Finite Group Databasebeta

← #6752312

Subgroups — #6752312

Order 512.

1533 conjugacy classes of subgroups.

The subgroup lattice has 767 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C211yesmaximal, characteristic, complemented
2#36187C211yesmaximal, characteristic
3#33048C211yesmaximal, characteristic, complemented
4#32222C214yesmaximal, complemented
5#32010C214yesmaximal, complemented
6#30304C211yesmaximal, characteristic, complemented
7#29842C211yesmaximal, characteristic, complemented
8#29218C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic, complemented
10#27973C211yesmaximal, characteristic, complemented
11#27891C211yesmaximal, characteristic, complemented
12#27877C211yesmaximal, characteristic, complemented
13#27876C211yesmaximal, characteristic, complemented
14#27381C214yesmaximal, complemented
15#27343C214yesmaximal, complemented
16#22219C211yesmaximal, characteristic, complemented
17#21981C211yesmaximal, characteristic, complemented
18#21102C211yesmaximal, characteristic, complemented
19#20824C211yesmaximal, characteristic, complemented
20C2×#555—21no—
21C2×#486C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2868C2211yescharacteristic
25#2859C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2856C2211yescharacteristic
30#2570—21no—
31#2563—21no—
32#2537—21no—
33#2537C2212yes—
34#2511—21no—
35#2507C2212yes—
36#2500—21no—
37#2500—21no—
38#2497C2212yes—
39#2491—21no—
40#2491—21no—
41#2491C2212yes—
42#2491C2212yes—
43#2487C2212yes—
44#2442—21no—
45#2441—21no—
46#2423—21no—
47#2417—21no—
48#2360—21no—
49#2354—21no—
50#2354C2212yes—
51#2352—21no—
52#2345C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C42×D4C2211yescharacteristic
60#2100C2214yes—
61#2083C2214yes—
62#2077C2214yes—
63#2075C2214yes—
64#2071C2214yes—
65#2067C2214yes—
66#2065C2214yes—
67#2057C2214yes—
68#2033C2214yes—
69#2033C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1987C2214yes—
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1948C2214yes—
78#1936C2214yes—
79#1926C2214yes—
80#1925C2214yes—
81#1924C2212yes—
82#1919C2214yes—
83#1918C2214yes—
84#1917C2212yes—
85#1895C2214yes—
86#1874C2214yes—
87#1870C2212yes—
88#1864C2214yes—
89#1864C2214yes—
90#1853C2214yes—
91#1851C2212yes—
92#1845C2211yescharacteristic
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1834C2211yescharacteristic
101#1809C2214yes—
102#1776C2214yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106#1744C2211yescharacteristic
107C22×M16C2312yes—
108#555—48no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2312yes—
113C2×#127C2311yescharacteristic
114C2×#125—21no—
115C2×#124—22no—
116C2×#124—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×#121—22no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121—21no—
123C2×#121—21no—
124C2×#121C2312yes—
125C2×#121C2312yes—
126C2×C4×Q8—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2312yes—
132C2×C4×D4C2311yescharacteristic
133C2×#117—21no—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117—22no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#486—24no—
143#431—24no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C4×M16—24no—
158C2×#97C2312yes—
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163#400—22no—
164#400—22no—
165#399—22no—
166#398—22no—
167#398—22no—
168#396—22no—
169#394—22no—
170#394—22no—
171#394—22no—
172#394—22no—
173#393—22no—
174#393—22no—
175#391—21no—
176#391—21no—
177#391—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390C2312yes—
185#390C2312yes—
186#390C2312yes—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#386—22no—
208#385—22no—
209#384—21no—
210#384—21no—
211#384—21no—
212#384—21no—
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2312yes—
216#384C2312yes—
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#382—21no—
222#382—21no—
223#382—21no—
224#382—21no—
225#380—21no—
226#380—21no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—22no—
231#380—22no—
232#380—22no—
233#380C2312yes—
234#380C2312yes—
235#380C2312yes—
236#380C2312yes—
237#380C2312yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C22⋊C4—22no—
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4C2311yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256C43C2311yescharacteristic
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8—22no—
