Finite Group Databasebeta

← #6856864

Subgroups — #6856864

Order 512.

1681 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36595C211yesmaximal, characteristic, complemented
2#34941C211yesmaximal, characteristic, complemented
3#33160C211yesmaximal, characteristic
4#31973C214yesmaximal, complemented
5#31150C214yesmaximal, complemented
6#29825C212yesmaximal, complemented
7#29348C212yesmaximal, complemented
8#29290C211yesmaximal, characteristic, complemented
9#28749C211yesmaximal, characteristic, complemented
10#28640C212yesmaximal, complemented
11#28286C212yesmaximal, complemented
12#26463C212yesmaximal, complemented
13#25321C212yesmaximal, complemented
14#25292C212yesmaximal, complemented
15Q8×C4⋊C8C211yesmaximal, characteristic, complemented
16#22145C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#486C2212yes—
20C2×#478C2214yes—
21C2×#476C2211yescharacteristic
22C2×#459C2212yes—
23C2×#459C2212yes—
24C2×#459C2212yes—
25C2×C8×Q8C2212yes—
26#2867C2211yescharacteristic
27#2858C2212yes—
28#2574—22no—
29#2567C2211yescharacteristic
30#2563C2211yescharacteristic
31#2552C2211yescharacteristic
32#2548—21no—
33#2548—21no—
34#2548C2212yes—
35#2544—22no—
36#2538C2211yescharacteristic
37#2508C2212yes—
38#2497—21no—
39#2454C2212yes—
40#2445—22no—
41#2445C2214yes—
42#2423—21no—
43#2396—22no—
44#2366—21no—
45#2366C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2212yes—
49C4×#125C2212yes—
50C4×C22⋊Q8—21no—
51#2293C2212yes—
52#2285—21no—
53C42×Q8C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2087C2214yes—
56#2073C2214yes—
57#2067C2214yes—
58#2065C2214yes—
59#2065C2214yes—
60#2065C2214yes—
61#2054C2214yes—
62#2054C2214yes—
63#2041C2214yes—
64#2025C2214yes—
65#1987C2211yescharacteristic
66#1987C2212yes—
67#1987C2212yes—
68#1982C2212yes—
69#1982C2212yes—
70#1982C2212yes—
71#1982C2212yes—
72#1982C2212yes—
73#1970C2214yes—
74#1957C2212yes—
75#1948C2211yescharacteristic
76#1947C2212yes—
77#1947C2212yes—
78#1939C2212yes—
79#1939C2214yes—
80#1934C2214yes—
81#1934C2214yes—
82#1934C2214yes—
83#1920C2212yes—
84#1879C2214yes—
85#1864C2212yes—
86#1862C2214yes—
87#1842C2211yescharacteristic
88#1842C2212yes—
89#1804C2214yes—
90#1802C2211yescharacteristic
91C4×#106C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C22×D8C2314yes—
96#559—48no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127—21no—
100C2×#127—21no—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127C2312yes—
104C2×#127C2312yes—
105C2×#127C2312yes—
106C2×#125C2312yes—
107C2×#124—22no—
108C2×#124—22no—
109C2×#124—22no—
110C2×#123—24no—
111C2×C22⋊Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2314yes—
124C2×#117—22no—
125C22×C42—21no—
126#486—44no—
127#478—48no—
128#476—24no—
129#459—28no—
130#459—44no—
131#459—44no—
132C8×Q8—44no—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#399—22no—
156#398—22no—
157#398—22no—
158#396—22no—
159#396—22no—
160#395—22no—
161#394—24no—
162#394—24no—
163#394—24no—
164#393—22no—
165#391—21no—
166#391—21no—
167#391—21no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390C2314yes—
175#390C2312yes—
176#390C2312yes—
177#390C2314yes—
178#390C2314yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#386—24no—
194#386—24no—
195#385—22no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382—22no—
201#382C2312yes—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C43C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8C2×D411yescharacteristic
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4C2414yes—
246C22×D4C2411yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—24no—
251C2×D8—28no—
252C2×D8—44no—
253C2×D8—44no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261#127—24no—
262#127—24no—
263#127—24no—
