Finite Group Databasebeta

← #8093541

Subgroups — #8093541

Order 512.

5418 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#53137C216yesmaximal, complemented
2#53098C216yesmaximal, complemented
3#53088C216yesmaximal, complemented
4#53085C212yesmaximal, complemented
5#53078C212yesmaximal, complemented
6#53041C216yesmaximal, complemented
7#53004C212yesmaximal, complemented
8#52993C212yesmaximal, complemented
9#52765C214yesmaximal, complemented
10#52529C2112yesmaximal, complemented
11#35660C213yesmaximal, complemented
12#35621C213yesmaximal, complemented
13#35564C213yesmaximal, complemented
14#35517C213yesmaximal, complemented
15#34641C211yesmaximal, characteristic, complemented
16#34639C211yesmaximal, characteristic, complemented
17C2×#2645C211yesmaximal, characteristic, complemented
18C22×2−1+4C2211yescharacteristic
19C2×#537C2213yes—
20C2×#536C2213yes—
21C2×#532C2213yes—
22C2×#532C2213yes—
23C2×#520C2213yes—
24#2841C22112yes—
25#2838C22112yes—
26#2838C22112yes—
27#2837C22112yes—
28#2834C2214yes—
29#2828C22112yes—
30#2827C22112yes—
31#2826C22112yes—
32#2825C2214yes—
33#2825C22112yes—
34#2824C22112yes—
35#2813C22112yes—
36#2807C22112yes—
37#2801C2214yes—
38#2796C22112yes—
39#2780C22112yes—
40#2778C22112yes—
41#2778C22112yes—
42#2760C2214yes—
43#2746C22112yes—
44#2742C22112yes—
45#2703C22112yes—
46#2693C22112yes—
47#2688C22112yes—
48#2658C2213yes—
49#2658C2211yescharacteristic
50#2657C2211yescharacteristic
51#2657C2213yes—
52#2656C2212yes—
53#2656C2216yes—
54#2645C2211yescharacteristic
55#2645C2212yes—
56#2645C2212yes—
57#2645C2216yes—
58#2645C2216yes—
59#2645C2213yes—
60#2644C2211yescharacteristic
61#2644C2213yes—
62#2643C2212yes—
63#2643C2216yes—
64#2547C2216yes—
65#2546C2216yes—
66#2545C2216yes—
67#2544C2216yes—
68#2543C22112yes—
69#2487C2216yes—
70#2486C2216yes—
71#2485C2216yes—
72#2485C2216yes—
73#2448C2216yes—
74#2447C2216yes—
75#2446C2216yes—
76#2445C2216yes—
77#2444C22112yes—
78#2443C22112yes—
79#2436C2216yes—
80#2434C2216yes—
81#2432C2216yes—
82#2431C2216yes—
83#2430C22112yes—
84#2430C22112yes—
85#2429C2216yes—
86#2428C2216yes—
87#2427C2216yes—
88#2426C2216yes—
89#2425C22112yes—
90#2425C22112yes—
91#2417C2216yes—
92#2417C2216yes—
93#2417C2216yes—
94#2416C22112yes—
95#2416C2216yes—
96#2415C2216yes—
97#2415C2216yes—
98#2415C2216yes—
99#2414C2216yes—
100#2409C2216yes—
101#2409C2216yes—
102#2407C2216yes—
103#2407C2216yes—
104#2401C2212yes—
105#2401C2216yes—
106C2×#401C2212yes—
107#2397C2212yes—
108#2397C2216yes—
109C2×#400C2216yes—
110#2311C2213yes—
111#2311C2213yes—
112#2297C2213yes—
113#2296C2213yes—
114C2×#391C2213yes—
115C2×#389C2213yes—
116C2×#376C2211yescharacteristic
117C2×2−1+4—412no—
118C22×#41C2311yescharacteristic
