Finite Group Databasebeta

← #7643386

Subgroups — #7643386

Order 512.

5238 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39766C214yesmaximal, complemented
2#39382C214yesmaximal, complemented
3#39185C214yesmaximal, complemented
4#38969C214yesmaximal, complemented
5#38129C218yesmaximal, complemented
6#37701C214yesmaximal, complemented
7#37700C214yesmaximal, complemented
8#37318C214yesmaximal, complemented
9#37317C214yesmaximal, complemented
10#36644C211yesmaximal, characteristic, complemented
11#36640C211yesmaximal, characteristic, complemented
12#35559C218yesmaximal, complemented
13#35319C212yesmaximal, complemented
14#35292C212yesmaximal, complemented
15#35288C214yesmaximal, complemented
16#35278C212yesmaximal, complemented
17#34226C212yesmaximal, complemented
18#34026C211yesmaximal, characteristic, complemented
19C2×#541C2214yes—
20C2×#540C2214yes—
21C2×#532C2212yes—
22C2×#532C2214yes—
23C2×C4×#41C2211yescharacteristic
24#2868C2211yescharacteristic
25#2867C2211yescharacteristic
26#2864C2214yes—
27#2859C2212yes—
28#2859C2212yes—
29#2858C2214yes—
30#2858C2212yes—
31#2857C2214yes—
32#2856C2214yes—
33#2856C2214yes—
34#2856C2214yes—
35#2851C2214yes—
36#2851C2214yes—
37#2796C22116yes—
38#2751C22116yes—
39#2714C22116yes—
40#2714C22116yes—
41#2711C22116yes—
42#2708C22116yes—
43#2694C22116yes—
44#2688C22116yes—
45#2644C2218yes—
46#2642C2214yes—
47#2641C2214yes—
48#2624C2218yes—
49#2624C2214yes—
50#2621C2214yes—
51#2619C2214yes—
52#2615C2214yes—
53#2595C22116yes—
54#2592C2214yes—
55#2580C2214yes—
56#2554C22116yes—
57#2540C2214yes—
58#2535C2214yes—
59#2516C22116yes—
60#2513C2218yes—
61#2506C2218yes—
62#2506C2218yes—
63#2496C2218yes—
64#2496C2218yes—
65#2490C2218yes—
66#2490C22116yes—
67#2467C22116yes—
68#2460C22116yes—
69#2454C2218yes—
70#2448C2218yes—
71#2447C2218yes—
72#2445C2218yes—
73#2445C2218yes—
74#2444C22116yes—
75#2428C2218yes—
76#2428C2218yes—
77#2427C2218yes—
78#2426C2218yes—
79#2420C2218yes—
80#2419C22116yes—
81#2414C2218yes—
82#2407C2218yes—
83#2367C2218yes—
84#2363C2218yes—
85#2357C2214yes—
86#2349C2214yes—
87#2341C2214yes—
88#2339C2214yes—
89#2328C2214yes—
90#2327C2214yes—
91#2324C2218yes—
92#2324C2218yes—
93#2324C2218yes—
94#2319C2218yes—
95#2319C2214yes—
96C4×C4⋊Q8C2214yes—
97C4×#127C2214yes—
98C4×#127C2214yes—
99C4×#127C2214yes—
100C4×#125C2214yes—
101C4×#125C2214yes—
102#2298C2214yes—
103#2297C2212yes—
104#2296C2214yes—
105#2296C2212yes—
106#2295C2212yes—
107#2295C2212yes—
108#2293C2212yes—
109#2293C2212yes—
110#2290C2212yes—
111#2290C2212yes—
112#2290C2214yes—
113#2285C2212yes—
114#2275C2214yes—
115C4×#117C2211yescharacteristic
116C4×#117C2212yes—
117C4×#117C2212yes—
118C2×C43C2211yescharacteristic
119C22×#41C2314yes—
120C22×#41C2311yescharacteristic
121#541—416no—
122#540—416no—
123#532—216no—
124#532—416no—
125C2×C4⋊Q8C2312yes—
126C2×C4⋊Q8C2314yes—
127C2×#127C2312yes—
128C2×#127C2314yes—
129C2×#127C2314yes—
130C2×#127C2314yes—
131C2×#126C23116yes—
132C2×#125C2314yes—
133C2×#125C2314yes—
134C2×#125C23116yes—
135C2×#124C2314yes—
136C2×#124C2318yes—
137C2×#123C23116yes—
138C2×C22⋊Q8C2318yes—
139C2×#121C2314yes—
140C2×#121C2314yes—
141C2×#121C2318yes—
142C2×#121C2318yes—
143C2×#121C23116yes—
144C2×#121C2318yes—
145C2×#121C23116yes—
146C4×#41—28no—
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2318yes—
