Finite Group Databasebeta

← #763816

Subgroups — #763816

Order 512.

6486 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#537C211yesmaximal, characteristic, complemented
2C2×#2539C212yesmaximal, complemented
3C2×#2406C212yesmaximal, complemented
4C2×#2404C212yesmaximal, complemented
5C22×#396C211yesmaximal, characteristic, complemented
6#15325C218yesmaximal, complemented
7#15315C218yesmaximal, complemented
8#14926C212yesmaximal, complemented
9#14716C212yesmaximal, complemented
10#14137C212yesmaximal, complemented
11#13776C211yesmaximal, characteristic, complemented
12C23×#41C2212yes—
13C2×#537—216no—
14C2×#537—28no—
15C22×#123C2214yes—
16C22×C22⋊Q8C2214yescomplemented
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#121C2214yescomplemented
20C22×#120C2214yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27#2539—216no—
28#2406—216no—
29#2404—216no—
30C2×#399C2214yes—
31C2×#396—28no—
32C2×#396C2218yes—
33C2×#395C2214yescomplemented
34C2×#394C2214yescomplemented
35C2×#393C2214yes—
36C2×#392C2214yes—
37C2×#392C2214yes—
38C2×#392C2214yes—
39C2×#390C2218yes—
40C2×#390C2214yescomplemented
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2218yes—
44C2×#385C2218yescomplemented
45C2×#381C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2211yescharacteristic
48#1444C2214yescomplemented
49#1439C2214yes—
50#1439C2218yescomplemented
51#1439C2218yescomplemented
52#1438C2218yes—
53#1438C2214yescomplemented
54#1436C2218yescomplemented
55C4⋊#95C2214yescomplemented
56C22⋊#95C2212yes—
57C26C2311yescharacteristic
58C22×#41—216no—
59C22×#41—28no—
60C22×#41—28no—
61C22×#41—28no—
62C22×#41—48no—
63C23×Q8C2312yes—
64C23×D4C2314yes—
65C23×D4C2312yes—
66C23×D4C2318yes—
67C23×D4C2312yes—
68C23×D4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2314yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76#537—2128no—
77C2×#123—216no—
78C2×#123—216no—
79C2×#123—48no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×#121—216no—
84C2×#121—216no—
85C2×#121—28no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—216no—
89C2×#121—216no—
90C2×#121D4116yescomplemented
91C2×#120—216no—
92C2×#120—216no—
93C2×#120—48no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4D418yescomplemented
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2318yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2318yes—
116C22×C22⋊C4C2318yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2318yes—
122C22×C42C2314yes—
123#399—216no—
124#396—232no—
125#395—216no—
126#394—216no—
127#393—216no—
128#392—216no—
129#392—216no—
130#392—48no—
131#390—232no—
132#390—48no—
133#386—216no—
134#386—216no—
135#386—232no—
136#385—232no—
137#381D4116yescomplemented
138C4×C22⋊C4—28no—
139C2×#95—24no—
140C2×#95C2314yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C25—22no—
149C25—24no—
150C25—24no—
151C25—24no—
152C25—24no—
153C25—42no—
154C25—42no—
155C25—42no—
156C25C2414yes—
157C25C2412yes—
158C25C2411yescharacteristic
159C2×#41—264no—
160C2×#41—464no—
161C2×#41—464no—
162C2×#41—432no—
163C2×#41—432no—
164C22×Q8—28no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—44no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—216no—
178C22×D4—216no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—216no—
186C22×D4—44no—
187C22×D4—48no—
188C22×D4—44no—
189C22×D4—48no—
190C22×D4—48no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—48no—
194C22×D4—44no—
195C22×D4C2×D4116yes—
196C22×D4C2×D418yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2×D414yes—
224C23×C4C2×D418yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2418yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236#123—464no—
237C22⋊Q8—2128no—
238#121—264no—
239#121—2128no—
240#121—264no—
241#120—464no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—416no—
259C2×C4⋊C4—416no—
260C2×C4⋊C4C2×D4116yes—
261C2×C4⋊C4C2×D418yes—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—216no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—416no—
289C2×C22⋊C4—416no—
290C2×C22⋊C4—416no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4C2×D4116yes—
294C2×C22⋊C4C2×D4116yes—
295C2×C42—28no—
296C2×C42—44no—
297C2×C42—44no—
298#95—28no—
299#95—28no—
300#95—216no—
301#95—216no—
302#95—44no—
303#95—44no—
304#95C2×D4116yes—
305#95C2×D4116yes—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—28no—
311C24—22no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—28no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—28no—
324C24—22no—
325C24—28no—
326C24—28no—
327C24—24no—
328C24—24no—
