Finite Group Databasebeta

← #6856265

Subgroups — #6856265

Order 512.

1928 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3#32905C211yesmaximal, characteristic, complemented
4#32448C214yesmaximal, complemented
5#31656C214yesmaximal, complemented
6#29718C212yesmaximal
7#28414C212yesmaximal, complemented
8#27978C212yesmaximal, complemented
9#27736C212yesmaximal, complemented
10#27681C212yesmaximal, complemented
11#25902C211yesmaximal, characteristic, complemented
12#25320C212yesmaximal, complemented
13#24676C212yesmaximal, complemented
14#24298C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22049C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#487C2212yescomplemented
23C2×#474C2212yescomplemented
24C2×#461C2212yes—
25C2×#450C2212yes—
26C2×#447C2211yescharacteristic
27C2×C4×D8C2212yes—
28#2857C2211yescharacteristic
29#2850C2212yes—
30#2596—22no—
31#2571C2211yescharacteristic, complemented
32#2547C2211yescharacteristic, complemented
33#2542—22no—
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2542C2212yes—
38#2542C2214yes—
39#2514C2212yescomplemented
40#2510—22no—
41#2494—21no—
42#2487—21no—
43#2448C2212yescomplemented
44C2×#399—22no—
45#2365C2211yescharacteristic, complemented
46#2354C2211yescharacteristic, complemented
47#2352—21no—
48#2347—21no—
49C4×C22⋊Q8C2212yes—
50C4×#121C2211yescharacteristic
51C2×#390C2212yes—
52C2×#384C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2214yescomplemented
55#2095C2214yescomplemented
56#2073C2214yes—
57#2072C2214yescomplemented
58#2071C2214yescomplemented
59#2057C2212yescomplemented
60#2055C2214yes—
61#2054C2214yes—
62#2024C2214yescomplemented
63#2023C2212yescomplemented
64#2012C2214yes—
65#1990C2212yescomplemented
66#1988C2212yescomplemented
67#1973C2214yes—
68#1959C2214yescomplemented
69#1947C2214yescomplemented
70#1939C2212yescomplemented
71#1936C2214yes—
72#1925C2214yes—
73#1917C2214yescomplemented
74#1880C2214yes—
75#1879C2212yes—
76#1877C2214yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1862C2212yes—
83#1852C2212yes—
84#1852C2211yescharacteristic
85#1836C2214yescomplemented
86#1803C2214yescomplemented
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1786C2211yescharacteristic
91#1770C2212yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2314yes—
99C22×D8C2312yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#124C2312yes—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—48no—
114C2×C22⋊Q8—48no—
115C2×C22⋊Q8C2312yes—
116C2×#121C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—44no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#487—44no—
139#474—44no—
140#461—44no—
141#450—28no—
142#447—24no—
143C4×D8—44no—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162#401—22no—
163#400—22no—
164#400—22no—
165#399—22no—
166#399—24no—
167#399—24no—
168#399—24no—
169#399—24no—
170#397—22no—
171#397—22no—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#390—22no—
176#390—22no—
177#390C2312yes—
178#390C2314yes—
179#390C2314yes—
180#390C2311yescharacteristic
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389C2311yescharacteristic
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—21no—
191#388—22no—
192#388—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#386—22no—
197#386—22no—
198#386—21no—
199#385—22no—
200#384—22no—
201#384—22no—
202#384C2311yescharacteristic
203#384C2314yes—
204#384C2314yes—
205#383—22no—
206#382—21no—
207#382C2312yes—
208#380—21no—
209#380—21no—
210#380—21no—
211#380—21no—
212#380—21no—
213#380C2311yescharacteristic
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C2×#95D412yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C25—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2412yes—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4#4114yes—
251C22×D4C2411yescharacteristic
252C22×D4C2412yes—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×SD16—28no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×D8—24no—
270C2×D8—42no—
271C2×D8—42no—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281#124—44no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—416no—
285C22⋊Q8—48no—
286C22⋊Q8—48no—
287C22⋊Q8—48no—
288C22⋊Q8—416no—
289C22⋊Q8—48no—
290#121—22no—