150C2×C4×D4C2314yes—
151C2×C4×D4C2314yes—
152C2×C4×D4C2312yes—
153C2×C4×D4C2312yes—
154C2×C4×D4C2311yescharacteristic
155C2×C4×D4C2311yescharacteristic
156C2×C4×D4C23116yes—
157C2×C4×D4C2318yes—
158C2×C4×D4C2318yes—
159C2×C4×D4C2318yes—
160C2×#117C2312yes—
161C2×#117C2312yes—
162C2×#117C2312yes—
163C2×#117C2311yescharacteristic
164C2×#117C2312yes—
165C2×#117C2314yes—
166C2×#117C2314yes—
167C22×C42C2312yes—
168C22×C42C2312yes—
169C22×C42C2312yes—
170C22×C42C2312yes—
171C22×C42C2311yescharacteristic
172#400C23116yes—
173#400C23116yes—
174#399C2318yes—
175#399C23116yes—
176#398C23116yes—
177#397C2318yes—
178#397C23116yes—
179#397C23116yes—
180#397C23116yes—
181#397C23116yes—
182#397C23116yes—
183#396C23116yes—
184#394C23116yes—
185#394C23116yes—
186#394C23116yes—
187#394C23116yes—
188#394C23116yes—
189#394C23116yes—
190#393C2318yes—
191#392C23116yes—
192#392C2318yes—
193#392C23116yes—
194#391C2318yes—
195#391C2318yes—
196#391C2314yes—
197#390C2314yes—
198#390C2318yes—
199#390C2314yes—
200#390C23116yes—
201#390C2318yes—
202#390C2318yes—
203#390C23116yes—
204#390C2318yes—
205#390C2318yes—
206#390C23116yes—
207#390C2318yes—
208#390C23116yes—
209#390C23116yes—
210#390C2318yes—
211#390C2314yes—
212#389C2318yes—
213#389C2318yes—
214#389C2318yes—
215#389C2318yes—
216#389C2318yes—
217#389C2318yes—
218#389C2318yes—
219#389C2318yes—
220#389C2314yes—
221#389C2314yes—
222#388C23116yes—
223#388C23116yes—
224#388C23116yes—
225#388C23116yes—
226#388C23116yes—
227#388C23116yes—
228#388C2314yes—
229#388C2314yes—
230#388C2314yes—
231#388C2314yes—
232#387C23116yes—
233#387C23116yes—
234#387C23116yes—
235#387C23116yes—
236#387C2314yes—
237#387C2314yes—
238#387C2314yes—
239#387C2314yes—
240#386C23116yes—
241#386C23116yes—
242#386C23116yes—
243#386C23116yes—
244#386C23116yes—
245#386C2314yes—
246#386C2314yes—
247#385C23116yes—
248#385C23116yes—
249#385C23116yes—
250#385C23116yes—
251#385C2314yes—
252#385C2314yes—
253#384C2314yes—
254#384C2314yes—
255#384C2314yes—
256#382C2314yes—
257#382C2314yes—
258#382C2314yes—
259#381C2318yes—
260#381C23116yes—
261#380C2318yes—
262#380C23116yes—
263#380C2314yes—
264#380C2318yes—
265#380C2318yes—
266#380C2314yes—
267C4×C4⋊C4C2314yes—
268C4×C4⋊C4C2314yes—
269C4×C4⋊C4C23116yes—
270C4×C4⋊C4C2318yes—
271C4×C4⋊C4C2318yes—
272C4×C4⋊C4C2312yes—
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2314yes—
275C4×C4⋊C4C2314yes—
276C4×C4⋊C4C2312yes—
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2314yes—
279C4×C22⋊C4C2311yescharacteristic
280C4×C22⋊C4C23116yes—
281C4×C22⋊C4C2311yescharacteristic
282C4×C22⋊C4C2318yes—
283C4×C22⋊C4C2312yes—
284C4×C22⋊C4C2314yes—
285C4×C22⋊C4C2312yes—
286C4×C22⋊C4C2312yes—
287C4×C22⋊C4C2314yes—
288C4×C22⋊C4C2314yes—
289#376C2314yes—
290#376C2312yes—
291#376C2312yes—
292#376C2312yes—
293C2×#95C2318yes—
294C43C2312yes—
295C43C2312yes—
296C43C2312yes—
297C43C2312yes—
298C43C2311yescharacteristic
299C43C2311yescharacteristic
300C2×#41—24no—
301C2×#41—216no—
302C2×#41—44no—
303C2×#41—416no—
304C22×Q8C2414yes—
305C22×Q8C2411yescharacteristic
306C22×D4C2414yes—
307C22×D4C2414yes—
308C22×D4C2414yes—
309C22×D4C2412yes—
310C22×D4C2411yescharacteristic
311C22×D4C24116yes—
312C22×D4C2418yes—
313C22×D4C24116yes—
314C23×C4C2412yes—
315C23×C4C2412yes—
316C23×C4C2412yes—
317C23×C4C2411yescharacteristic
