Finite Group Databasebeta

← #647571

Subgroups — #647571

Order 512.

4806 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic, complemented
2C22×#390C211yesmaximal, characteristic, complemented
3#17007C212yesmaximal, complemented
4#16810C212yesmaximal, complemented
5#16662C214yesmaximal, complemented
6#15112C212yesmaximal, complemented
7#15091C212yesmaximal, complemented
8#14820C212yesmaximal, complemented
9#14811C212yesmaximal, complemented
10#14716C212yesmaximal, complemented
11#14715C212yesmaximal, complemented
12#14683C211yesmaximal, characteristic, complemented
13#14311C214yesmaximal, complemented
14#13967C214yesmaximal, complemented
15C24×D4C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C22×#120C2214yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic, complemented
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C23×C42C2211yescharacteristic
24C2×#397C2214yescomplemented
25C2×#392C2214yes—
26C2×#392C2214yes—
27C2×#392C2214yescomplemented
28C4×#121—28no—
29C2×#390—28no—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic, complemented
35C2×#389C2211yescharacteristic, complemented
36C2×#388C2212yescomplemented
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#386C2218yescomplemented
40C2×#383C2212yescomplemented
41C2×#380C2211yescharacteristic
42C2×#379C2214yes—
43C2×C4×C4⋊C4C2211yescharacteristic, complemented
44C2×C4×C22⋊C4C2212yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1440C2218yescomplemented
48#1439C2214yescomplemented
49#1439C2218yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1438C2214yes—
54#1438C2214yes—
55#1438C2218yescomplemented
56#1438C2218yescomplemented
57#1437C2218yescomplemented
58#1434C2212yescomplemented
59#1434C2212yescomplemented
60#1434C2212yescomplemented
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2214yes—
65C26C2312yes—
66C23×D4—28no—
67C23×D4—24no—
68C23×D4—44no—
69C23×D4—44no—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2318yes—
74C23×D4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C2×#121—28no—
81C2×#121—24no—
82C2×#120—216no—
83C2×#120—48no—
84C2×#120—48no—
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—44no—
88C2×C4×D4—48no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—48no—
100C22×C22⋊C4—48no—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2318yes—
114C22×C22⋊C4C2318yes—
115C22×C42—44no—
116C22×C42D418yescomplemented
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#397—48no—
123#392—216no—
124#392—48no—
125#392—48no—
126#390—28no—
127#390—28no—
128#390—28no—
129#390—28no—
130#390—24no—
131#389—24no—
132#388—28no—
133#386—28no—
134#386—416no—
135#386—48no—
136#383—44no—
137#380—24no—
138#379—48no—
139C4×C4⋊C4—24no—
140C4×C22⋊C4—28no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2318yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C25—24no—
154C25—28no—
155C25—28no—
156C25—24no—
157C25—28no—
158C25—24no—
159C25—44no—
160C25—44no—
161C25—44no—
162C25C2414yes—
163C25C2412yes—
164C25C2411yescharacteristic
165C22×D4—232no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—216no—
173C22×D4—216no—
174C22×D4—22no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—42no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—464no—
185C22×D4—416no—
186C22×D4—416no—
187C22×D4—48no—
188C22×D4—48no—
189C22×D4—48no—
190C22×D4—48no—
191C22×D4—48no—
192C22×D4#4118yes—
193C22×D4#4118yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2418yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225#121—232no—
226#120—464no—
227C4×D4—416no—
228C4×D4—432no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—416no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
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—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—432no—
271C2×C22⋊C4—432no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—416no—
284C2×C42—44no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42C2×D414yes—
288C2×C42C2×D414yes—
289C2×C42C2×D414yes—
290C2×C42C2×D414yes—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—44no—
299#95—44no—
300#95—48no—
301#95—48no—
302#95—48no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—216no—
308C24—22no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—28no—
313C24—24no—
314C24—24no—
315C24—28no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—28no—
325C24—28no—
326C24—432no—
327C24—432no—
328C24—416no—
329C24—416no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24—48no—
334C24—48no—
335C24—48no—
