Finite Group Databasebeta

← #6486808

Subgroups — #6486808

Order 512.

1607 conjugacy classes of subgroups.

The subgroup lattice has 777 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic, complemented
2#34226C211yesmaximal, characteristic, complemented
3#33086C214yesmaximal, complemented
4#30023C214yesmaximal, complemented
5#29542C214yesmaximal, complemented
6#29418C214yesmaximal, complemented
7#29190C211yesmaximal, characteristic, complemented
8#29177C211yesmaximal, characteristic, complemented
9#28655C211yesmaximal, characteristic, complemented
10#28615C211yesmaximal, characteristic, complemented
11#27890C211yesmaximal, characteristic, complemented
12#27875C211yesmaximal, characteristic, complemented
13#25673C211yesmaximal, characteristic, complemented
14#25599C211yesmaximal, characteristic
15#22269C211yesmaximal, characteristic, complemented
16#22247C211yesmaximal, characteristic, complemented
17#22219C211yesmaximal, characteristic, complemented
18#22219C211yesmaximal, characteristic, complemented
19#20782C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#493C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#486C2211yescharacteristic
25#2859C2211yescharacteristic
26#2858C2211yescharacteristic
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2644C2212yes—
30#2624C2212yes—
31#2513—21no—
32#2490—21no—
33#2454—21no—
34#2448—21no—
35#2447—21no—
36#2427—21no—
37#2426—21no—
38#2420—21no—
39#2367—21no—
40#2367—21no—
41#2363—21no—
42#2363—21no—
43#2357—21no—
44#2349—21no—
45#2341—21no—
46#2339—21no—
47#2324C2212yes—
48#2324C2212yes—
49#2324C2212yes—
50#2319C2212yes—
51#2319C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58#2285C2211yescharacteristic
59#2275C2212yes—
60C4×#117C2211yescharacteristic
61#2088C2214yes—
62#2077C2214yes—
63#2067C2214yes—
64#2065C2214yes—
65#2054C2214yes—
66#2034C2214yes—
67#2033C2214yes—
68#1999C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1957C2211yescharacteristic
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1926C2214yes—
78#1926C2214yes—
79#1926C2214yes—
80#1925C2214yes—
81#1920C2214yes—
82#1919C2214yes—
83#1918C2214yes—
84#1918C2214yes—
85#1879C2214yes—
86#1879C2214yes—
87#1879C2214yes—
88#1879C2214yes—
89#1864C2214yes—
90#1864C2214yes—
91#1864C2214yes—
92#1862C2214yes—
93#1844C2211yescharacteristic
94#1844C2211yescharacteristic
95#1840C2212yes—
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1832C2212yes—
99#1818C2212yes—
100#1798C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1744C2211yescharacteristic
106#1744C2211yescharacteristic
107C22×#41—21no—
108C22×D8C2314yes—
109#532—24no—
110#532—24no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#127C2311yescharacteristic
115C2×#125—21no—
116C2×#124—21no—
117C2×#124—21no—
118C2×#124—21no—
119C2×C22⋊Q8—21no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121—21no—
123C2×#121C2312yes—
124C2×#121C2312yes—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2312yes—
139C2×#117—21no—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#500—24no—
144#493—24no—
145#486—24no—
146#486—24no—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164#399—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#390—22no—
177#390—22no—
178#390—22no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
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#390C2312yes—
194#390C2312yes—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#386—22no—
215#386—22no—
216#386—22no—
217#385—22no—
218#385—22no—
219#385—22no—
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#382—21no—
225#382—21no—
226#381—22no—
227#381—22no—
228#380—21no—
229#380—21no—
230#380—22no—
231#380—22no—
232#380C2312yes—
233#380C2312yes—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248C4×C22⋊C4—22no—
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253#376C2311yescharacteristic
254#376C2311yescharacteristic
255C2×#95—22no—
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C2×#41—28no—
273C2×#41—82no—
274C2×#41—82no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×D4—22no—
278C22×D4—22no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—21no—
283C22×D4C2412yes—
284C22×D4C2412yes—
285C23×C4—22no—
286C23×C4—22no—
287C23×C4—22no—
