Finite Group Databasebeta

← #6806458

Subgroups — #6806458

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36595C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic, complemented
3#33230C211yesmaximal, characteristic, complemented
4#32109C214yesmaximal, complemented
5#32102C214yesmaximal, complemented
6#30219C211yesmaximal, characteristic, complemented
7#30218C211yesmaximal, characteristic
8#29193C211yesmaximal, characteristic, complemented
9#29192C211yesmaximal, characteristic, complemented
10#28750C211yesmaximal, characteristic, complemented
11#28750C211yesmaximal, characteristic, complemented
12#28749C211yesmaximal, characteristic
13#28508C211yesmaximal, characteristic, complemented
14#25660C214yesmaximal, complemented
15#25285C214yesmaximal, complemented
16#22221C211yesmaximal, characteristic, complemented
17#22220C211yesmaximal, characteristic, complemented
18#21646C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×#559C2212yes—
21C2×#489C2214yes—
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#474C2214yes—
26#2868C2211yescharacteristic
27#2867C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2582—21no—
33#2567C2212yes—
34#2552—21no—
35#2550—21no—
36#2550—21no—
37#2548—21no—
38#2548—21no—
39#2548C2212yes—
40#2544C2212yes—
41#2544C2212yes—
42#2507C2212yes—
43#2507C2212yes—
44#2451—21no—
45#2451—21no—
46#2445—21no—
47#2445—21no—
48#2445—21no—
49#2445—21no—
50#2369C2212yes—
51#2366—21no—
52#2366—21no—
53#2355—21no—
54Q8×C4⋊C4—21no—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2093C2214yes—
63#2068C2214yes—
64#2067C2214yes—
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#1928C2214yes—
90#1925C2214yes—
91#1917C2212yes—
92#1917C2212yes—
93#1863C2214yes—
94#1863C2214yes—
95#1862C2214yes—
96#1862C2214yes—
97#1845C2211yescharacteristic
98#1845C2211yescharacteristic
99#1834C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107#559—48no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127—21no—
114C2×#127—21no—
115C2×#127—21no—
116C2×#127—21no—
117C2×#127C2311yescharacteristic
118C2×#127C2311yescharacteristic
119C2×#127C2312yes—
120C2×#127C2312yes—
121C2×#125—21no—
122C2×#125—21no—
123C2×#124—22no—
124C2×#124C2312yes—
125C2×#124C2312yes—
126C2×#124C2312yes—
127C2×#124C2312yes—
128C2×#123—22no—
129C2×#123—22no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8D411yescharacteristic
134C2×C4×Q8D411yescharacteristic
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×D4—22no—
138C2×C4×D4—21no—
139C2×C4×D4—21no—
140C2×C4×D4—21no—
141C2×C4×D4—21no—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144#489—48no—
145#487—24no—
146#486—24no—
147#486—24no—
148#474—48no—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#97C2312yes—
167C2×#97C2312yes—
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172#398—22no—
173#398—22no—
174#398—22no—
175#398—22no—
176#395—22no—
177#395—22no—
178#394—22no—
179#394—22no—
180#394—22no—
181#394—22no—
182#394—22no—
183#394—22no—
184#391—22no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391—22no—
189#391—22no—
190#391C2312yes—
191#390—22no—
192#390—22no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
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#390C2312yes—
206#390C2312yes—
207#389—22no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—22no—
218#389—22no—
219#389C2312yes—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#389C2312yes—
226#387—22no—
227#387—22no—
228#386—22no—
229#386—22no—
230#386—22no—
231#386—22no—
232#384—22no—
233#384—21no—
234#384—21no—
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2312yes—
240#382—21no—
241#382—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2311yescharacteristic
258C4×C4⋊C4C2311yescharacteristic
259C4×C4⋊C4C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8C2×D411yescharacteristic
275C22×Q8C2412yes—
276C22×D4—22no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—22no—
282C22×D4—22no—
283C22×D4—22no—
284C22×D4C2412yes—
285C22×D4C2412yes—
286C23×C4—22no—
287C23×C4—22no—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—22no—
293C4⋊Q8—28no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298C4⋊Q8#4114yes—
299#127—22no—
