Finite Group Databasebeta

← #6490751

Subgroups — #6490751

Order 512.

1714 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#36631C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#33112C214yesmaximal, complemented
4#30065C214yesmaximal, complemented
5#29261C214yesmaximal, complemented
6#29207C211yesmaximal, characteristic, complemented
7#29207C211yesmaximal, characteristic, complemented
8#28750C211yesmaximal, characteristic, complemented
9#28750C211yesmaximal, characteristic, complemented
10#27908C211yesmaximal, characteristic, complemented
11#27907C211yesmaximal, characteristic
12#25905C211yesmaximal, characteristic
13#25904C211yesmaximal, characteristic, complemented
14#24068C214yesmaximal, complemented
15#22221C211yesmaximal, characteristic, complemented
16#22220C211yesmaximal, characteristic, complemented
17#22219C211yesmaximal, characteristic, complemented
18#22218C211yesmaximal, characteristic, complemented
19#20777C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#529C2212yes—
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#485C2214yes—
27C2×#482C2214yes—
28#2861C2212yes—
29#2860—21no—
30#2860—21no—
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2854—21no—
36#2854—21no—
37#2850C2212yes—
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#125C2212yes—
45C4×#125C2212yes—
46C4×#124C2212yes—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8—21no—
49C4×C22⋊Q8—21no—
50C4×C22⋊Q8—21no—
51C4×#121—21no—
52C4×#121—21no—
53#2296C2212yes—
54#2295—21no—
55#2295—21no—
56#2293—21no—
57#2293—21no—
58#2285C2211yescharacteristic
59C42×Q8—21no—
60C42×Q8—21no—
61C4×#117C2212yes—
62C2×C43C2211yescharacteristic
63#2097C2214yes—
64#2092C2214yes—
65#2059C2214yes—
66#2059C2214yes—
67#2042C2214yes—
68#2042C2214yes—
69#2026C2214yes—
70#2025C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1983C2214yes—
76#1983C2214yes—
77#1957C2211yescharacteristic
78#1957C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1949C2214yes—
82#1949C2214yes—
83#1928C2214yes—
84#1927C2214yes—
85#1926C2214yes—
86#1925C2214yes—
87#1875C2214yes—
88#1873C2214yes—
89#1865C2214yes—
90#1864C2214yes—
91#1863C2214yes—
92#1862C2214yes—
93#1845C2211yescharacteristic
94#1845C2211yescharacteristic
95#1844C2211yescharacteristic
96#1844C2211yescharacteristic
97#1839C2212yes—
98#1831C2212yes—
99#1817C2212yes—
100#1797C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1744C2211yescharacteristic
106#1744C2211yescharacteristic
107#533—24no—
108#533—24no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#127C2311yescharacteristic
113#529—48no—
114C2×#126—22no—
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×#124C2312yes—
118C2×#124C2312yes—
119C2×#123—22no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8C2312yes—
123C2×#121—21no—
124C2×#121—21no—
125C2×#121C2312yes—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8C2312yes—
133C2×C4×D4—22no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4C2312yes—
137C2×#117—22no—
138C2×#117C2312yes—
139C2×#117C2312yes—
140C22×C42D411yescharacteristic
141C22×C42D411yescharacteristic
142C22×C42D411yescharacteristic
143C22×C42D411yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#487—24no—
148#487—24no—
149#486—24no—
150#486—24no—
151#485—48no—
152#482—48no—
153C2×#107C2314yes—
154C2×#107C2314yes—
155C2×#106C2314yes—
156C2×#106C2314yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2312yes—
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391C2312yes—
178#391C2312yes—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390C2312yes—
188#390C2312yes—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#388—22no—
206#388—22no—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#386—22no—
218#386—22no—
219#385—22no—
220#385—22no—
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#382C2312yes—
226#382C2312yes—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—21no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—22no—
244C4×C4⋊C4—22no—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C22⋊C4—22no—
250C4×C22⋊C4—22no—
251C4×C22⋊C4—22no—
252C4×C22⋊C4—22no—
253#376—22no—
254C43—22no—
255C43D411yescharacteristic
256C43D411yescharacteristic
257C43D411yescharacteristic
258C43D411yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8C2412yes—