288C23×C4—22no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2411yescharacteristic
292C2×D8—28no—
293C2×D8—44no—
294C2×D8—44no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8#4114yes—
303#127—22no—
304#127—22no—
305#127—22no—
306#127#4114yes—
307#125—82no—
308#124—28no—
309#124—82no—
310#124—82no—
311C22⋊Q8—82no—
312#121—28no—
313#121—24no—
314#121—24no—
315#121—44no—
316#121—82no—
317#121—82no—
318C4×Q8—82no—
319C4×Q8—82no—
320C4×Q8—82no—
321C4×D4—28no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—82no—
328C4×D4—82no—
329C4×D4—82no—
330C4×D4—82no—
331C4×D4—82no—
332C4×D4—82no—
333#117—42no—
334#117—42no—
335#117—82no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408#107—28no—
409#106—44no—
410#106—44no—
411C4⋊C8—42no—
412C4⋊C8—42no—
413#102—28no—
414#102—28no—
415#102—44no—
416#102—44no—
417#102—44no—
418#102—44no—
419C22⋊C8—42no—
420C22⋊C8—42no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445C24—22no—
446C24—22no—
447C24C22×D411yescharacteristic
448#41—816no—
449C2×Q8—24no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×D4—28no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—44no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—82no—
477C2×D4—82no—
478C2×D4—82no—
479C2×D4—82no—
480C2×D4C2×#4114yes—
481C22×C4—22no—
482C22×C4—28no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C2511yescharacteristic
523D8—88no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4C2×#4114yes—
564C22⋊C4—28no—
565C22⋊C4—28no—
566C22⋊C4—41no—
567C22⋊C4—41no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C22⋊C4—82no—
587C22⋊C4—82no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—21no—
599C42—21no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—82no—
605C42—82no—
606C42—82no—
607C42—82no—
608C42C2×#4112yes—
609C42C2×#4112yes—
610C23—24no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23—42no—
628C23—42no—
629C23—42no—
630C23—42no—
631C23C23×D411yescharacteristic
632Q8—48no—
633Q8—84no—
634D4—28no—
635D4—48no—
636D4—48no—
637D4—44no—
638D4—44no—
639D4—44no—
640D4—88no—
641D4—84no—
642D4—84no—
643D4—164no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—24no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—21no—
675C2×C4—21no—
676C2×C4—21no—
677C2×C4—21no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—88no—
700C2×C4—164no—
701C2×C4—164no—
702C2×C4—164no—
703C2×C4#52512yes—
704C2×C4#52512yes—
705C2×C4#57612yes—
706C2×C4#57612yes—
707C2×C4#57712yes—
708C2×C4#57712yes—
709C2×C4C22×#4111yescharacteristic
710C2×C4C22×#4111yescharacteristic
711C2×C4#58512yes—
712C2×C4#58512yes—
713C8—42no—
714C8—42no—
715C8—42no—
716C8—42no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—44no—
723C22—41no—
724C22—44no—
725C22—41no—
726C22—44no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—82no—
736C22—82no—
737C22—82no—
738C22C2×#52511yescharacteristic
739C22#344911yescharacteristic
740C22#344911yescharacteristic
741C22C2×#57611yescharacteristic
742C22C2×#57611yescharacteristic
743C22C2×#57711yescharacteristic
744C22C2×#57711yescharacteristic
745C4—41no—
746C4—41no—
747C4—41no—
748C4—41no—
749C4—41no—
750C4—41no—
751C4—41no—
752C4—41no—
753C4—41no—
754C4—44no—
755C4—41no—
756C4—41no—
757C4—82no—
758C4—82no—
759C4—82no—
760C4—82no—
761C4—82no—
762C4#312214yes—
763C4#312514yes—
764C4#352112yes—
765C4#352412yes—
766C2—81no—
767C2—84no—
768C2—162no—
769C2#3521111yescharacteristic
770C2C2×#312211yescharacteristic
771C2C2×#312511yescharacteristic
772C2#6100611yescharacteristic
773C2#6100611yescharacteristic
774C2#6100611yescharacteristic
775C2#6100611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#25599C2
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. #35288C2semidirect
2. #34226C2semidirect
3. #33086C2semidirect
4. #30023C2semidirect
5. #29542C2semidirect
6. #29418C2semidirect
7. #29190C2semidirect
8. #29177C2semidirect
9. #28655C2semidirect
10. #28615C2semidirect
11. #27890C2semidirect
12. #27875C2semidirect
13. #25673C2semidirect
15. #22269C2semidirect
16. #22247C2semidirect
17. #22219C2semidirect
18. #22219C2semidirect
19. #20782C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#342262
#330862
#300232
#295422
#294182
#291902
#291772
#286552
#286152
#278902
#278752
#256732
#255992
#222692
#222472
#222192
#207822