272C22×D4—22no—
273C22×D4—22no—
274C22×D4—21no—
275C22×D4—21no—
276C22×D4C2×D411yescharacteristic
277C22×D4C2412yes—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C23×C4C2411yescharacteristic
286C2×M16—24no—
287C2×M16—42no—
288C2×M16—42no—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—44no—
296C4⋊Q8—44no—
297C4⋊Q8—82no—
298#127—22no—
299#127—22no—
300#125—82no—
301#124—48no—
302#124—48no—
303C22⋊Q8—44no—
304C22⋊Q8—82no—
305#121—44no—
306#121—48no—
307#121—44no—
308#121—44no—
309#121—44no—
310#121—82no—
311#121—82no—
312C4×Q8—82no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—44no—
316C4×D4—42no—
317C4×D4—44no—
318C4×D4—44no—
319#117—48no—
320#117—44no—
321#117—44no—
322#117—82no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—82no—
349C2×C4⋊C4—82no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401#107—44no—
402#107—44no—
403#106—44no—
404#106—44no—
405C4⋊C8—24no—
406C4⋊C8—42no—
407#102—28no—
408#102—44no—
409#102—44no—
410#102—44no—
411C22⋊C8—24no—
412C22⋊C8—42no—
413C22⋊C8—42no—
414#97—24no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8#4114yes—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—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#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439C24—22no—
440C24C22×D411yescharacteristic
441C24C22×D411yescharacteristic
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—41no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—82no—
464C2×D4—82no—
465C2×D4—82no—
466C2×D4—82no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4AGL(1,ℤ/8)14yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C2511yescharacteristic
513M16—48no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8C2×#4114yes—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—41no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—41no—
566C22⋊C4—41no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—21no—
589C42—21no—
590C42—21no—
591C42—21no—
592C42—22no—
593C42—22no—
594C42—44no—
595C42—44no—
596C42—82no—
597C42—82no—
598C42#13714yes—
599C42C2×#4112yes—
600C422−1+412yes—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23C2×AGL(1,ℤ/8)12yes—
622C23C23×D411yescharacteristic
623Q8—84no—
624Q8—164no—
625D4—44no—
626D4—44no—
627D4—44no—
628D4—84no—
629D4—84no—
630D4—164no—
631D4—164no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—28no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
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—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—88no—
689C2×C4—164no—
690C2×C4#54712yes—
691C2×C4#54712yes—
692C2×C4C2×AGL(1,ℤ/8)12yes—
693C2×C4C2×AGL(1,ℤ/8)12yes—
694C2×C4C2×#13712yes—
695C2×C4C2×#13712yes—
696C2×C4C2×#13712yes—
697C2×C4#57612yes—
698C2×C4#57712yes—
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×2−1+411yescharacteristic
701C2×C4#58512yes—
702C2×C4#58512yes—
703C8—24no—
704C8—42no—
705C8—42no—
706C8—42no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22#344911yescharacteristic
729C22C2×#54711yescharacteristic
730C22#350011yescharacteristic
731C22C22×AGL(1,ℤ/8)11yescharacteristic
732C22C22×#13711yescharacteristic
733C22C2×#57611yescharacteristic
734C22C2×#57711yescharacteristic
735C4—22no—
736C4—22no—
737C4—22no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4#334514yes—
753C4#335714yes—
754C4#349312yes—
755C4#356512yes—
756C2—42no—
757C2—81no—
758C2—162no—
759C2#3787111yescharacteristic
760C2#6096911yescharacteristic
761C2#6100611yescharacteristic
762C2C2×#334511yescharacteristic
763C2C2×#335711yescharacteristic
764C2#6253511yescharacteristic
765C2#6254211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36189C2semidirect
3. #33048C2semidirect
4. #32222C2semidirect
5. #32010C2semidirect
6. #30304C2semidirect
7. #29842C2semidirect
8. #29218C2semidirect
9. #29190C2semidirect
10. #27973C2semidirect
11. #27891C2semidirect
12. #27877C2semidirect
13. #27876C2semidirect
14. #27381C2semidirect
15. #27343C2semidirect
16. #22219C2semidirect
17. #21981C2semidirect
18. #21102C2semidirect
19. #20824C2semidirect

Maximal subgroups

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

subgroupindex
#361892
#361872
#330482
#322222
#320102
#303042
#298422
#292182
#291902
#279732
#278912
#278772
#278762
#273812
#273432
#222192
#219812
#211022
#208242