Finite Group Databasebeta

← #6806460

Subgroups — #6806460

Order 512.

1682 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#36595C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic, complemented
3#33230C211yesmaximal, characteristic, complemented
4#32107C214yesmaximal, complemented
5#32104C214yesmaximal, complemented
6#30219C211yesmaximal, characteristic, complemented
7#30218C211yesmaximal, characteristic
8#29191C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic, complemented
10#28750C211yesmaximal, characteristic, complemented
11#28750C211yesmaximal, characteristic, complemented
12#28749C211yesmaximal, characteristic
13#28509C211yesmaximal, characteristic, complemented
14#25659C214yesmaximal, complemented
15#25284C214yesmaximal, complemented
16#22219C211yesmaximal, characteristic, complemented
17#22218C211yesmaximal, characteristic, complemented
18#21646C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×#559C2212yes—
21C2×#557—21no—
22C2×#489C2214yes—
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#476C2214yes—
27#2868C2211yescharacteristic
28#2867C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2858C2211yescharacteristic
33#2568—21no—
34#2565—21no—
35#2565—21no—
36#2548—21no—
37#2548—21no—
38#2548C2212yes—
39#2546—21no—
40#2546C2212yes—
41#2544C2212yes—
42#2544C2212yes—
43#2506—21no—
44#2506—21no—
45#2506C2212yes—
46#2506C2212yes—
47#2445—21no—
48#2445—21no—
49#2445—21no—
50#2445—21no—
51#2366—21no—
52#2366—21no—
53#2365—21no—
54#2365C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2094C2214yes—
63#2066C2214yes—
64#2065C2214yes—
65#2042C2214yes—
66#2042C2214yes—
67#2042C2214yes—
68#2042C2214yes—
69#2041C2214yes—
70#2041C2214yes—
71#2041C2214yes—
72#1990C2211yescharacteristic
73#1985C2212yes—
74#1983C2214yes—
75#1983C2214yes—
76#1982C2214yes—
77#1959C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1957C2211yescharacteristic
83#1957C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1940C2212yes—
88#1936C2214yes—
89#1927C2214yes—
90#1926C2214yes—
91#1917C2212yes—
92#1917C2212yes—
93#1865C2214yes—
94#1865C2214yes—
95#1864C2214yes—
96#1864C2214yes—
97#1844C2211yescharacteristic
98#1844C2211yescharacteristic
99#1834C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107#559—48no—
108#557—48no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2311yescharacteristic
115C2×#127—21no—
116C2×#127—21no—
117C2×#127—21no—
118C2×#127—21no—
119C2×#127C2311yescharacteristic
120C2×#127C2311yescharacteristic
121C2×#127C2312yes—
122C2×#127C2312yes—
123C2×#125—22no—
124C2×#125—22no—
125C2×#125—21no—
126C2×#125—21no—
127C2×#124—22no—
128C2×#124C2312yes—
129C2×#124C2312yes—
130C2×#124C2312yes—
131C2×#124C2312yes—
132C2×#123—22no—
133C2×#123—22no—
134C2×C4×Q8—22no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8D411yescharacteristic
139C2×C4×Q8D411yescharacteristic
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×D4—22no—
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4—21no—
146C2×C4×D4—21no—
147C2×C4×D4C2312yes—
148C2×C4×D4C2312yes—
149#489—48no—
150#487—24no—
151#486—24no—
152#486—24no—
153#476—48no—
154C2×#106C2314yes—
155C2×#106C2314yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×#97C2312yes—
172C2×#97C2312yes—
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177#400—22no—
178#400—22no—
179#398—22no—
180#398—22no—
181#398—22no—
182#398—22no—
183#394—22no—
184#394—22no—
185#394—22no—
186#394—22no—
187#394—22no—
188#394—22no—
189#391—22no—
190#391—22no—
191#391—21no—
192#391—21no—
193#391—21no—
194#391C2312yes—
195#390—22no—
196#390—22no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390—21no—
203#390—21no—
204#390—21no—
205#390—21no—
206#390—21no—
207#390—21no—
208#390—21no—
209#390C2312yes—
210#390C2312yes—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#387—22no—
228#387—22no—
229#387—22no—
230#387—22no—
231#386—22no—
232#386—22no—
233#386—22no—
234#386—22no—
235#384—21no—
236#384—21no—
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2311yescharacteristic
240#384C2311yescharacteristic
241#384C2312yes—
242#382—21no—
243#382—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2311yescharacteristic
259C4×C4⋊C4C2311yescharacteristic
260C4×C4⋊C4C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263C43C2311yescharacteristic
264#340C2314yes—
265#340C2314yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×Q8C2×D411yescharacteristic
276C22×Q8C2412yes—
277C22×D4—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—22no—
283C22×D4—22no—
284C22×D4—22no—
285C22×D4C2412yes—
286C22×D4C2412yes—
287C23×C4—22no—
288C23×C4—22no—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—22no—
294C4⋊Q8—28no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299C4⋊Q8—48no—