291#121—22no—
292C4×Q8—44no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—44no—
301#117—44no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—48no—
355C2×C22⋊C4—48no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369#107—42no—
370#107—44no—
371C4⋊C8—24no—
372C4⋊C8—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376#102—24no—
377#102—28no—
378#102—42no—
379#102—42no—
380#102—42no—
381#102—44no—
382C22⋊C8—22no—
383C22⋊C8—24no—
384C22⋊C8—22no—
385#97—44no—
386C4×C8—24no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—24no—
405C24—22no—
406C24—21no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24C22×D411yescharacteristic
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—816no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—48no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—48no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4C2×#4114yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—41no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—816no—
479C22×C4AGL(1,ℤ/8)14yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C42−1+412yes—
491C22×C4C2511yescharacteristic
492SD16—88no—
493D8—48no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—48no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4AGL(1,ℤ/8)14yes—
535C22⋊C4—44no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—24no—
555C42—24no—
556C42—42no—
557C42—42no—
558C42—44no—
559C42—42no—
560C42—82no—
561C42H(5,𝔽2)14yes—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—24no—
567C23—48no—
568C23—48no—
569C23—42no—
570C23—48no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23#54812yes—
588C23C2×AGL(1,ℤ/8)12yes—
589C23#57812yes—
590C23C23×D411yescharacteristic
591Q8—44no—
592Q8—44no—
593Q8—88no—
594Q8—88no—
595D4—24no—
596D4—24no—
597D4—48no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—84no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—88no—
651C2×C4—84no—
652C2×C4—88no—
653C2×C4—84no—
654C2×C4—88no—
655C2×C4#53512yes—
656C2×C4#53512yes—
657C2×C4#54812yes—
658C2×C4C2×AGL(1,ℤ/8)14yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4C2×AGL(1,ℤ/8)12yes—
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)12yes—
664C2×C4C2×2−1+411yescharacteristic
665C2×C4#58512yes—
666C8—42no—
667C8—44no—
668C8—42no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—816no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22#336714yes—
691C22C2×#53511yescharacteristic
692C22C2×#54811yescharacteristic
693C22#348511yescharacteristic
694C22C22×AGL(1,ℤ/8)11yescharacteristic
695C22C22×AGL(1,ℤ/8)11yescharacteristic
696C22C2×#57811yescharacteristic
697C22#358611yescharacteristic
698C4—24no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—41no—
706C4—81no—
707C4—82no—
708C4—81no—
709C4—82no—
710C4—84no—
711C4—81no—
712C4—82no—
713C4#321714yes—
714C4#350512yes—
715C4#354112yes—
716C2—24no—
717C2—84no—
718C2—161no—
719C2#3972211yescharacteristic
720C2C2×#321711yescharacteristic
721C2#6081411yescharacteristic
722C2C2×#336711yescharacteristic
723C2#6131711yescharacteristic
724C2#6171711yescharacteristic
725C2#6173711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36586C2semidirect
2. C2×#2542C2semidirect
3. #32905C2semidirect
4. #32448C2semidirect
5. #31656C2semidirect
7. #28414C2semidirect
8. #27978C2semidirect
9. #27736C2semidirect
10. #27681C2semidirect
11. #25902C2semidirect
12. #25320C2semidirect
13. #24676C2semidirect
14. #24298C2semidirect
15. #22629C2semidirect
16. #22049C2semidirect
22. C2×#487C22semidirect
23. C2×#474C22semidirect
31. #2571C22semidirect
32. #2547C22semidirect
39. #2514C22semidirect
43. #2448C22semidirect
45. #2365C22semidirect
46. #2354C22semidirect
54. #2100C22semidirect
55. #2095C22semidirect
57. #2072C22semidirect
58. #2071C22semidirect
59. #2057C22semidirect
62. #2024C22semidirect
63. #2023C22semidirect
65. #1990C22semidirect
66. #1988C22semidirect
68. #1959C22semidirect
69. #1947C22semidirect
70. #1939C22semidirect
73. #1917C22semidirect
85. #1836C22semidirect
86. #1803C22semidirect
91. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#25422
#329052
#324482
#316562
#297182
#284142
#279782
#277362
#276812
#259022
#253202
#246762
#242982
#226292
#220492
C4×#4472