Finite Group Databasebeta

← #6497752

Subgroups — #6497752

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic, complemented
2#35288C211yesmaximal, characteristic
3#33290C211yesmaximal, characteristic, complemented
4#33277C211yesmaximal, characteristic, complemented
5#33207C211yesmaximal, characteristic, complemented
6#33042C211yesmaximal, characteristic
7#30307C211yesmaximal, characteristic, complemented
8#30278C211yesmaximal, characteristic, complemented
9#29878C211yesmaximal, characteristic, complemented
10#29842C211yesmaximal, characteristic
11#28944C214yesmaximal, complemented
12#28842C214yesmaximal, complemented
13#28068C214yesmaximal, complemented
14#28037C214yesmaximal, complemented
15#22246C211yesmaximal, characteristic, complemented
16#22219C211yesmaximal, characteristic, complemented
17C4×#500C211yesmaximal, characteristic, complemented
18C4×#486C211yesmaximal, characteristic, complemented
19C8×#117C211yesmaximal, characteristic, complemented
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2862C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2858C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2856C2211yescharacteristic
34#2645—21no—
35#2644—21no—
36#2625—21no—
37#2624—21no—
38#2514—21no—
39#2513C2212yes—
40#2509C2212yes—
41#2508C2212yes—
42#2495C2212yes—
43#2494C2212yes—
44#2490C2212yes—
45#2489—21no—
46#2454C2212yes—
47#2453—21no—
48#2448—21no—
49#2447—21no—
50#2427—21no—
51#2426—21no—
52#2421—21no—
53#2420C2212yes—
54#2324—21no—
55#2323—21no—
56#2320—21no—
57#2319—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2092C2214yes—
64#2092C2214yes—
65#2087C2214yes—
66#2085C2214yes—
67#2076C2214yes—
68#2067C2214yes—
69#2059C2214yes—
70#2033C2214yes—
71#1991C2214yes—
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1942C2214yes—
81#1941C2214yes—
82#1934C2214yes—
83#1934C2214yes—
84#1934C2214yes—
85#1934C2214yes—
86#1926C2214yes—
87#1920C2214yes—
88#1919C2214yes—
89#1893C2214yes—
90#1864C2214yes—
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1835C2214yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1807C2214yes—
97C4×#107C2214yes—
98C8×C4⋊C4C2212yes—
99C8×C4⋊C4C2212yes—
100C4×#102C2214yes—
101C8×C22⋊C4C2212yes—
102C8×C22⋊C4C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C42×C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×C8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×#127C2311yescharacteristic
112C2×#125—21no—
113C2×#125C2312yes—
114C2×#124—21no—
115C2×#124—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×#121—21no—
120C2×#121—21no—
121C2×#121C2312yes—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×#117D411yescharacteristic
129C2×#117D411yescharacteristic
130C22×C42C2311yescharacteristic
131#500—24no—
132#498—24no—
133#486—24no—
134#486—24no—
135#431—24no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8—24no—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2312yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156#401—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#399—22no—
162#398—22no—
163#397—22no—
164#397—22no—
165#397—22no—
166#397—22no—
167#397—22no—
168#397—22no—
169#396—22no—
170#394—22no—
171#394—22no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#390—22no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#389—22no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#388—22no—
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#386—22no—
223#385—22no—
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#383—22no—
229#382—21no—
230#382—22no—
231#382C2312yes—
232#382C2312yes—
233#382C2312yes—
234#381—22no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—22no—
239#380C2312yes—
240#380C2312yes—
241#380C2312yes—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2311yescharacteristic
251C4×C4⋊C4C2311yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4C2311yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256#376D411yescharacteristic
257#376D411yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×D4—22no—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4C2412yes—
279C23×C4—22no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—42no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296#127—22no—
297#127—22no—
298#125—44no—
299#125—82no—
300#124—82no—
301#124—82no—
302C22⋊Q8—44no—
303C22⋊Q8—82no—
304C22⋊Q8—82no—
305#121—44no—
306#121—44no—
307#121—82no—
308C4×Q8—44no—
309C4×Q8—82no—
310C4×Q8—82no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—82no—
314#117—24no—
315#117—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C22⋊C4C2×D411yescharacteristic
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—28no—
387#107—44no—
388#106—28no—
389#106—44no—
390C4⋊C8—24no—
391#102—28no—
392#102—28no—
393#102—28no—
394#102—44no—
395C22⋊C8—24no—
396C4×C8—24no—
397C4×C8—24no—
398C4×C8—24no—
399C4×C8—22no—
400C4×C8—22no—
401C4×C8—42no—
402C4×C8—42no—
403C4×C8#4114yes—
404C4×C8#4114yes—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
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#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433C24—22no—
434C24C22×D411yescharacteristic
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—44no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—82no—
456C2×D4—82no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—44no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4#13514yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C2511yescharacteristic
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—48no—
501C2×C8C2×#4114yes—
502C2×C8C2×#4114yes—
503C2×C82−1+414yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—21no—
566C42—21no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—44no—
572C42—82no—
573C42—82no—
574C42#13514yes—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23C2×#13512yes—
591C23C23×D411yescharacteristic
592Q8—84no—
593Q8—84no—
594Q8—84no—
595D4—44no—
596D4—44no—
597D4—84no—
598D4—84no—
599D4—164no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—28no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—164no—
656C2×C4#53212yes—
657C2×C4#53212yes—
658C2×C4C2×#13512yes—
659C2×C4C2×#13512yes—
660C2×C4C2×#13512yes—
661C2×C4C2×#13512yes—
662C2×C4C2×#13512yes—
663C2×C4#57612yes—
664C2×C4#57712yes—
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C2×2−1+412yes—
668C2×C4#58512yes—
669C8—24no—
670C8—24no—
671C8—24no—
672C8—42no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22C2×#53211yescharacteristic
688C22#349911yescharacteristic
689C22#350011yescharacteristic
690C22C22×#13511yescharacteristic
691C22C22×#13511yescharacteristic
692C22C2×#57611yescharacteristic
693C22C2×#57711yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#316814yes—
713C4#317914yes—
714C4#346112yes—
715C4#349512yes—
716C2—42no—
717C2—162no—
718C2#3531811yescharacteristic
719C2C2×#316811yescharacteristic
720C2C2×#317911yescharacteristic
721C2#6247011yescharacteristic
722C2#6247411yescharacteristic
723C2#6252711yescharacteristic
724C2#6253111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35291C2semidirect
3. #33290C2semidirect
4. #33277C2semidirect
5. #33207C2semidirect
7. #30307C2semidirect
8. #30278C2semidirect
9. #29878C2semidirect
11. #28944C2semidirect
12. #28842C2semidirect
13. #28068C2semidirect
14. #28037C2semidirect
15. #22246C2semidirect
16. #22219C2semidirect
17. C4×#500C2semidirect
18. C4×#486C2semidirect
19. C8×#117C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352882
#332902
#332772
#332072
#330422
#303072
#302782
#298782
#298422
#289442
#288422
#280682
#280372
#222462
#222192
C4×#5002
C4×#4862
C8×#1172