4—48no—
548D4—48no—
549D4—88no—
550D4—816no—
551D4—816no—
552D4—816no—
553D4—816no—
554D4—816no—
555D4—816no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—216no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4#53412yes—
597C2×C4#53414yes—
598C2×C4C2×H(5,𝔽2)14yes—
599C2×C4C2×H(5,𝔽2)12yes—
600C8—44no—
601C8—44no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—22no—
615C22—22no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—416no—
620C22—432no—
621C22—432no—
622C22—416no—
623C22—416no—
624C22—42no—
625C22—42no—
626C22—41no—
627C22—42no—
628C22—41no—
629C22—42no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22—1616no—
642C22#318818yes—
643C22#319214yes—
644C22C2×#53412yes—
645C22C2×#53411yescharacteristic
646C22C22×#13511yescharacteristic
647C22C22×AGL(1,ℤ/8)12yes—
648C22C2×#57511yescharacteristic
649C4—22no—
650C4—24no—
651C4—42no—
652C4—41no—
653C4—41no—
654C4—84no—
655C4—84no—
656C4—84no—
657C4#353214yes—
658C2—28no—
659C2—24no—
660C2—42no—
661C2—44no—
662C2—42no—
663C2—164no—
664C2#3743611yescharacteristic
665C2C2×#318812yes—
666C2#6073411yescharacteristic
667C2C2×#319211yescharacteristic
668C2#6075012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2847C2semidirect
2. #32707C2semidirect
3. #29891C2semidirect
4. #29732C2semidirect
5. #29594C2semidirect
6. #27603C2semidirect
7. #24553C2semidirect
8. #24378C2semidirect
9. #24005C2semidirect
10. #23918C2semidirect
11. C2×#1779C2semidirect
14. C22×#121C22semidirect
15. C22×#121C22semidirect
17. C2×#481C22semidirect
18. C2×#466C22semidirect
19. C2×#447C22semidirect
20. C22×C22⋊C8C22semidirect
21. C22×C22⋊C8C22semidirect
28. #2847C22semidirect
29. #2847C22semidirect
34. C2×#380C22semidirect
35. C2×#380C22semidirect
36. #2077C22semidirect
37. #2076C22semidirect
38. #2076C22semidirect
39. #2034C22semidirect
40. #2034C22semidirect
41. #2034C22semidirect
42. #2033C22semidirect
43. #1999C22semidirect
44. #1999C22semidirect
45. #1999C22semidirect
46. #1893C22semidirect
47. #1874C22semidirect
48. #1874C22semidirect
49. #1853C22semidirect
50. #1852C22semidirect
51. #1824C22semidirect
52. #1807C22semidirect
53. #1786C22semidirect
56. #1779C22semidirect
57. #1779C22semidirect
75. C22×D8C23semidirect
109. C2×#107C23semidirect
110. C2×#106C23semidirect
111. C2×#102C23semidirect
112. C2×#102C23semidirect
113. C2×#102C23semidirect
149. #340C23semidirect
150. #340C23semidirect
151. #340C23semidirect
152. #340C23semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
#327072
#298912
#297322
#295942
#276032
#245532
#243782
#240052
#239182
C2×#17792