329C24—416no—
330C24—416no—
331C24—416no—
332C24—44no—
333C24—44no—
334C24—44no—
335C24—44no—
336C24—44no—
337C24—44no—
338C24—416no—
339C24—88no—
340C24—88no—
341C24—88no—
342C24C22×D418yes—
343C24C22×D412yes—
344C24C22×D414yes—
345C24C22×D414yes—
346C24H(5,𝔽2)14yes—
347C242−1+414yes—
348C24C2511yescharacteristic
349#41—4256no—
350C2×Q8—216no—
351C2×Q8—416no—
352C2×Q8—416no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×D4—216no—
356C2×D4—232no—
357C2×D4—232no—
358C2×D4—232no—
359C2×D4—216no—
360C2×D4—216no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—48no—
364C2×D4—416no—
365C2×D4—432no—
366C2×D4—432no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—432no—
372C2×D4—48no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—88no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—816no—
379C2×D4—816no—
380C2×D4—816no—
381C2×D4—816no—
382C2×D4—816no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—232no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—232no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—416no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—416no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4#120116yes—
444C22×C4#127116yes—
445C22×C4C22×D4116yes—
446C22×C4C22×D414yes—
447C22×C4C22×D418yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4H(5,𝔽2)14yes—
455C4⋊C4—216no—
456C4⋊C4—232no—
457C4⋊C4—216no—
458C4⋊C4—416no—
459C4⋊C4—432no—
460C4⋊C4—416no—
461C22⋊C4—232no—
462C22⋊C4—232no—
463C22⋊C4—232no—
464C22⋊C4—432no—
465C22⋊C4—416no—
466C22⋊C4—432no—
467C22⋊C4—432no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—816no—
471C42—88no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—216no—
489C23—216no—
490C23—28no—
491C23—416no—
492C23—416no—
493C23—416no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—48no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—44no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—864no—
526C23—88no—
527C23—88no—
528C23—88no—
529C23—88no—
530C23C2×#12014yes—
531C23C2×#12714yes—
532C23#53514yes—
533C23D42116yes—
534C23C23×D411yescharacteristic
535C23C23×D412yes—
536C23C23×D412yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+412yes—
540C23#58512yes—
541C23#58512yes—
542Q8—432no—
543D4—432no—
544D4—464no—
545D4—816no—
546D4—816no—
547D4—864no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—232no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—216no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—216no—
567C2×C4—216no—
568C2×C4—416no—
569C2×C4—464no—
570C2×C4—416no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—88no—
594C2×C4C2×#12018yes—
595C2×C4C2×#12718yes—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—232no—
602C22—24no—
603C22—48no—
604C22—44no—
605C22—48no—
606C22—44no—
607C22—44no—
608C22—48no—
609C22—416no—
610C22—416no—
611C22—416no—
612C22—832no—
613C22—82no—
614C22—84no—
615C22—84no—
616C22—82no—
617C22—84no—
618C22—82no—
619C22—88no—
620C22—88no—
621C22—88no—
622C22—84no—
623C22—82no—
624C22#243318yes—
625C22C22×#12011yescharacteristic
626C22C22×#12711yescharacteristic
627C22C2×#53511yescharacteristic
628C22C2×D4214yes—
629C22#348411yescharacteristic
630C22#348511yescharacteristic
631C22#348611yescharacteristic
632C22#348711yescharacteristic
633C22#348814yes—
634C22#348914yes—
635C22#348914yes—
636C22#349614yes—
637C22#353614yes—
638C22#356014yes—
639C4—28no—
640C4—28no—
641C4—44no—
642C4—48no—
643C4—416no—
644C4—44no—
645C4—82no—
646C4—82no—
647C2—28no—
648C2—88no—
649C2—84no—
650C2C2×#243311yescharacteristic
651C2#3534711yescharacteristic
652C2#3556511yescharacteristic
653C2#3563411yescharacteristic
654C2#3565911yescharacteristic
655C2#3670711yescharacteristic
656C2#3696111yescharacteristic
657C2#4390718yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#537C2semidirect
2. C2×#2539C2semidirect
3. C2×#2406C2semidirect
4. C2×#2404C2semidirect
5. C22×#396C2semidirect
6. #15325C2semidirect
7. #15315C2semidirect
8. #14926C2semidirect
9. #14716C2semidirect
10. #14137C2semidirect
11. #13776C2semidirect
16. C22×C22⋊Q8C22semidirect
19. C22×#121C22semidirect
33. C2×#395C22semidirect
34. C2×#394C22semidirect
40. C2×#390C22semidirect
44. C2×#385C22semidirect
48. #1444C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
53. #1438C22semidirect
54. #1436C22semidirect
55. C4⋊#95C22semidirect
90. C2×#121D4semidirect
99. C22×C4⋊C4D4semidirect
137. #381D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5372
C2×#25392
C2×#24062
C2×#24042
C22×#3962
#153252
#153152
#149262
#147162
#141372
#137762