119C22×#41C2313yes—
120C22×#41C2311yescharacteristic
121C22×#41C2312yes—
122C22×#41C2314yes—
123C22×#41C2312yes—
124C23×Q8C2311yescharacteristic
125C23×Q8C2312yes—
126C23×Q8C2312yes—
127#537—412no—
128#536—412no—
129#532—412no—
130#532—412no—
131C2×C4⋊Q8C2313yes—
132C2×C4⋊Q8C2313yes—
133C2×C4⋊Q8C2316yes—
134C2×#127C2313yes—
135C2×#127C2313yes—
136C2×#124C2313yes—
137C2×#124C2313yes—
138C2×#124C2313yes—
139C2×#124C2313yes—
140C2×#124C2316yes—
141C2×#123C23112yes—
142C2×#123C23112yes—
143C2×C22⋊Q8C2316yes—
144C2×C22⋊Q8C2316yes—
145C2×C22⋊Q8C2316yes—
146C2×C22⋊Q8C2316yes—
147C2×#121C2316yes—
148C2×#121C2316yes—
149C2×#121C2316yes—
150C2×#121C2316yes—
151C2×#121C2316yes—
152C2×#121C23112yes—
153C2×#121C2316yes—
154C2×#121C23112yes—
155#520—412no—
156C2×C4×Q8C2316yes—
157C2×C4×D4C2316yes—
158C2×C4×D4C2316yes—
159C2×C4×D4C2316yes—
160C2×C4×D4C2316yes—
161C2×C4×D4C23112yes—
162C2×C4×D4C2316yes—
163C2×#117C2313yes—
164C2×#117C2316yes—
165C22×C4⋊C4C2313yes—
166C22×C4⋊C4C2316yes—
167C22×C4⋊C4C2316yes—
168C22×C42C2313yes—
169#401C2314yes—
170#401C23112yes—
171#401C2312yes—
172#401C2316yes—
173#400C2316yes—
174#400C23112yes—
175#400C2316yes—
176#400C23112yes—
177#400C23112yes—
178#400C23112yes—
179#400C2316yes—
180#400C23112yes—
181#400C2316yes—
182#399C23112yes—
183#399C2316yes—
184#399C2312yes—
185#399C2316yes—
186#399C2316yes—
187#399C2316yes—
188#399C2316yes—
189#398C23112yes—
190#398C23112yes—
191#398C23112yes—
192#398C23112yes—
193#397C23112yes—
194#397C23112yes—
195#397C2316yes—
196#397C23112yes—
197#397C23112yes—
198#397C23112yes—
199#397C23112yes—
200#397C2316yes—
201#397C2316yes—
202#397C2316yes—
203#396C2312yes—
204#396C23112yes—
205#396C23112yes—
206#396C23112yes—
207#396C23112yes—
208#396C2316yes—
209#395C23112yes—
210#395C23112yes—
211#394C23112yes—
212#394C23112yes—
213#394C23112yes—
214#394C23112yes—
215#394C23112yes—
216#394C23112yes—
217#394C23112yes—
218#394C23112yes—
219#394C23112yes—
220#393C23112yes—
221#393C23112yes—
222#392C2316yes—
223#392C23112yes—
224#392C2312yes—
225#392C2316yes—
226#392C23112yes—
227#392C2316yes—
228#392C2316yes—
229#392C23112yes—
230#392C2316yes—
231#391C2316yes—
232#391C2316yes—
233#391C2316yes—
234#391C2316yes—
235#391C2313yes—
236#391C2313yes—
237#391C2316yes—
238#391C2316yes—
239#390C2316yes—
240#390C2316yes—
241#390C23112yes—
242#390C2313yes—
243#390C2313yes—
244#389C23112yes—
245#389C23112yes—
246#389C23112yes—
247#389C2313yes—
248#389C2313yes—
249#389C2313yes—
250#389C2313yes—
251#389C2316yes—
252#389C2316yes—
253#389C2316yes—
254#389C2316yes—
255#388C23112yes—
256#388C23112yes—
257#388C23112yes—
258#388C23112yes—
259#388C23112yes—
260#388C23112yes—
261#387C23112yes—
262#387C23112yes—