318C23×C4C2411yescharacteristic
319C23×C4C2411yescharacteristic
320C23×C4C2418yes—
321C23×C4C2414yes—
322C23×C4C24116yes—
323C4⋊Q8—28no—
324C4⋊Q8—216no—
325#127—28no—
326#127—216no—
327#127—48no—
328#127—48no—
329#126—432no—
330#125—216no—
331#125—48no—
332#125—432no—
333#124—216no—
334#124—416no—
335#123—432no—
336C22⋊Q8—416no—
337#121—216no—
338#121—216no—
339#121—432no—
340#121—416no—
341#121—416no—
342#121—432no—
343#121—416no—
344C4×Q8—24no—
345C4×Q8—24no—
346C4×Q8—416no—
347C4×D4—28no—
348C4×D4—28no—
349C4×D4—24no—
350C4×D4—24no—
351C4×D4—216no—
352C4×D4—216no—
353C4×D4—416no—
354C4×D4—416no—
355C4×D4—432no—
356C4×D4—416no—
357#117—24no—
358#117—28no—
359#117—28no—
360#117—28no—
361#117—48no—
362#117—44no—
363#117—48no—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2411yescharacteristic
368C2×C4⋊C4C2418yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C2418yes—
372C2×C4⋊C4C24116yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2414yes—
375C2×C4⋊C4C2414yes—
376C2×C4⋊C4C24116yes—
377C2×C4⋊C4C2414yes—
378C2×C4⋊C4C2414yes—
379C2×C4⋊C4C2414yes—
380C2×C4⋊C4C24116yes—
381C2×C4⋊C4C24116yes—
382C2×C4⋊C4C24116yes—
383C2×C22⋊C4C2412yes—
384C2×C22⋊C4C2412yes—
385C2×C22⋊C4C2412yes—
386C2×C22⋊C4C2414yes—
387C2×C22⋊C4C2414yes—
388C2×C22⋊C4C2418yes—
389C2×C22⋊C4C2418yes—
390C2×C22⋊C4C2412yes—
391C2×C22⋊C4C2412yes—
392C2×C22⋊C4C2414yes—
393C2×C22⋊C4C2414yes—
394C2×C22⋊C4C2418yes—
395C2×C22⋊C4C24116yes—
396C2×C22⋊C4C24116yes—
397C2×C22⋊C4C24116yes—
398C2×C22⋊C4C24116yes—
399C2×C22⋊C4C24116yes—
400C2×C22⋊C4C24116yes—
401C2×C22⋊C4C24116yes—
402C2×C22⋊C4C24116yes—
403C2×C22⋊C4C24116yes—
404C2×C42—28no—
405C2×C42—28no—
406C2×C42—44no—
407C2×C42—44no—
408C2×C42#4118yes—
409C2×C42C2412yes—
410C2×C42C2412yes—
411C2×C42C2412yes—
412C2×C42C2414yes—
413C2×C42C2414yes—
414C2×C42C2412yes—
415C2×C42C2414yes—
416C2×C42C2412yes—
417C2×C42C2412yes—
418C2×C42C2412yes—
419C2×C42C2412yes—
420C2×C42C2412yes—
421C2×C42C2412yes—
422C2×C42C2418yes—
423C2×C42C24116yes—
424C2×C42C2414yes—
425C2×C42C2414yes—
426C2×C42C2411yescharacteristic
427C2×C42C2411yescharacteristic
428C2×C42C2411yescharacteristic
429C2×C42C2411yescharacteristic
430C2×C42C2412yes—
431#95C2412yes—
432#95C2418yes—
433#95C24116yes—
434#95C2412yes—
435#95C2414yes—
436#95C2414yes—
437#95C2414yes—
438#95C2414yes—
439#95C2414yes—
440#95C2414yes—
441#95C2414yes—
442#95C2414yes—
443#95C2414yes—
444#95C24116yes—
445#95C2418yes—
446#95C24116yes—
447#95C24116yes—
448#95C24116yes—
449#95C24116yes—
450C24C2512yes—
451C24C2518yes—
452C24C2511yescharacteristic
453#41—416no—
454#41—832no—
455C2×Q8—22no—
456C2×Q8—28no—
457C2×Q8—216no—
458C2×Q8—42no—
459C2×D4—24no—
460C2×D4—216no—
461C2×D4—22no—
462C2×D4—28no—
463C2×D4—28no—
464C2×D4—28no—
465C2×D4—216no—
466C2×D4—232no—
467C2×D4—28no—
468C2×D4—216no—
469C2×D4—28no—
470C2×D4—42no—
471C2×D4—44no—
472C2×D4—416no—
473C2×D4—416no—
474C2×D4—416no—
475C2×D4—416no—
476C2×D4—416no—
477C2×D4—416no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—28no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—28no—
488C22×C4—28no—
489C22×C4—216no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—416no—
493C22×C4—416no—
494C22×C4—416no—
495C22×C4—416no—
496C22×C4C2×#4114yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4114yes—
499C22×C4C2512yes—
500C22×C4C2514yes—
