Finite Group Databasebeta

← #6802693

Subgroups — #6802693

Order 512.

1943 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36580C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic, complemented
3#32902C211yesmaximal, characteristic
4#32659C214yesmaximal, complemented
5#32612C214yesmaximal, complemented
6#29760C212yesmaximal, complemented
7#28312C212yesmaximal, complemented
8#28272C212yesmaximal, complemented
9#28114C212yesmaximal, complemented
10#27909C211yesmaximal, characteristic, complemented
11#27647C211yesmaximal, characteristic, complemented
12#25300C212yesmaximal, complemented
13#25057C212yesmaximal, complemented
14#24186C212yesmaximal, complemented
15#22510C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic, complemented
19C22×C22⋊Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#483C2212yes—
23C2×#480C2212yes—
24C2×#480C2212yes—
25C2×#459C2212yes—
26C2×#450C2211yescharacteristic
27C2×#447C2212yes—
28C2×#446C2212yescomplemented
29C2×C4×SD16C2212yes—
30#2856C2211yescharacteristic
31#2850C2212yes—
32#2592C2212yescomplemented
33#2582C2211yescharacteristic, complemented
34#2580C2212yescomplemented
35#2548C2211yescharacteristic, complemented
36#2538—21no—
37#2538—21no—
38#2538—22no—
39#2538C2214yes—
40#2538C2212yes—
41#2538C2212yes—
42#2458—22no—
43#2448—21no—
44#2443—22no—
45#2436—21no—
46C2×#396—22no—
47#2367—21no—
48#2361—21no—
49#2355C2211yescharacteristic, complemented
50C4×C22⋊Q8C2211yescharacteristic
51C4×#121C2212yes—
52C2×#390C2212yes—
53C2×#389C2212yes—
54C2×#384C2211yescharacteristic
55C2×#380—21no—
56#2093C2214yescomplemented
57#2092C2214yes—
58#2085C2214yescomplemented
59#2084C2212yescomplemented
60#2072C2212yescomplemented
61#2068C2214yescomplemented
62#2067C2214yes—
63#2065C2214yescomplemented
64#2054C2214yes—
65#2042C2214yescomplemented
66#2012C2214yes—
67#1985C2214yescomplemented
68#1984C2212yescomplemented
69#1973C2214yes—
70#1972C2214yes—
71#1957C2212yescomplemented
72#1948C2212yescomplemented
73#1943C2214yescomplemented
74#1936C2214yescomplemented
75#1925C2214yes—
76#1921C2214yescomplemented
77#1879C2214yes—
78#1879C2214yes—
79#1877C2212yes—
80#1874C2212yes—
81#1872C2211yescharacteristic
82#1864C2212yes—
83#1864C2214yescomplemented
84#1853C2211yescharacteristic
85#1845C2212yescomplemented
86#1823C2212yes—
87#1803C2212yescomplemented
88#1802C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1786C2211yescharacteristic
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×D8C2314yes—
100#559—44no—
101C2×#127C2311yescharacteristic
102C2×#124—21no—
103C2×#124—21no—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—48no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×#121—21no—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×C4×Q8—44no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D412yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—44no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4D412yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#483—44no—
143#480—28no—
144#480—44no—
145#459—44no—
146#450—24no—
147#447—44no—
148#446—44no—
149C4×SD16—44no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#106C2314yes—
153C2×#106C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×C22⋊C8C2311yescharacteristic
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2311yescharacteristic
168C2×#97C2314yes—
169C2×C4×C8C2312yes—
170#399—22no—
171#398—22no—
172#397—22no—
173#396—24no—
174#396—24no—
175#396—24no—
176#396—24no—
177#394—22no—
178#393—22no—
179#393—22no—
180#391—21no—
181#391—21no—
182#391C2312yes—
183#391C2311yescharacteristic
184#390—21no—
185#390—21no—
186#390—21no—
187#390—22no—
188#390—22no—
189#390C2312yes—
190#390C2314yes—
191#390C2312yes—
192#390C2312yes—
193#390C2314yes—
194#389—21no—
195#389—22no—
196#389—22no—
197#389C2314yes—
198#389C2312yes—
199#389C2311yescharacteristic
200#389C2314yes—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—21no—
208#386—22no—
209#386—22no—
210#386—22no—
211#385—21no—
212#385—22no—
213#385—22no—
214#384—21no—
215#384—21no—
216#384C2311yescharacteristic
217#384C2312yes—
218#384C2312yes—
219#382C2312yes—
220#380—22no—
221#380—22no—
222#380—24no—
223#380C2311yescharacteristic
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—21no—
229C4×C22⋊C4—22no—
230C2×#95—22no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239#336C2312yes—
240C25—21no—
241C22×Q8—24no—
242C22×Q8—42no—
243C22×Q8—42no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D412yes—
247C22×Q8C2411yescharacteristic
248C22×D4—24no—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—42no—
256C22×D4C2412yes—
257C22×D4C2412yes—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—44no—
264C23×C4—44no—
265C23×C4—44no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—42no—
274C2×D8—28no—
275C2×D8—44no—
276C2×D8—44no—
277C22×C8C2414yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280#127—24no—
281#124—44no—
282#124—44no—
283#124—44no—
284#123—48no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—416no—
288C22⋊Q8—48no—
289C22⋊Q8—48no—
290#121—24no—
291#121—24no—
292#121—44no—
