Finite Group Databasebeta

← #763959

Subgroups — #763959

Order 512.

6114 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#534C211yesmaximal, characteristic
2C2×#2591C212yesmaximal, complemented
3C22×#396C211yesmaximal, characteristic, complemented
4#15667C218yesmaximal, complemented
5#15268C218yesmaximal, complemented
6#14913C212yesmaximal, complemented
7#14719C212yesmaximal, complemented
8#14137C212yesmaximal, complemented
9#13867C212yesmaximal, complemented
10#13867C212yesmaximal, complemented
11#13775C211yesmaximal, characteristic, complemented
12C24×D4C2212yes—
13C2×#534—216no—
14C2×#534—28no—
15C22×#123C2214yes—
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×#120C2214yes—
19C23×C4⋊C4C2211yescharacteristic, complemented
20C23×C4⋊C4C2211yescharacteristic, complemented
21C23×C4⋊C4C2211yescharacteristic, complemented
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25#2591—216no—
26C2×#400C2214yescomplemented
27C2×#396—28no—
28C2×#396C2218yescomplemented
29C2×#394C2214yescomplemented
30C2×#394C2214yescomplemented
31C2×#390C2214yescomplemented
32C2×#388C2218yescomplemented
33C2×#386C2214yescomplemented
34C2×#386C2214yescomplemented
35C2×#386C2214yescomplemented
36C2×#386C2214yescomplemented
37C2×#379C2212yes—
38C2×#379C2214yescomplemented
39C2×#379C2214yescomplemented
40C2×C4×C22⋊C4C2212yescomplemented
41C22×#95C2211yescharacteristic, complemented
42#1442C2214yescomplemented
43#1441C2214yescomplemented
44#1440C2218yescomplemented
45#1439C2218yescomplemented
46#1439C2218yescomplemented
47#1438C2218yes—
48#1438C2218yes—
49#1438C2214yescomplemented
50#1437C2218yescomplemented
51C4⋊#95C2214yescomplemented
52C22⋊#95C2214yescomplemented
53C22⋊#95C2214yescomplemented
54C22⋊#95C2212yescomplemented
55C26C2312yes—
56C26C2311yescharacteristic
57C23×D4—28no—
58C23×D4—216no—
59C23×D4—28no—
60C23×D4—28no—
61C23×D4—48no—
62C23×D4C2314yes—
63C23×D4C2314yes—
64C23×D4C2312yes—
65C23×D4C2312yes—
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73#534—2128no—
74C2×#123—216no—
75C2×#123—216no—
76C2×#123—216no—
77C2×#121—216no—
78C2×#121—28no—
79C2×#121—28no—
80C2×#121—28no—
81C2×#121D4116yes—
82C2×#120—216no—
83C2×#120—216no—
84C2×#120—216no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4D418yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2318yes—
114C22×C42C2314yes—
115#400—216no—
116#396—232no—
117#394—216no—
118#394—48no—
119#390—48no—
120#388—232no—
121#386—216no—
122#386—216no—
123#386—48no—
124#386—48no—
125#379—28no—
126#379—48no—
127#379—48no—
128C4×C22⋊C4—28no—
129C2×#95—24no—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2318yes—
133C2×#95C2318yes—
134C2×#95C2314yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2314yes—
138C25—24no—
139C25—24no—
140C25—28no—
141C25—24no—
142C25—24no—
143C25—24no—
144C25—42no—
145C25—42no—
146C25—44no—
147C25—42no—
148C25—42no—
149C25—44no—
150C25—44no—
151C25—44no—
152C25—44no—
153C25C2×D414yes—
154C25#4118yes—
155C25C2414yes—
156C25C2412yes—
157C25C2411yescharacteristic
158C22×D4—264no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—464no—
181C22×D4—464no—
182C22×D4—432no—
183C22×D4—432no—
184C22×D4C2×D4116yes—
185C22×D4C2×D418yes—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4C2×D414yes—
213C23×C4C2×D418yes—
214C23×C4C2×D414yes—
215C23×C4C2418yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223#123—2128no—
224#121—264no—
225#121—264no—
226#120—2128no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—416no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4C2×D418yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—232no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—416no—
269C2×C22⋊C4—416no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4C2×D4116yes—
282C2×C22⋊C4C2×D4116yes—
283C2×C42—28no—
284C2×C42—44no—
285C2×C42—44no—
286#95—28no—
287#95—28no—
288#95—216no—
289#95—216no—
290#95—44no—
291#95—48no—
292#95—48no—
293#95—44no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—28no—
305C24—28no—
306C24—232no—
307C24—216no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—28no—
313C24—28no—
314C24—24no—
315C24—24no—
316C24—28no—
317C24—28no—
318C24—44no—
319C24—44no—
320C24—44no—
321C24—44no—
322C24—416no—
323C24—416no—
324C24—416no—
325C24—416no—
326C24—48no—
327C24—416no—
328C24—416no—
329C24—48no—
330C24—88no—
331C24—88no—
332C24—88no—
333C24—88no—
334C24—84no—
335C24—84no—