263#386C23112yes—
264#386C23112yes—
265#386C23112yes—
266#386C23112yes—
267#386C23112yes—
268#386C23112yes—
269#386C23112yes—
270#386C23112yes—
271#386C23112yes—
272#386C23112yes—
273#386C23112yes—
274#386C2316yes—
275#386C2316yes—
276#385C23112yes—
277#385C23112yes—
278#385C23112yes—
279#385C23112yes—
280#385C23112yes—
281#385C2316yes—
282#385C2316yes—
283#381C23112yes—
284#380C2316yes—
285#380C2316yes—
286#380C2316yes—
287#380C2316yes—
288#380C23112yes—
289#380C23112yes—
290#380C23112yes—
291C4×C4⋊C4C2316yes—
292C4×C4⋊C4C2316yes—
293C4×C22⋊C4C2316yes—
294C4×C22⋊C4C2316yes—
295#376C2313yes—
296#376C2311yescharacteristic
297#376C2313yes—
298#376C2311yescharacteristic
299C2×#95C2313yes—
300C2×#95C2311yescharacteristic
301C2×#95C2316yes—
302C2×#95C2316yes—
303C2×#95C2316yes—
3042−1+4—464no—
305C2×#41—212no—
306C2×#41—212no—
307C2×#41—212no—
308C2×#41—212no—
309C2×#41—212no—
310C2×#41—424no—
311C2×#41—412no—
312C2×#41—412no—
313C22×Q8—212no—
314C22×Q8—412no—
315C22×Q8—412no—
316C22×Q8C2413yes—
317C22×Q8C2412yes—
318C22×Q8C2411yescharacteristic
319C22×Q8C2412yes—
320C22×Q8C2411yescharacteristic
321C22×Q8C2414yes—
322C22×D4C2416yes—
323C22×D4C2413yes—
324C22×D4C2412yes—
325C22×D4C2412yes—
326C22×D4C2411yescharacteristic
327C22×D4C2411yescharacteristic
328C22×D4C2416yes—
329C22×D4C2416yes—
330C22×D4C2416yes—
331C22×D4C2416yes—
332C22×D4C24112yes—
333C22×D4C2414yes—
334C23×C4C2416yes—
335C23×C4C2412yes—
336C23×C4C2412yes—
337C23×C4C2412yes—
338C23×C4C2416yes—
339C23×C4C2413yes—
340C23×C4C2413yes—
341C23×C4C24112yes—
342C23×C4C2414yes—
343C23×C4C2414yes—
344C23×C4C2411yescharacteristic
345C4⋊Q8—46no—
346C4⋊Q8—46no—
347C4⋊Q8—412no—
348#127—46no—
349#127—46no—
350#124—46no—
351#124—46no—
352#124—46no—
353#124—46no—
354#124—412no—
355#123—424no—
356#123—812no—
357C22⋊Q8—412no—
358C22⋊Q8—412no—
359C22⋊Q8—412no—
360C22⋊Q8—412no—
361#121—412no—
362#121—412no—
363#121—412no—
364#121—412no—
365#121—412no—
366#121—412no—
367#121—424no—
368#121—812no—
369C4×Q8—412no—
370C4×D4—412no—
371C4×D4—412no—
372C4×D4—412no—
373C4×D4—412no—
374C4×D4—412no—
375C4×D4—424no—
376#117—212no—
377#117—412no—
378C2×C4⋊C4—212no—
379C2×C4⋊C4—412no—
380C2×C4⋊C4—86no—
381C2×C4⋊C4C2416yes—
382C2×C4⋊C4C2413yes—
383C2×C4⋊C4C2416yes—
384C2×C4⋊C4C2413yes—
385C2×C4⋊C4C24112yes—
386C2×C4⋊C4C24112yes—
387C2×C4⋊C4C24112yes—
388C2×C4⋊C4C2416yes—
389C2×C4⋊C4C2416yes—
390C2×C4⋊C4C2416yes—
391C2×C4⋊C4C2416yes—
392C2×C4⋊C4C24112yes—
393C2×C4⋊C4C24112yes—
394C2×C4⋊C4C24112yes—
395C2×C4⋊C4C2416yes—
396C2×C4⋊C4C2416yes—
397C2×C4⋊C4C2416yes—
398C2×C4⋊C4C2416yes—
399C2×C22⋊C4C2416yes—
400C2×C22⋊C4C24112yes—
401C2×C22⋊C4C2416yes—
402C2×C22⋊C4C2416yes—
403C2×C22⋊C4C2416yes—