264#127—24no—
265#127—42no—
266#127—42no—
267#127—44no—
268#127—44no—
269#127—82no—
270#127—82no—
271#125—24no—
272#125—42no—
273#124—48no—
274#124—48no—
275#124—84no—
276#123—416no—
277C22⋊Q8—44no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—44no—
283C4×Q8—82no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—41no—
289C4×D4—48no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—48no—
293C4×D4—84no—
294C4×D4#4114yes—
295#117—84no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—44no—
357#106—24no—
358#106—24no—
359C4⋊C8—24no—
360C4⋊C8—22no—
361C4⋊C8—24no—
362C4⋊C8—22no—
363C4⋊C8—44no—
364C4⋊C8—42no—
365C4⋊C8—44no—
366#102—24no—
367#102—28no—
368#102—28no—
369#102—44no—
370#102—42no—
371#102—44no—
372#102—44no—
373#102—44no—
374#102—44no—
375C4×C8—24no—
376C4×C8—44no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—21no—
394C24—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×Q8AGL(1,ℤ/8)14yes—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—24no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4C2×#4112yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—42no—
443C22×C4—48no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464D8—88no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4C2×#4112yes—
507C4⋊C4H(5,𝔽2)12yes—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—48no—
511C22⋊C4—42no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—21no—
533C42—42no—
534C42—41no—
535C42—42no—
536C42—42no—
537C42—42no—
538C42—44no—
539C42—44no—
540C42#13514yes—
541C42C2×#4112yes—
542C42H(5,𝔽2)12yes—
543C422−1+412yes—
544C422−1+412yes—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23C23×D411yescharacteristic
560Q8—28no—
561Q8—84no—
562Q8—84no—
563D4—22no—
564D4—22no—
565D4—44no—
566D4—44no—
567D4—41no—
568D4—44no—
569D4—41no—
570D4—44no—
571D4—88no—
572D4—88no—
573D4—88no—
574D4—164no—
575D4—164no—
576D4—164no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—168no—
622C2×C4#53712yes—
623C2×C4#53712yes—
624C2×C4#54812yes—
625C2×C4#54812yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13512yes—
628C2×C4C2×#13512yes—
629C2×C4C2×AGL(1,ℤ/8)14yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57612yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4#58511yescharacteristic
638C8—42no—
639C8—44no—
640C8—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22C2×#53711yescharacteristic
656C22C2×#54811yescharacteristic
657C22#348811yescharacteristic
658C22C22×#13511yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C2×#57611yescharacteristic
661C22#358511yescharacteristic
662C4—22no—
663C4—24no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—41no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4#324714yes—
678C4#336114yes—
679C4#350711yescharacteristic
680C4#350811yescharacteristic
681C4#355011yescharacteristic
682C4#355411yescharacteristic
683C2—82no—
684C2—164no—
685C2#3972611yescharacteristic
686C2C2×#324711yescharacteristic
687C2#6092211yescharacteristic
688C2C2×#336111yescharacteristic
689C2#6132111yescharacteristic
690C2#6194911yescharacteristic
691C2#6200111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#33160C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36595C2semidirect
2. #34941C2semidirect
4. #31973C2semidirect
5. #31150C2semidirect
6. #29825C2semidirect
7. #29348C2semidirect
8. #29290C2semidirect
9. #28749C2semidirect
10. #28640C2semidirect
11. #28286C2semidirect
12. #26463C2semidirect
13. #25321C2semidirect
14. #25292C2semidirect
15. Q8×C4⋊C8C2semidirect
16. #22145C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#365952
#349412
#331602
#319732
#311502
#298252
#293482
#292902
#287492
#286402
#282862
#264632
#253212
#252922
Q8×C4⋊C82
#221452
C4×#4762