Finite Group Databasebeta

← #6986983

Subgroups — #6986983

Order 512.

1747 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36493C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic, complemented
3#32772C212yesmaximal, complemented
4#32032C214yesmaximal, complemented
5#32002C214yesmaximal, complemented
6#30188C212yesmaximal, complemented
7#30173C212yesmaximal, complemented
8#29921C211yesmaximal, characteristic, complemented
9#29156C212yesmaximal, complemented
10#28597C212yesmaximal, complemented
11#26283C212yesmaximal, complemented
12#25903C211yesmaximal, characteristic, complemented
13#25652C212yesmaximal, complemented
14#25341C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic, complemented
16#23757C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#536—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#486C2212yes—
23C2×#475C2212yes—
24C2×C8×D4C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2856C2211yescharacteristic
28#2850C2212yes—
29#2621—22no—
30#2547—21no—
31#2547C2212yes—
32#2546—21no—
33#2546C2212yes—
34#2541C2211yescharacteristic
35#2540C2212yes—
36#2538C2211yescharacteristic
37#2535C2212yes—
38#2486—22no—
39#2486C2214yes—
40#2471—22no—
41#2456—22no—
42#2448—21no—
43#2431—21no—
44#2366—21no—
45#2363—21no—
46#2355C2211yescharacteristic
47#2310—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C4×#121C2212yes—
50#2285C2211yescharacteristic
51#2101C2214yes—
52#2100C2211yescharacteristic
53#2095C2211yescharacteristic
54#2094C2212yes—
55#2085C2214yes—
56#2083C2211yescharacteristic
57#2077C2214yes—
58#2076C2212yes—
59#2072C2211yescharacteristic
60#2071C2212yes—
61#2058C2212yes—
62#2042C2214yes—
63#2042C2214yes—
64#2036C2212yes—
65#2035C2214yes—
66#2034C2214yes—
67#2033C2214yes—
68#2033C2214yes—
69#2024C2212yes—
70#2023C2212yes—
71#2023C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1984C2212yes—
75#1982C2212yes—
76#1948C2212yes—
77#1921C2214yes—
78#1920C2214yes—
79#1893C2214yes—
80#1893C2214yes—
81#1892C2214yes—
82#1874C2214yes—
83#1872C2212yes—
84#1869C2214yes—
85#1864C2214yes—
86#1864C2214yes—
87#1853C2212yes—
88#1817C2212yes—
89#1807C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×C8C2314yes—
98#559—44no—
99#536—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#127—21no—
107C2×#127C2311yescharacteristic
108C2×#124—21no—
109C2×#124—22no—
110C2×#124C2312yes—
111C2×#123—24no—
112C2×#123—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2311yescharacteristic
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121—21no—
119C2×#121C2312yes—
120C2×#121C2312yes—
121C2×#121C2312yes—
122C4×#41—44no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D412yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117D411yescharacteristic
135C2×#117D412yes—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140#486—44no—
141#475—44no—
142C8×D4—44no—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C4×C8C2312yes—
159#399—22no—
160#398—24no—
161#398—24no—
162#397—22no—
163#397—24no—
164#397—24no—
165#396—22no—
166#392—22no—
167#392—22no—
168#392—22no—
169#391—21no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391C2311yescharacteristic
175#390—21no—
176#390—21no—
177#390—22no—
178#390—22no—
179#390—21no—
180#390—21no—
181#390C2312yes—
182#389—21no—
183#389—22no—
184#389—22no—
185#389C2311yescharacteristic
186#389C2314yes—
187#389C2314yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—21no—
193#386—22no—
194#386—22no—
195#386—22no—
196#386—22no—
197#386—22no—
198#385—21no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382—21no—
203#382C2312yes—
204#381—24no—
205#380—22no—
206#380—22no—
207#380—21no—
208#380C2312yes—
209#380C2314yes—
210#380C2311yescharacteristic
211#380C2312yes—
212#380C2314yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—42no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2411yescharacteristic
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C23×C4C2412yes—
261C22×C8—44no—
262C22×C8—44no—