404C2×C22⋊C4C2416yes—
405C2×C22⋊C4C2416yes—
406C2×C22⋊C4C2416yes—
407C2×C22⋊C4C2416yes—
408C2×C22⋊C4C2416yes—
409C2×C22⋊C4C24112yes—
410C2×C22⋊C4C24112yes—
411C2×C22⋊C4C2416yes—
412C2×C22⋊C4C24112yes—
413C2×C22⋊C4C24112yes—
414C2×C22⋊C4C24112yes—
415C2×C22⋊C4C24112yes—
416C2×C22⋊C4C2416yes—
417C2×C22⋊C4C2416yes—
418C2×C22⋊C4C2416yes—
419C2×C22⋊C4C24112yes—
420C2×C22⋊C4C24112yes—
421C2×C42—212no—
422C2×C42C2416yes—
423C2×C42C2413yes—
424C2×C42C2413yes—
425C2×C42C2413yes—
426C2×C42C2413yes—
427C2×C42C24112yes—
428#95C2413yes—
429#95C2411yescharacteristic
430#95C2413yes—
431#95C2413yes—
432#95C2413yes—
433#95C2413yes—
434#95C24112yes—
435#95C24112yes—
436#95C24112yes—
437#95C24112yes—
438#95C24112yes—
439#95C24112yes—
440#95C2416yes—
441#95C2416yes—
442#95C2416yes—
443#95C2416yes—
444#95C2416yes—
445#95C2416yes—
446#95C2416yes—
447#95C2416yes—
448#95C24112yes—
449#95C24112yes—
450#95C24112yes—
451#95C24112yes—
452C24C2516yes—
453C24C2512yes—
454C24C2512yes—
455C24C2511yescharacteristic
456#41—464no—
457#41—432no—
458#41—432no—
459#41—448no—
460#41—416no—
461#41—416no—
462C2×Q8—26no—
463C2×Q8—26no—
464C2×Q8—212no—
465C2×Q8—26no—
466C2×Q8—26no—
467C2×Q8—212no—
468C2×Q8—26no—
469C2×Q8—412no—
470C2×Q8—432no—
471C2×Q8—432no—
472C2×Q8—416no—
473C2×D4—212no—
474C2×D4—212no—
475C2×D4—212no—
476C2×D4—212no—
477C2×D4—212no—
478C2×D4—26no—
479C2×D4—26no—
480C2×D4—26no—
481C2×D4—26no—
482C2×D4—26no—
483C2×D4—412no—
484C2×D4—46no—
485C2×D4—46no—
486C2×D4—46no—
487C2×D4—46no—
488C2×D4—46no—
489C2×D4—46no—
490C2×D4—46no—
491C2×D4—46no—
492C2×D4—46no—
493C2×D4—46no—
494C2×D4—46no—
495C2×D4—46no—
496C2×D4—412no—
497C2×D4—412no—
498C2×D4—412no—
499C22×C4—212no—
500C22×C4—26no—
501C22×C4—26no—
502C22×C4—212no—
503C22×C4—212no—
504C22×C4—212no—
505C22×C4—212no—
506C22×C4—212no—
507C22×C4—212no—
508C22×C4—412no—
509C22×C4—412no—
510C22×C4—46no—
511C22×C4—43no—
512C22×C4—46no—
513C22×C4—43no—
514C22×C4—43no—
515C22×C4—43no—
516C22×C4—412no—
517C22×C4—412no—
518C22×C4—412no—
519C22×C4H(5,𝔽2)112yes—
520C22×C4C2513yes—
521C22×C4C2513yes—
522C22×C4C2513yes—
523C22×C4C2513yes—
524C22×C4C25112yes—
525C22×C4C2512yes—
526C22×C4C2512yes—
527C22×C4C2514yes—
528C22×C4C2516yes—
529C22×C4C2516yes—
530C22×C4C2516yes—
531C22×C4C2511yescharacteristic
532C22×C4C2511yescharacteristic
533C4⋊C4—212no—
534C4⋊C4—212no—
535C4⋊C4—26no—
536C4⋊C4—26no—
537C4⋊C4—412no—
538C4⋊C4—412no—
539C4⋊C4—412no—
540C4⋊C4—46no—
541C4⋊C4—46no—
542C4⋊C4—46no—
543C4⋊C4—46no—
544C4⋊C4—46no—
545C4⋊C4—46no—
546C4⋊C4—46no—
547C4⋊C4—412no—
548C4⋊C4—412no—
549C4⋊C4—412no—
550C4⋊C4—46no—
551C22⋊C4—212no—
552C22⋊C4—412no—
553C22⋊C4—412no—
554C22⋊C4—46no—
555C22⋊C4—46no—
556C22⋊C4—46no—
557C22⋊C4—412no—
558C22⋊C4—46no—