336C24—84no—
337C24—84no—
338C24—88no—
339C24—88no—
340C24—816no—
341C24—816no—
342C24—816no—
343C24—816no—
344C24#124116yes—
345C24C22×D418yes—
346C24C22×D414yes—
347C24C22×D412yes—
348C24C22×D414yes—
349C24C2×#4114yes—
350C24C2×#4114yes—
351C24C2×#4114yes—
352C24H(5,𝔽2)14yes—
353C24C2511yescharacteristic
354C2×D4—232no—
355C2×D4—232no—
356C2×D4—232no—
357C2×D4—232no—
358C2×D4—216no—
359C2×D4—216no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—416no—
365C2×D4—4256no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—88no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—816no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—216no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—416no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4#120116yes—
441C22×C4C22×D418yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C4H(5,𝔽2)14yes—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C22⋊C4—232no—
454C22⋊C4—232no—
455C22⋊C4—232no—
456C22⋊C4—432no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—416no—
460C22⋊C4—816no—
461C22⋊C4—816no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—816no—
465C42—88no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23—28no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—232no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—216no—
493C23—216no—
494C23—416no—
495C23—416no—
496C23—432no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—864no—
520C23—88no—
521C23—88no—
522C23—88no—
523C23—88no—
524C23—816no—
525C23—816no—
526C23—816no—
527C23—816no—
528C23—864no—
529C23—864no—
530C23—816no—
531C23—816no—
532C23C2×#12014yes—
533C23C2×#12418yes—
534C23C2×#12414yes—
535C23#53514yes—
536C23#546116yes—
537C23C23×D411yescharacteristic
538C23C23×D412yes—
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)12yes—
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)12yes—
543C23#58512yes—
544C23#58512yes—
545D4—464no—
546D4—464no—
547D4—816no—
548D4—816no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—416no—
566C2×C4—432no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—88no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4C2×#12018yes—
597C22—232no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—28no—
605C22—28no—
606C22—28no—
607C22—416no—
608C22—416no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—416no—
614C22—416no—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—832no—
619C22—864no—
620C22—88no—
621C22—88no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22#243518yes—
627C22C22×#12011yescharacteristic
628C22C22×#12411yescharacteristic
629C22C2×#53511yescharacteristic
630C22C2×#54614yes—
631C22#348411yescharacteristic
632C22#348411yescharacteristic
633C22#348511yescharacteristic
634C22#348511yescharacteristic
635C22#348814yes—
636C22#349614yes—
637C22#350514yes—
638C22#350514yes—
639C22#353714yes—
640C22#356014yes—
641C4—28no—
642C4—44no—
643C4—48no—
644C4—44no—
645C4—44no—
646C4—88no—
647C4—82no—
648C4—82no—
649C2—28no—
650C2—48no—
651C2—88no—
652C2C2×#243511yescharacteristic
653C2#3534711yescharacteristic
654C2#3537711yescharacteristic
655C2#3537711yescharacteristic
656C2#3551211yescharacteristic
657C2#3671711yescharacteristic
658C2#3696911yescharacteristic
659C2#4302218yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#534C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2591C2semidirect
3. C22×#396C2semidirect
4. #15667C2semidirect
5. #15268C2semidirect
6. #14913C2semidirect
7. #14719C2semidirect
8. #14137C2semidirect
9. #13867C2semidirect
10. #13867C2semidirect
11. #13775C2semidirect
19. C23×C4⋊C4C22semidirect
20. C23×C4⋊C4C22semidirect
21. C23×C4⋊C4C22semidirect
26. C2×#400C22semidirect
28. C2×#396C22semidirect
29. C2×#394C22semidirect
30. C2×#394C22semidirect
31. C2×#390C22semidirect
32. C2×#388C22semidirect
33. C2×#386C22semidirect
34. C2×#386C22semidirect
35. C2×#386C22semidirect
36. C2×#386C22semidirect
38. C2×#379C22semidirect
39. C2×#379C22semidirect
40. C2×C4×C22⋊C4C22semidirect
41. C22×#95C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect
44. #1440C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
49. #1438C22semidirect
50. #1437C22semidirect
51. C4⋊#95C22semidirect
52. C22⋊#95C22semidirect
53. C22⋊#95C22semidirect
54. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5342
C2×#25912
C22×#3962
#156672
#152682
#149132
#147192
#141372
#138672
#137752