336C24—416no—
337C24—416no—
338C24—816no—
339C24—88no—
340C24—88no—
341C24—816no—
342C24C22×D412yes—
343C24C2×#4118yes—
344C24C2×#4114yes—
345C24C2×#4114yes—
346C24H(5,𝔽2)14yes—
347C24H(5,𝔽2)14yes—
348C24C2511yescharacteristic
349C2×D4—28no—
350C2×D4—216no—
351C2×D4—216no—
352C2×D4—232no—
353C2×D4—216no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—432no—
357C2×D4—44no—
358C2×D4—4128no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—416no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—432no—
367C2×D4—432no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—816no—
371C2×D4—816no—
372C2×D4—816no—
373C2×D4—816no—
374C2×D4—816no—
375C22×C4—232no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—816no—
432C22×C4—816no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)14yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C4H(5,𝔽2)14yes—
444C4⋊C4—28no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C22⋊C4—216no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—432no—
455C22⋊C4—48no—
456C22⋊C4—416no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—416no—
460C22⋊C4—416no—
461C22⋊C4—816no—
462C22⋊C4—816no—
463C22⋊C4—816no—
464C42—28no—
465C42—28no—
466C42—44no—
467C42—44no—
468C42—48no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—216no—
484C23—216no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—432no—
489C23—432no—
490C23—432no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—48no—
504C23—44no—
505C23—48no—
506C23—44no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—44no—
511C23—48no—
512C23—48no—
513C23—44no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—864no—
523C23—864no—
524C23—816no—
525C23—816no—
526C23—816no—
527C23—816no—
528C23#52514yes—
529C23#53514yes—
530C23#53514yes—
531C23#538116yes—
532C23C23×D411yescharacteristic
533C23C22×#4112yes—
534C23C22×#4112yes—
535C23C2×H(5,𝔽2)12yes—
536C23C2×H(5,𝔽2)12yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)12yes—
540D4—416no—
541D4—432no—
542D4—432no—
543D4—432no—
544D4—864no—
545C2×C4—232no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—416no—
567C2×C4—416no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
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—816no—
594C2×C4—816no—
595C2×C4#52514yes—
596C2×C4#53414yes—
597C2×C4#53514yes—
598C2×C4#53514yes—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—232no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—48no—
610C22—416no—
611C22—416no—
612C22—416no—
613C22—48no—
614C22—48no—
615C22—864no—
616C22—84no—
617C22—84no—
618C22—82no—
619C22—84no—
620C22—88no—
621C22—88no—
622C22—88no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—84no—
627C22#263118yes—
628C22C2×#52511yescharacteristic
629C22C2×#52511yescharacteristic
630C22C2×#53411yescharacteristic
631C22C2×#53511yescharacteristic
632C22C2×#53511yescharacteristic
633C22C2×#53511yescharacteristic
634C22C2×#53511yescharacteristic
635C22C2×#53814yes—
636C22#348114yes—
637C22#349014yes—
638C22#351914yes—
639C22#351914yes—
640C22#353714yes—
641C22#354314yes—
642C4—24no—
643C4—24no—
644C4—24no—
645C4—24no—
646C4—42no—
647C4—44no—
648C4—42no—
649C4—82no—
650C4—88no—
651C4—82no—
652C4—84no—
653C2—28no—
654C2—84no—
655C2—88no—
656C2C2×#263111yescharacteristic
657C2#3496311yescharacteristic
658C2#3505011yescharacteristic
659C2#3507611yescharacteristic
660C2#3523711yescharacteristic
661C2#3745011yescharacteristic
662C2#3761411yescharacteristic
663C2#3856418yes—
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. C2×C4×#121C2semidirect
2. C22×#390C2semidirect
3. #17007C2semidirect
4. #16810C2semidirect
5. #16662C2semidirect
6. #15112C2semidirect
7. #15091C2semidirect
8. #14820C2semidirect
9. #14811C2semidirect
10. #14716C2semidirect
11. #14715C2semidirect
12. #14683C2semidirect
13. #14311C2semidirect
14. #13967C2semidirect
19. C22×C4×D4C22semidirect
24. C2×#397C22semidirect
27. C2×#392C22semidirect
34. C2×#390C22semidirect
35. C2×#389C22semidirect
36. C2×#388C22semidirect
39. C2×#386C22semidirect
40. C2×#383C22semidirect
43. C2×C4×C4⋊C4C22semidirect
44. C2×C4×C22⋊C4C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1440C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
55. #1438C22semidirect
56. #1438C22semidirect
57. #1437C22semidirect
58. #1434C22semidirect
59. #1434C22semidirect
60. #1434C22semidirect
116. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C22×#3902
#170072
#168102
#166622
#151122
#150912
#148202
#148112
#147162
#147152
#146832
#143112
#139672