559C22⋊C4—46no—
560C22⋊C4—46no—
561C22⋊C4—46no—
562C22⋊C4—46no—
563C22⋊C4—46no—
564C22⋊C4—46no—
565C22⋊C4—46no—
566C22⋊C4—412no—
567C22⋊C4—412no—
568C22⋊C4—46no—
569C22⋊C4—412no—
570C22⋊C4—412no—
571C22⋊C4—412no—
572C22⋊C4—412no—
573C42—26no—
574C42—26no—
575C42—26no—
576C42—26no—
577C42—212no—
578C42—412no—
579C23—21no—
580C23—23no—
581C23—26no—
582C23—26no—
583C23—26no—
584C23—26no—
585C23—26no—
586C23—22no—
587C23—22no—
588C23—26no—
589C23—26no—
590C23—26no—
591C23—26no—
592C23—26no—
593C23—26no—
594C23C2×H(5,𝔽2)16yes—
595C23C2611yescharacteristic
596Q8—416no—
597Q8—48no—
598Q8—44no—
599Q8—48no—
600Q8—44no—
601Q8—412no—
602D4—416no—
603D4—424no—
604D4—48no—
605D4—48no—
606D4—412no—
607D4—44no—
608D4—44no—
609D4—812no—
610D4—812no—
611D4—812no—
612D4—812no—
613D4—1612no—
614C2×C4—23no—
615C2×C4—23no—
616C2×C4—23no—
617C2×C4—23no—
618C2×C4—26no—
619C2×C4—26no—
620C2×C4—26no—
621C2×C4—212no—
622C2×C4—26no—
623C2×C4—26no—
624C2×C4—26no—
625C2×C4—26no—
626C2×C4—212no—
627C2×C4—26no—
628C2×C4—212no—
629C2×C4—23no—
630C2×C4—23no—
631C2×C4—26no—
632C2×C4—26no—
633C2×C4—212no—
634C2×C4—23no—
635C2×C4—23no—
636C2×C4—23no—
637C2×C4—26no—
638C2×C4—23no—
639C2×C4—23no—
640C2×C4—23no—
641C2×C4—416no—
642C2×C4—416no—
643C2×C4—48no—
644C2×C4—48no—
645C2×C4—48no—
646C2×C4—424no—
647C2×C4—812no—
648C2×C4—86no—
649C2×C4—86no—
650C2×C4—1612no—
651C2×C4#560116yes—
652C2×C4C2×H(5,𝔽2)16yes—
653C2×C4C2×H(5,𝔽2)16yes—
654C22—26no—
655C22—26no—
656C22—46no—
657C22—46no—
658C22—46no—
659C22—46no—
660C22—46no—
661C22—46no—
662C22—46no—
663C22—46no—
664C22—46no—
665C22—46no—
666C22—42no—
667C22—42no—
668C22—46no—
669C22C2×#56014yes—
670C22C22×H(5,𝔽2)13yes—
671C22H(7,𝔽2)11yescharacteristic
672C222−1+613yes—
673C4—412no—
674C4—43no—
675C4—43no—
676C4—46no—
677C4—46no—
678C4—43no—
679C4—43no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—46no—
684C4C2×#56014yes—
685C4C2×#56014yes—
686C2—24no—
687C2—86no—
688C2—82no—
689C2—82no—
690C2C22×#56011yescharacteristic
691C2#6287813yes—
692C2#6288813yes—
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. #53137C2semidirect
2. #53098C2semidirect
3. #53088C2semidirect
4. #53085C2semidirect
5. #53078C2semidirect
6. #53041C2semidirect
7. #53004C2semidirect
8. #52993C2semidirect
9. #52765C2semidirect
10. #52529C2semidirect
11. #35660C2semidirect
12. #35621C2semidirect
13. #35564C2semidirect
14. #35517C2semidirect
15. #34641C2semidirect
16. #34639C2semidirect
17. C2×#2645C2semidirect

Maximal subgroups

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

subgroupindex
#531372
#530982
#530882
#530852
#530782
#530412
#530042
#529932
#527652
#525292
#356602
#356212
#355642
#355172
#346412
#346392
C2×#26452