501C22×C4C2512yes—
502C22×C4C25116yes—
503C22×C4C2514yes—
504C22×C4C2518yes—
505C22×C4C2514yes—
506C22×C4C2512yes—
507C22×C4C2511yescharacteristic
508C22×C4C2511yescharacteristic
509C22×C4C2511yescharacteristic
510C22×C4C2514yes—
511C22×C4C2511yescharacteristic
512C22×C4C2511yescharacteristic
513C22×C4C2511yescharacteristic
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—216no—
523C4⋊C4—232no—
524C4⋊C4—232no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—416no—
531C4⋊C4—416no—
532C4⋊C4—416no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—216no—
537C22⋊C4—216no—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—48no—
545C22⋊C4—416no—
546C22⋊C4—416no—
547C22⋊C4—416no—
548C22⋊C4—416no—
549C22⋊C4—416no—
550C22⋊C4—416no—
551C22⋊C4—416no—
552C22⋊C4—416no—
553C22⋊C4—416no—
554C42—24no—
555C42—24no—
556C42—24no—
557C42—28no—
558C42—24no—
559C42—24no—
560C42—24no—
561C42—24no—
562C42—24no—
563C42—24no—
564C42—24no—
565C42—28no—
566C42—28no—
567C42—44no—
568C42—44no—
569C42—48no—
570C42—416no—
571C42C2×#4114yes—
572C42C2×#4114yes—
573C42C2×#4114yes—
574C42C2×#4114yes—
575C42H(5,𝔽2)18yes—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—28no—
581C23—22no—
582C23—22no—
583C23—216no—
584C23—24no—
585C23—28no—
586C23—216no—
587C23—28no—
588C23—24no—
589C23C22×#4112yes—
590C23C2611yescharacteristic
591Q8—44no—
592Q8—416no—
593D4—44no—
594D4—48no—
595D4—416no—
596D4—416no—
597D4—416no—
598D4—832no—
599D4—816no—
600D4—1616no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—216no—
617C2×C4—216no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—216no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—216no—
630C2×C4—416no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—816no—
635C2×C4—1616no—
636C2×C4C22×#4112yes—
637C2×C4C22×#4112yes—
638C2×C4C22×#4112yes—
639C2×C4C22×#4112yes—
640C2×C4C22×#4112yes—
641C2×C4C22×#4112yes—
642C2×C4C2×H(5,𝔽2)18yes—
643C2×C4C2×H(5,𝔽2)14yes—
644C2×C4#58514yes—
645C2×C4#58514yes—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—44no—
650C22—42no—
651C22—48no—
652C22—42no—
653C22—48no—
654C22—416no—
655C22—416no—
656C22—44no—
657C22—44no—
658C22—42no—
659C22—48no—
660C22C23×#4111yescharacteristic
661C22C22×H(5,𝔽2)12yes—
662C22C2×#58512yes—
663C22C2×#58512yes—
664C4—22no—
665C4—22no—
666C4—28no—
667C4—22no—
668C4—42no—
669C4—48no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—416no—
675C4—42no—
676C4#345514yes—
677C4#350814yes—
678C4#351214yes—
679C2—42no—
680C2—82no—
681C2—88no—
682C2C2×#345511yescharacteristic
683C2C2×#350811yescharacteristic
684C2C2×#351211yescharacteristic
685C2#6283614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #39766C2semidirect
2. #39382C2semidirect
3. #39185C2semidirect
4. #38969C2semidirect
5. #38129C2semidirect
6. #37701C2semidirect
7. #37700C2semidirect
8. #37318C2semidirect
9. #37317C2semidirect
10. #36644C2semidirect
11. #36640C2semidirect
12. #35559C2semidirect
13. #35319C2semidirect
14. #35292C2semidirect
15. #35288C2semidirect
16. #35278C2semidirect
17. #34226C2semidirect
18. #34026C2semidirect

Maximal subgroups

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

subgroupindex
#397662
#393822
#391852
#389692
#381292
#377012
#377002
#373182
#373172
#366442
#366402
#355592
#353192
#352922
#352882
#352782
#342262
#340262