Finite Group Databasebeta

← #6800390

Subgroups — #6800390

Order 512.

1281 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39606C211yesmaximal, characteristic
2#35181C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32341C212yesmaximal, complemented
5#32316C212yesmaximal, complemented
6#31474C212yesmaximal, complemented
7#31467C212yesmaximal, complemented
8#29979C212yesmaximal, complemented
9#28807C212yesmaximal, complemented
10#27970C212yesmaximal, complemented
11#27968C211yesmaximal, characteristic, complemented
12#27744C212yesmaximal, complemented
13#27655C212yesmaximal
14#27650C211yesmaximal, characteristic
15#26518C212yesmaximal, complemented
16#26119C212yesmaximal, complemented
17#22628C211yesmaximal, characteristic, complemented
18#22088C212yesmaximal
19#21578C211yesmaximal, characteristic
20C2×#482C2212yes—
21C2×#480C2211yescharacteristic
22#2857C2211yescharacteristic
23#2851—22no—
24#2768—22no—
25#2748—22no—
26#2646—21no—
27#2646—21no—
28#2629C2212yes—
29#2571—21no—
30#2571C2212yes—
31#2570—22no—
32#2570C2212yes—
33#2570C2212yes—
34#2552—21no—
35#2552C2212yes—
36#2552C2211yescharacteristic
37#2542C2211yescharacteristic
38#2538—21no—
39#2507C2212yes—
40#2502—21no—
41#2454C2212yes—
42#2423—21no—
43#2417—21no—
44#2365C2211yescharacteristic
45D4×C4⋊C4C2211yescharacteristic
46#2324C2212yes—
47#2317C2212yes—
48#2317C2212yes—
49C4×#121C2211yescharacteristic
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2212yes—
54#2101C2212yes—
55#2100C2212yes—
56#2100C2212yes—
57#2100C2212yes—
58#2099C2212yes—
59#2099C2212yes—
60#2099C2212yes—
61#2088C2212yes—
62#2088C2212yes—
63#2087C2212yes—
64#2086C2212yes—
65#2085C2212yes—
66#2084C2211yescharacteristic
67#2083C2211yescharacteristic
68#2083C2212yes—
69#2073C2212yes—
70#2071C2211yescharacteristic
71#2071C2211yescharacteristic
72#2071C2212yes—
73#2059C2212yes—
74#2059C2212yes—
75#1993C2212yes—
76#1992C2212yes—
77#1991C2212yes—
78#1987C2212yes—
79#1949C2212yes—
80#1949C2212yes—
81#1947C2212yes—
82#1947C2212yes—
83#1944C2212yes—
84#1942C2212yes—
85#1940C2212yes—
86#1936C2212yes—
87#1934C2212yes—
88#1926C2212yes—
89#1921C2212yes—
90#1921C2212yes—
91#1921C2212yes—
92#1921C2212yes—
93#1920C2212yes—
94#1917C2212yes—
95#1916C2212yes—
96#1874C2211yescharacteristic
97#1862C2212yes—
98#1862C2212yes—
99#1853C2211yescharacteristic
100#1852C2212yes—
101#1836C2212yes—
102#1836C2212yes—
103#1834C2212yes—
104#1825C2212yes—
105#1825C2212yes—
106#1825C2212yes—
107#1824C2212yes—
108#1823C2211yescharacteristic
109#1804C2212yes—
110#1804C2212yes—
111#1802C2212yes—
112#1798C2212yes—
113#1797C2211yescharacteristic
114#1776C2212yes—
115#1771C2212yes—
116#1762C2212yes—
117#1750C2212yes—
118#1749C2211yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×#125—22no—
121C2×#125C2312yes—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×#121C2311yescharacteristic
126C2×C4×Q8—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×#117C2312yes—
131C2×#117C2312yes—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#482—28no—
138#480—24no—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#400—22no—
160#400—22no—
161#400—22no—
162#399—22no—
163#399—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#398—22no—
172#398—22no—
173#396—22no—
174#396—22no—
175#396—22no—
176#391—21no—
177#391—21no—
178#390—22no—
179#390C2312yes—
180#390C2311yescharacteristic
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2311yescharacteristic
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#388—22no—
198#388—21no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#386—21no—
206#386—22no—
207#385—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#385—22no—
213#385—22no—
214#384—22no—
215#384—21no—
216#384D411yescharacteristic
217#384D411yescharacteristic
218#384C2312yes—
219#384C2312yes—
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2312yes—
224#382—22no—
225#382—22no—
226#382D412yes—
227#382C2312yes—
228#382C2312yes—
229#382C2312yes—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—22no—
235#380D412yes—
236#380D411yescharacteristic
237#380D411yescharacteristic
238#380C2312yes—
239#380C2311yescharacteristic
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2311yescharacteristic
250C4×C4⋊C4C2312yes—
251C4×C22⋊C4—21no—
252C2×#95—21no—
253#340C2312yes—
254#340C2312yes—
255#340C2312yes—