278C22×D4—22no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C2412yes—
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8#4114yes—
299C4⋊Q8#4114yes—
300#127—22no—
301#127—22no—
302#126—48no—
303#125—24no—
304#125—24no—
305#125—44no—
306#124—28no—
307#124—44no—
308#123—48no—
309C22⋊Q8—28no—
310C22⋊Q8—28no—
311C22⋊Q8—28no—
312#121—28no—
313#121—44no—
314#121—44no—
315C4×Q8—28no—
316C4×Q8—28no—
317C4×Q8—44no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×D4—44no—
323C4×D4—44no—
324C4×D4—48no—
325C4×D4—44no—
326#117—24no—
327#117—24no—
328#117—28no—
329#117—48no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C42—22no—
372C2×C42—24no—
373C2×C42—24no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—42no—
384C2×C42—42no—
385C2×C42—42no—
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×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414#107—44no—
415#107—44no—
416#106—44no—
417#106—44no—
418C4⋊C8—24no—
419C4⋊C8—24no—
420#103—28no—
421#103—28no—
422#103—44no—
423#103—44no—
424#102—28no—
425#102—28no—
426#102—44no—
427#102—44no—
428C22⋊C8—24no—
429C22⋊C8—24no—
430C4×C8—22no—
431C4×C8—22no—
432C4×C8—22no—
433C4×C8—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#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95—22no—
450C24—22no—
451C24C22×D411yescharacteristic
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
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×C4H(5,𝔽2)14yes—
520C22×C4C2511yescharacteristic
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—
532C2×C8—22no—
533C4⋊C4—28no—
534C4⋊C4—28no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4C2×#4114yes—
562C4⋊C4C2×#4114yes—
563C22⋊C4—28no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C42—28no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—21no—
593C42—21no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—41no—
598C42—41no—
599C42—41no—
600C42—41no—
601C42—41no—
602C42—41no—
603C42—44no—
604C42—44no—
605C42—44no—
606C42—44no—
607C42—44no—
608C42—44no—
609C42#13514yes—
610C42C2×#4112yes—
611C42C2×#4112yes—
612C422−1+414yes—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23#53512yes—
627C23C23×D411yescharacteristic
628Q8—48no—
629Q8—48no—
630Q8—44no—
631Q8—44no—
632D4—44no—
633D4—44no—
634D4—88no—
635D4—84no—
636D4—84no—
637C2×C4—21no—
638C2×C4—21no—
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—24no—
648C2×C4—24no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—24no—
652C2×C4—24no—
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—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—44no—
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—88no—
695C2×C4#52512yes—
696C2×C4#52512yes—
697C2×C4#53512yes—
698C2×C4#53612yes—
699C2×C4#53612yes—
700C2×C4C2×#13512yes—
701C2×C4C2×#13512yes—
702C2×C4C2×#13512yes—
703C2×C4#57512yes—
704C2×C4#57512yes—
705C2×C4#57512yes—
706C2×C4C22×#4111yescharacteristic
707C2×C4C22×#4111yescharacteristic
708C2×C4C2×H(5,𝔽2)12yes—
709C2×C4C2×2−1+412yes—
710C8—42no—
711C8—42no—
712C8—42no—
713C8—42no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22C2×#52511yescharacteristic
729C22C2×#53511yescharacteristic
730C22C2×#53611yescharacteristic
731C22C22×#13511yescharacteristic
732C22C2×#57511yescharacteristic
733C22C2×#57511yescharacteristic
734C22C2×#57511yescharacteristic
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—22no—
740C4—41no—
741C4—41no—
742C4—44no—
743C4—41no—
744C4—44no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4#310214yes—
754C4#311114yes—
755C4#352212yes—
756C4#352212yes—
757C2—42no—
758C2—162no—
759C2#3524911yescharacteristic
760C2C2×#310211yescharacteristic
761C2C2×#311111yescharacteristic
762C2#6078211yescharacteristic
763C2#6080111yescharacteristic
764C2#6084611yescharacteristic
765C2#6086211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#533C2semidirect
3. #33112C2semidirect
4. #30065C2semidirect
5. #29261C2semidirect
6. #29207C2semidirect
7. #29207C2semidirect
8. #28750C2semidirect
9. #28750C2semidirect
10. #27908C2semidirect
13. #25904C2semidirect
14. #24068C2semidirect
15. #22221C2semidirect
16. #22220C2semidirect
17. #22219C2semidirect
18. #22218C2semidirect
19. #20777C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#5332
#331122
#300652
#292612
#292072
#287502
#279082
#279072
#259052
#259042
#240682
#222212
#222202
#222192
#222182
#207772