300C4⋊Q8#4114yes—
301#127—22no—
302#127—22no—
303#127—22no—
304#127—22no—
305#127—44no—
306#127—44no—
307#127—44no—
308#127—44no—
309#127—82no—
310#127—82no—
311#125—44no—
312#125—48no—
313#125—48no—
314#125—44no—
315#124—28no—
316#124—44no—
317#124—44no—
318#124—44no—
319#124—48no—
320#123—48no—
321#123—48no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—42no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×Q8—44no—
328C4×Q8—48no—
329C4×Q8—44no—
330C4×D4—28no—
331C4×D4—44no—
332C4×D4—48no—
333C4×D4—82no—
334C4×D4—82no—
335C4×D4—82no—
336C4×D4—82no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
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⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
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⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C2×D411yescharacteristic
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×C42C2×D411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410#106—44no—
411#106—44no—
412C4⋊C8—24no—
413C4⋊C8—42no—
414C4⋊C8—42no—
415#103—28no—
416#103—44no—
417#103—44no—
418#103—44no—
419#102—28no—
420#102—28no—
421#102—44no—
422#102—44no—
423#102—44no—
424#102—44no—
425#102—44no—
426#102—44no—
427#97—24no—
428#97—42no—
429C4×C8—22no—
430C4×C8—22no—
431C4×C8—22no—
432C4×C8—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451C24—22no—
452C24—22no—
453C2×Q8—22no—
454C2×Q8—24no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—24no—
458C2×Q8—22no—
459C2×Q8—24no—
460C2×Q8—41no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—24no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—44no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—44no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—82no—
493C2×D4—82no—
494C2×D4—82no—
495C2×D4—82no—
496C2×D4—82no—
497C2×D4—82no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4—82no—
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×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C2511yescharacteristic
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—
536C2×C8—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—44no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4C2×#4114yes—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—44no—
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—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—44no—
605C42—44no—
606C42—44no—
607C42—44no—
608C42AGL(1,ℤ/8)14yes—
609C42C2×#4112yes—
610C42H(5,𝔽2)14yes—
611C422−1+412yes—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23C23×D411yescharacteristic
627Q8—44no—
628Q8—48no—
629Q8—44no—
630Q8—44no—
631Q8—88no—
632D4—44no—
633D4—44no—
634D4—44no—
635D4—44no—
636D4—88no—
637D4—84no—
638D4—84no—
639D4—84no—
640D4—84no—
641D4—164no—
642D4—164no—
643D4—164no—
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—24no—
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—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—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—164no—
695C2×C4—164no—
696C2×C4#53512yes—
697C2×C4#53512yes—
698C2×C4#54812yes—
699C2×C4#54812yes—
700C2×C4C2×AGL(1,ℤ/8)12yes—
701C2×C4C2×AGL(1,ℤ/8)12yes—
702C2×C4C2×AGL(1,ℤ/8)12yes—
703C2×C4#57512yes—
704C2×C4#57712yes—
705C2×C4#57712yes—
706C2×C4C22×#4111yescharacteristic
707C2×C4C2×H(5,𝔽2)12yes—
708C2×C4C2×2−1+411yescharacteristic
709C2×C4#58512yes—
710C8—42no—
711C8—42no—
712C8—42no—
713C8—42no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22C2×#53511yescharacteristic
729C22C2×#54811yescharacteristic
730C22#350011yescharacteristic
731C22C22×AGL(1,ℤ/8)11yescharacteristic
732C22C2×#57511yescharacteristic
733C22C2×#57711yescharacteristic
734C22C2×#57711yescharacteristic
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—41no—
744C4—44no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4#337114yes—
754C4#337314yes—
755C4#351712yes—
756C4#355412yes—
757C2—162no—
758C2—162no—
759C2#3917411yescharacteristic
760C2#6080311yescharacteristic
761C2#6081811yescharacteristic
762C2C2×#337311yescharacteristic
763C2C2×#337111yescharacteristic
764C2#6254111yescharacteristic
765C2#6254411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2867C22
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
2. #36560C2semidirect
3. #33230C2semidirect
4. #32107C2semidirect
5. #32104C2semidirect
6. #30219C2semidirect
8. #29191C2semidirect
9. #29190C2semidirect
10. #28750C2semidirect
11. #28750C2semidirect
13. #28509C2semidirect
14. #25659C2semidirect
15. #25284C2semidirect
16. #22219C2semidirect
17. #22218C2semidirect
18. #21646C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365952
#365602
#332302
#321072
#321042
#302192
#302182
#291912
#291902
#287502
#287492
#285092
#256592
#252842
#222192
#222182
#216462
#213402