256#340C2312yes—
257#340C2312yes—
258#340C2312yes—
259#340C2312yes—
260#340C2312yes—
261#340C2312yes—
262#340C2312yes—
263#340C2312yes—
264#340C2312yes—
265#340C2312yes—
266#340C2312yes—
267#340C2312yes—
268#340C2312yes—
269#340C2312yes—
270#340C2312yes—
271#340C2312yes—
272#340C2312yes—
273#340C2312yes—
274#340C2312yes—
275#336C2311yescharacteristic
276#336C2311yescharacteristic
277#336C2312yes—
278#336C2312yes—
279#336C2312yes—
280#336C2312yes—
281#336C2312yes—
282C22×Q8—21no—
283C22×D4—21no—
284C22×D4C2411yescharacteristic
285C23×C4—21no—
286C23×C4—21no—
287C23×C4—21no—
288C23×C4—22no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2411yescharacteristic
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C4⋊Q8—24no—
297#125—42no—
298#125—42no—
299#125—84no—
300C22⋊Q8—82no—
301C22⋊Q8—82no—
302C22⋊Q8—82no—
303#121—22no—
304#121#4114yes—
305C4×Q8—82no—
306C4×D4—24no—
307C4×D4—42no—
308C4×D4—42no—
309C4×D4—82no—
310#117—24no—
311#117—24no—
312#117—42no—
313#117—42no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—42no—
343C2×C4⋊C4—82no—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—24no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—82no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—24no—
387#107—24no—
388#107—42no—
389#107—42no—
390#106—24no—
391#106—42no—
392C4⋊C8—24no—
393C4⋊C8—42no—
394C4⋊C8—42no—
395#102—24no—
396#102—24no—
397#102—24no—
398#102—42no—
399C22⋊C8—22no—
400C22⋊C8—24no—
401C22⋊C8—22no—
402#97—24no—
403#97—42no—
404C4×C8—42no—
405#95—22no—
406#95—22no—
407#95—21no—
408#95—21no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—21no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95C2×D412yes—
427#95C2×D412yes—
428#95C2×D412yes—
429#95C2×D412yes—
430C24—21no—
431C24C22×D411yescharacteristic
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—22no—
439C2×D4—81no—
440C2×D4—81no—
441C2×D4C2×#4112yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4112yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—41no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—81no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—81no—
531C4⋊C4—82no—
532C4⋊C4—81no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4C2×#4112yes—
544C22⋊C4—24no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—82no—
551C22⋊C4—81no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—81no—
555C22⋊C4—82no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—44no—
564C42—42no—
565C42—44no—
566C42—82no—
567C42—82no—
568C422−1+414yes—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23#54812yes—
583C23C23×D411yescharacteristic
584Q8—84no—
585D4—24no—
586D4—42no—
587D4—84no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—82no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—164no—
646C2×C4#53612yes—
647C2×C4#53612yes—
648C2×C4#54812yes—
649C2×C4#57812yes—
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×2−1+412yes—
652C2×C4C2×2−1+411yescharacteristic
653C2×C4#58512yes—
654C8—42no—
655C8—42no—
656C8—42no—
657C8—42no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22C2×#53611yescharacteristic
673C22C2×#54811yescharacteristic
674C22#348511yescharacteristic
675C22C2×#57811yescharacteristic
676C22#358511yescharacteristic
677C22#358511yescharacteristic
678C22#358511yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—41no—
689C4—82no—
690C4—82no—
691C4—81no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—81no—
696C4—82no—
697C4#351512yes—
698C4#356312yes—
699C2—42no—
700C2—82no—
701C2#3912211yescharacteristic
702C2#6086811yescharacteristic
703C2#6087411yescharacteristic
704C2#6133211yescharacteristic
705C2#6134811yescharacteristic
706C2#6179511yescharacteristic
707C2#6181011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35181C2semidirect
3. #33149C2semidirect
4. #32341C2semidirect
5. #32316C2semidirect
6. #31474C2semidirect
7. #31467C2semidirect
8. #29979C2semidirect
9. #28807C2semidirect
10. #27970C2semidirect
11. #27968C2semidirect
12. #27744C2semidirect
15. #26518C2semidirect
16. #26119C2semidirect
17. #22628C2semidirect

Maximal subgroups

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

subgroupindex
#396062
#351812
#331492
#323412
#323162
#314742
#314672
#299792
#288072
#279702
#279682
#277442
#276552
#276502
#265182
#261192
#226282
#220882
#215782