300#127—22no—
301#127—22no—
302#127—22no—
303#127—44no—
304#127—44no—
305#127—44no—
306#127—44no—
307#127—82no—
308#127—82no—
309#125—82no—
310#125—82no—
311#124—28no—
312#124—44no—
313#124—44no—
314#124—44no—
315#124—48no—
316#123—48no—
317#123—48no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—42no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8—82no—
324C4×Q8—82no—
325C4×D4—28no—
326C4×D4—44no—
327C4×D4—48no—
328C4×D4—82no—
329C4×D4—82no—
330C4×D4—82no—
331C4×D4—82no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
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—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
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⋊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×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
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×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—44no—
406#107—44no—
407C4⋊C8—24no—
408C4⋊C8—42no—
409C4⋊C8—42no—
410#103—28no—
411#103—44no—
412#103—44no—
413#103—44no—
414#102—28no—
415#102—28no—
416#102—44no—
417#102—44no—
418#102—44no—
419#102—44no—
420#102—44no—
421#102—44no—
422#97—24no—
423#97—42no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446C24—22no—
447C24—22no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—24no—
453C2×Q8—22no—
454C2×Q8—24no—
455C2×Q8—41no—
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—82no—
460C2×Q8—82no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—24no—
468C2×D4—24no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—44no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—82no—
488C2×D4—82no—
489C2×D4—82no—
490C2×D4—82no—
491C2×D4—82no—
492C2×D4—82no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4C2×#4114yes—
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—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—44no—
600C42—44no—
601C42—82no—
602C42—82no—
603C42#13714yes—
604C42C2×#4112yes—
605C42H(5,𝔽2)14yes—
606C422−1+412yes—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23C23×D411yescharacteristic
622Q8—44no—
623Q8—48no—
624Q8—44no—
625Q8—44no—
626Q8—164no—
627D4—44no—
628D4—44no—
629D4—44no—
630D4—44no—
631D4—88no—
632D4—84no—
633D4—84no—
634D4—84no—
635D4—84no—
636D4—164no—
637D4—164no—
638D4—164no—
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—24no—
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—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—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—42no—
689C2×C4—164no—
690C2×C4—164no—
691C2×C4#53512yes—
692C2×C4#53512yes—
693C2×C4#54812yes—
694C2×C4#54812yes—
695C2×C4C2×#13712yes—
696C2×C4C2×#13712yes—
697C2×C4C2×#13712yes—
698C2×C4#57512yes—
699C2×C4#57612yes—
700C2×C4#57612yes—
701C2×C4C22×#4111yescharacteristic
702C2×C4C2×H(5,𝔽2)12yes—
703C2×C4C2×2−1+411yescharacteristic
704C2×C4#58512yes—
705C8—42no—
706C8—42no—
707C8—42no—
708C8—42no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22C2×#53511yescharacteristic
724C22C2×#54811yescharacteristic
725C22#350011yescharacteristic
726C22C22×#13711yescharacteristic
727C22C2×#57511yescharacteristic
728C22C2×#57611yescharacteristic
729C22C2×#57611yescharacteristic
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—44no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4#337014yes—
749C4#337514yes—
750C4#351712yes—
751C4#355412yes—
752C2—162no—
753C2—162no—
754C2#3917411yescharacteristic
755C2#6080411yescharacteristic
756C2#6081711yescharacteristic
757C2C2×#337511yescharacteristic
758C2C2×#337011yescharacteristic
759C2#6253711yescharacteristic
760C2#6254611yescharacteristic
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. #36563C2semidirect
3. #33230C2semidirect
4. #32109C2semidirect
5. #32102C2semidirect
6. #30219C2semidirect
8. #29193C2semidirect
9. #29192C2semidirect
10. #28750C2semidirect
11. #28750C2semidirect
13. #28508C2semidirect
14. #25660C2semidirect
15. #25285C2semidirect
16. #22221C2semidirect
17. #22220C2semidirect
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
#365632
#332302
#321092
#321022
#302192
#302182
#291932
#291922
#287502
#287492
#285082
#256602
#252852
#222212
#222202
#216462
#213402