293#121—82no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—42no—
304C4×D4—42no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—44no—
309C4×D4—84no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
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—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—44no—
364C2×C22⋊C4—48no—
365C2×C22⋊C4—84no—
366C2×C42—24no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—44no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379#107—28no—
380#107—44no—
381#106—42no—
382#106—44no—
383C4⋊C8—24no—
384C4⋊C8—44no—
385#103—42no—
386#103—44no—
387#102—24no—
388#102—24no—
389#102—28no—
390#102—42no—
391#102—44no—
392#102—44no—
393#102—44no—
394C22⋊C8—24no—
395C22⋊C8—22no—
396C22⋊C8#4114yes—
397#97—44no—
398C4×C8—24no—
399#95—24no—
400#95—24no—
401#95—21no—
402#95—21no—
403#95—22no—
404#95—21no—
405#95—22no—
406#95—21no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412C24—24no—
413C24—22no—
414C24—22no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24C22×D411yescharacteristic
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—416no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—48no—
447C2×D4—48no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—81no—
458C2×D4—81no—
459C2×D42−1+414yes—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—816no—
496C22×C4AGL(1,ℤ/8)14yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C2×#4112yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C4C2511yescharacteristic
509SD16—48no—
510D8—88no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8—22no—
516C2×C8—24no—
517C2×C8—22no—
518C2×C8—24no—
519C2×C8C2×#4114yes—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—81no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—81no—
551C22⋊C4—44no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—82no—
565C22⋊C4—84no—
566C22⋊C4—84no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C42—22no—
570C42—22no—
571C42—24no—
572C42—22no—
573C42—24no—
574C42—42no—
575C42—42no—
576C42—42no—
577C42—42no—
578C42—82no—
579C42H(5,𝔽2)14yes—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—24no—
585C23—48no—
586C23—48no—
587C23—48no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—84no—
602C23—84no—
603C23—84no—
604C23—84no—
605C23#55012yes—
606C23C2×AGL(1,ℤ/8)12yes—
607C23#57612yes—
608C23C23×D411yescharacteristic
609Q8—48no—
610Q8—44no—
611Q8—44no—
612Q8—42no—
613Q8—42no—
614D4—24no—
615D4—24no—
616D4—48no—
617D4—42no—
618D4—44no—
619D4—42no—
620D4—44no—
621D4—88no—
622D4—84no—
623D4—162no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—28no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—44no—
649C2×C4—48no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—44no—
671C2×C4—88no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4—88no—
675C2×C4—88no—
676C2×C4—164no—
677C2×C4#53412yes—
678C2×C4#53412yes—
679C2×C4#55012yes—
680C2×C4C2×AGL(1,ℤ/8)12yes—
681C2×C4C2×AGL(1,ℤ/8)12yes—
682C2×C4#57512yes—
683C2×C4C22×#4111yescharacteristic
684C2×C4C2×H(5,𝔽2)12yes—
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4C2×2−1+412yes—
687C8—24no—
688C8—42no—
689C8—44no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—816no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22#339914yes—
712C22C2×#53411yescharacteristic
713C22C2×#55011yescharacteristic
714C22#348611yescharacteristic
715C22C22×AGL(1,ℤ/8)11yescharacteristic
716C22C2×#57511yescharacteristic
717C22C2×#57611yescharacteristic
718C22#358611yescharacteristic
719C4—22no—
720C4—22no—
721C4—22no—
722C4—42no—
723C4—42no—
724C4—41no—
725C4—42no—
726C4—42no—
727C4—82no—
728C4—81no—
729C4—81no—
730C4—84no—
731C4—82no—
732C4—81no—
733C4—81no—
734C4#319414yes—
735C4#352012yes—
736C4#353612yes—
737C2—24no—
738C2—84no—
739C2—162no—
740C2#3913311yescharacteristic
741C2C2×#319411yescharacteristic
742C2#6076211yescharacteristic
743C2C2×#339911yescharacteristic
744C2#6140311yescharacteristic
745C2#6182711yescharacteristic
746C2#6183511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36580C2semidirect
2. C2×#2538C2semidirect
4. #32659C2semidirect
5. #32612C2semidirect
6. #29760C2semidirect
7. #28312C2semidirect
8. #28272C2semidirect
9. #28114C2semidirect
10. #27909C2semidirect
11. #27647C2semidirect
12. #25300C2semidirect
13. #25057C2semidirect
14. #24186C2semidirect
15. #22510C2semidirect
16. #22081C2semidirect
17. C4×#450C2semidirect
18. C2×#559C22semidirect
28. C2×#446C22semidirect
32. #2592C22semidirect
33. #2582C22semidirect
34. #2580C22semidirect
35. #2548C22semidirect
49. #2355C22semidirect
56. #2093C22semidirect
58. #2085C22semidirect
59. #2084C22semidirect
60. #2072C22semidirect
61. #2068C22semidirect
63. #2065C22semidirect
65. #2042C22semidirect
67. #1985C22semidirect
68. #1984C22semidirect
71. #1957C22semidirect
72. #1948C22semidirect
73. #1943C22semidirect
74. #1936C22semidirect
76. #1921C22semidirect
83. #1864C22semidirect
85. #1845C22semidirect
87. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#365802
C2×#25382
#329022
#326592
#326122
#297602
#283122
#282722
#281142
#279092
#276472
#253002
#250572
#241862
#225102
#220812
C4×#4502