Finite Group Databasebeta

← #640971

Subgroups — #640971

Order 512.

5490 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#532C211yesmaximal, characteristic
2C2×#2406C212yesmaximal, complemented
3C2×#2618C212yesmaximal, complemented
4C22×#387C211yesmaximal, characteristic, complemented
5#16507C212yesmaximal, complemented
6#16505C214yesmaximal, complemented
7#16503C212yesmaximal, complemented
8#14926C212yesmaximal, complemented
9#14913C212yesmaximal, complemented
10#14812C214yesmaximal, complemented
11#14801C214yesmaximal, complemented
12#14685C211yesmaximal, characteristic, complemented
13#14271C212yesmaximal, complemented
14#13956C212yesmaximal, complemented
15C23×#41C2212yes—
16C2×#532—28no—
17C2×#532—216no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C22×#124C2212yes—
21C22×#123C2214yes—
22C22×#121C2212yes—
23C22×#121C2212yes—
24C22×C4×D4C2212yes—
25C22×C4×D4C2212yes—
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C23×C42C2211yescharacteristic
29#2618—216no—
30#2406—216no—
31C2×#398C2214yes—
32C2×#396C2218yes—
33C2×#395C2214yes—
34C2×#394C2214yes—
35C2×#394C2214yes—
36C2×#387—28no—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#385C2218yes—
45C2×#381C2212yes—
46C2×#380C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1446C2214yes—
52#1445C2214yes—
53#1444C2214yes—
54#1442C2214yes—
55#1440C2218yes—
56#1439C2218yes—
57#1439C2218yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2218yes—
68C22×#41—216no—
69C22×#41—28no—
70C22×#41—28no—
71C22×#41—28no—
72C22×#41—48no—
73C23×Q8C2312yes—
74C23×D4C2314yes—
75C23×D4C2312yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86#532—2128no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—24no—
89C2×#127—28no—
90C2×#127—24no—
91C2×#124—216no—
92C2×#124—28no—
93C2×#123—216no—
94C2×#123—216no—
95C2×#123—48no—
96C2×#121—216no—
97C2×#121—28no—
98C2×#121—28no—
99C2×#121—28no—
100C2×#121D4116yes—
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C2×C4×D4—216no—
104C2×C4×D4—28no—
105C2×C4×D4—28no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4D418yescomplemented
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2318yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2318yes—
124C22×C22⋊C4C2318yes—
125C22×C22⋊C4C2318yes—
126C22×C42—28no—
127C22×C42D418yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#398—216no—
133#396—232no—
134#395—216no—
135#394—216no—
136#394—216no—
137#387—28no—
138#387—28no—
139#387—28no—
140#387—28no—
141#386—216no—
142#386—216no—
143#386—48no—
144#385—232no—
145#381—28no—
146#380—28no—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95—24no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C25C2414yes—
168C25C2411yescharacteristic
169C2×#41—264no—
170C2×#41—464no—
171C2×#41—464no—
172C2×#41—432no—
173C2×#41—432no—
174C22×Q8—28no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—44no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—28no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4C2×D4116yes—
199C22×D4C2×D418yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D418yes—
237C23×C4#4114yes—
238C23×C4C2418yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2414yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C4⋊Q8—232no—
251#127—232no—
252#124—264no—
253#123—464no—
254#121—264no—
255#121—264no—
256C4×D4—264no—
257C4×D4—264no—
258C2×C4⋊C4—232no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—216no—
272C2×C4⋊C4—216no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—416no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—216no—
292C2×C22⋊C4—216no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4C2×D4116yes—
302C2×C42—232no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—216no—
323#95—216no—
324#95—28no—
325#95—44no—
326#95—48no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—28no—
338C24—28no—
339C24—28no—
340C24—28no—
341C24C22×D418yes—
342C24C22×D412yes—
343C24C22×D414yes—
344C24C22×D412yes—
345C24C2×#4112yes—
346C24H(5,𝔽2)14yes—
347C24C2511yescharacteristic
348#41—4256no—
349C2×Q8—216no—
350C2×Q8—416no—
351C2×Q8—416no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×D4—216no—
355C2×D4—216no—
356C2×D4—232no—
357C2×D4—232no—
358C2×D4—232no—
359C2×D4—216no—
360C2×D4—416no—
361C2×D4—432no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—48no—
374C2×D4—48no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—232no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4#120116yes—
454C22×C4#121116yes—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D418yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C42−1+414yes—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—432no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—88no—
479C22⋊C4—232no—
480C22⋊C4—232no—
481C22⋊C4—432no—
482C22⋊C4—432no—
483C22⋊C4—416no—
484C22⋊C4—816no—
485C22⋊C4—816no—
486C42—28no—
487C42—28no—
488C42—28no—
489C42—28no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—216no—
506C23—216no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—48no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—48no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23C2×#12014yes—
526C23C2×#12114yes—
527C23#53514yes—
528C23D4218yes—
529C23#54618yes—
530C23C23×D411yescharacteristic
531C23C23×D412yes—
532C23C23×D411yescharacteristic
533C23C22×#4111yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23C2×2−1+412yes—
536C23#58512yes—
537C23#58512yes—
538C23#58512yes—
539Q8—432no—
540D4—432no—
541D4—432no—
542D4—432no—
543D4—464no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—232no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—416no—
573C2×C4—416no—
574C2×C4—464no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—416no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4C2×#12014yes—
606C2×C4C2×#12014yes—
607C2×C4C2×#12114yes—
608C2×C4C2×#12114yes—
609C2×C4#53614yes—
610C22—24no—
611C22—216no—
612C22—216no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—48no—
619C22—416no—
620C22—416no—
621C22—48no—
622C22—48no—
623C22—88no—
624C22—88no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22#240818yes—
630C22C22×#12011yescharacteristic
631C22C22×#12111yescharacteristic
632C22C2×#53511yescharacteristic
633C22C2×#53611yescharacteristic
634C22#344911yescharacteristic
635C22#344911yescharacteristic
636C22#344912yes—
637C22#344911yescharacteristic
638C22#345512yes—
639C22C2×D4212yes—
640C22C2×#54612yes—
641C22#348514yes—
642C22#348914yes—
643C22#349712yes—
644C22#350712yes—
645C22#355914yes—
646C22#356014yes—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—24no—
651C4—24no—
652C4—44no—
653C4—44no—
654C4—48no—
655C4—42no—
656C4—82no—
657C4—88no—
658C4—82no—
659C2—28no—
660C2—88no—
661C2#3463311yescharacteristic
662C2C2×#240811yescharacteristic
663C2#3508811yescharacteristic
664C2#3517811yescharacteristic
665C2#3530511yescharacteristic
666C2#3753911yescharacteristic
667C2#3754711yescharacteristic
668C2#3767914yes—
669C2#3769814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#532C2
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×#2406C2semidirect
3. C2×#2618C2semidirect
4. C22×#387C2semidirect
5. #16507C2semidirect
6. #16505C2semidirect
7. #16503C2semidirect
8. #14926C2semidirect
9. #14913C2semidirect
10. #14812C2semidirect
11. #14801C2semidirect
12. #14685C2semidirect
13. #14271C2semidirect
14. #13956C2semidirect
110. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5322
C2×#24062
C2×#26182
C22×#3872
#165072
#165052
#165032
#149262
#149132
#148122
#148012
#146852
#142712
#139562