263C22×C8—44no—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—48no—
273#127—24no—
274#127—44no—
275#124—44no—
276#124—44no—
277#124—48no—
278#123—48no—
279#123—416no—
280C22⋊Q8—22no—
281C22⋊Q8—41no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C22⋊Q8—44no—
285#121—24no—
286#121—42no—
287#121—44no—
288#121—44no—
289#121—44no—
290#121—82no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—42no—
294C4×D4—24no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—84no—
301#117—44no—
302#117—42no—
303#117—44no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—24no—
364#107—42no—
365#106—44no—
366#106—44no—
367C4⋊C8—24no—
368#103—42no—
369#103—42no—
370#103—44no—
371#102—44no—
372#102—44no—
373#102—44no—
374#102—44no—
375C22⋊C8—42no—
376C22⋊C8—42no—
377C22⋊C8—44no—
378C4×C8—24no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—22no—
394C24—22no—
395C24C22×D411yescharacteristic
396C24C22×D412yes—
397#41—416no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—816no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—81no—
430C2×D4—81no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4AGL(1,ℤ/8)14yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—42no—
461C22×C4—41no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4#13514yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2511yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—88no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C42−1+412yes—
520C22⋊C4—24no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—81no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—42no—
547C42—82no—
548C42H(5,𝔽2)14yes—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—22no—
556C23—21no—
557C23—22no—
558C23—21no—
559C23—22no—
560C23—21no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23C2×#13512yes—
569C23C2×AGL(1,ℤ/8)14yes—
570C23C23×D411yescharacteristic
571Q8—44no—
572Q8—42no—
573Q8—42no—
574Q8—88no—
575Q8—88no—
576D4—28no—
577D4—48no—
578D4—44no—
579D4—44no—
580D4—88no—
581D4—84no—
582D4—162no—
583D4—164no—
584D4—164no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—84no—
635C2×C4—88no—
636C2×C4—164no—
637C2×C4#53412yes—
638C2×C4#53412yes—
639C2×C4C2×#13512yes—
640C2×C4C2×#13512yes—
641C2×C4C2×AGL(1,ℤ/8)12yes—
642C2×C4#57712yes—
643C2×C4C2×H(5,𝔽2)12yes—
644C2×C4C2×2−1+411yescharacteristic
645C2×C4#58512yes—
646C2×C4#58511yescharacteristic
647C8—42no—
648C8—44no—
649C8—42no—
650C22—22no—
651C22—24no—
652C22—22no—
653C22—24no—
654C22—22no—
655C22—24no—
656C22—42no—
657C22—41no—
658C22—42no—
659C22—41no—
660C22—42no—
661C22—42no—
662C22—41no—
663C22—41no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22C2×#53411yescharacteristic
672C22#348911yescharacteristic
673C22#350011yescharacteristic
674C22C22×#13511yescharacteristic
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C2×#57711yescharacteristic
677C22#358611yescharacteristic
678C4—22no—
679C4—22no—
680C4—22no—
681C4—42no—
682C4—42no—
683C4—41no—
684C4—42no—
685C4—42no—
686C4—84no—
687C4—82no—
688C4—81no—
689C4—82no—
690C4—81no—
691C4—81no—
692C4—82no—
693C4#318814yes—
694C4#355412yes—
695C4#356212yes—
696C2—42no—
697C2—44no—
698C2—162no—
699C2#4291311yescharacteristic
700C2C2×#318811yescharacteristic
701C2#6075811yescharacteristic
702C2#6210111yescharacteristic
703C2#6212811yescharacteristic
704C2#6253111yescharacteristic
705C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35316C2semidirect
3. #32772C2semidirect
4. #32032C2semidirect
5. #32002C2semidirect
6. #30188C2semidirect
7. #30173C2semidirect
8. #29921C2semidirect
9. #29156C2semidirect
10. #28597C2semidirect
11. #26283C2semidirect
12. #25903C2semidirect
13. #25652C2semidirect
14. #25341C2semidirect
15. #24301C2semidirect
16. #23757C2semidirect
17. #22469C2semidirect

Maximal subgroups

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

subgroupindex
#364932
#353162
#327722
#320322
#320022
#301882
#301732
#299212
#291562
#285972
#262832
#259032
#256522
#253412
#243012
#237572
#224692