Finite Group Databasebeta

← #6468285

Subgroups — #6468285

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29220C211yesmaximal, characteristic, complemented
4#29207C211yesmaximal, characteristic, complemented
5#29192C211yesmaximal, characteristic, complemented
6#29190C211yesmaximal, characteristic, complemented
7#29178C211yesmaximal, characteristic, complemented
8#29176C211yesmaximal, characteristic, complemented
9#29167C214yesmaximal, complemented
10#29143C211yesmaximal, characteristic, complemented
11#28514C211yesmaximal, characteristic, complemented
12#24841C214yesmaximal, complemented
13#23784C214yesmaximal, complemented
14#21617C214yesmaximal, complemented
15#21410C211yesmaximal, characteristic
16#21410C211yesmaximal, characteristic
17#20824C211yesmaximal, characteristic, complemented
18#20824C211yesmaximal, characteristic, complemented
19C4×#423C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×C4×#41—21no—
22C2×#478C2214yes—
23C2×#423C2212yes—
24C2×C4×M16C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26#2864—21no—
27#2864—21no—
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2851—21no—
33#2851—21no—
34#2851C2212yes—
35#2851C2212yes—
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×#127C2211yescharacteristic
39C4×#127C2211yescharacteristic
40C4×#125—21no—
41C4×#125—21no—
42C4×#125C2212yes—
43C4×#125C2212yes—
44C4×#124—21no—
45C4×#124—21no—
46C4×#121—21no—
47C4×#121—21no—
48C4×#121C2212yes—
49C4×#121C2212yes—
50#2297—21no—
51#2296—21no—
52#2296—21no—
53#2293—21no—
54#2293C2212yes—
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C42×D4—21no—
59C42×D4C2212yes—
60C2×C43C2211yescharacteristic
61#2101C2214yes—
62#2092C2214yes—
63#2088C2214yes—
64#2086C2214yes—
65#2067C2214yes—
66#2065C2214yes—
67#2059C2214yes—
68#2025C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1989C2214yes—
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1949C2214yes—
79#1920C2214yes—
80#1918C2214yes—
81#1895C2214yes—
82#1895C2214yes—
83#1864C2214yes—
84#1862C2214yes—
85#1845C2211yescharacteristic
86#1845C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1809C2214yes—
94#1809C2214yes—
95#1804C2214yes—
96#1776C2214yes—
97#1776C2214yes—
98#1767C2212yes—
99#1767C2212yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1762C2214yes—
105#1762C2214yes—
106#1728C2212yes—
107C22×#41—21no—
108C22×M16C2312yes—
109C22×M16C2312yes—
110#532—24no—
111#532Q818yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#125—21no—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×#124—21no—
119C2×#121—21no—
120C2×#121C2312yes—
121C4×#41—48no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117—21no—
137C2×#117—22no—
138C22×C42—22no—
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42D411yescharacteristic
142C22×C42D411yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146#478—216no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151#423—28no—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C4×M16—24no—
161C4×M16—24no—
162C2×#97C2312yes—
163C2×#97C2312yes—
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168#391—21no—
169#391—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#386—22no—
209#386—22no—
210#385—22no—
211#385—22no—
212#385—22no—
213#385—22no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#382—21no—
221#382—21no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382C2312yes—
226#382C2312yes—
227#380—21no—
228#380—21no—
229#380—21no—
230#380C2312yes—
231#380C2312yes—
232#380C2312yes—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C22⋊C4—22no—
248C4×C22⋊C4—22no—
249C4×C22⋊C4—22no—
250C4×C22⋊C4—22no—
251C4×C22⋊C4—22no—
252C4×C22⋊C4—22no—
253C43D411yescharacteristic
254C43D411yescharacteristic
255C43D411yescharacteristic
256C43D411yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C2×#41—28no—
274C2×#41—44no—
275C2×#41—44no—
276C22×Q8—21no—
277C22×D4—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2412yes—
281C23×C4—22no—
282C23×C4—22no—
283C23×C4—22no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2411yescharacteristic
291C2×M16—24no—
292C2×M16—24no—
293C2×M16—24no—
294C2×M16—24no—
295C2×M16—42no—
296C2×M16—42no—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C4⋊Q8—22no—
302C4⋊Q8C2×Q814yes—
303#127—22no—
304#127C2×Q814yes—
305#125—28no—
306#125—44no—
307#125—44no—
308#125—44no—
309#124—28no—
310#121—28no—
311#121—24no—
312#121C2×Q818yes—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×D4—28no—
316C4×D4—24no—
317C4×D4—28no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—48no—
323C4×D4—48no—
324C4×D4—44no—
325C4×D4C2×Q818yes—
326#117—48no—
327#117—44no—
328#117—44no—
329#117—44no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
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—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
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⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
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—24no—
372C2×C42—24no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—42no—
379C2×C42—42no—
380C2×C42—42no—
381C2×C42—42no—
382C2×C42—48no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
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×Q814yes—
406C2×C42#4114yes—
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413#107—28no—
414#107—44no—
415#106—28no—
416#106—44no—
417C4⋊C8—24no—
418C4⋊C8—24no—
419C4⋊C8—24no—
420C4⋊C8#4118yescomplemented
421#102—28no—
422#102—28no—
423#102—44no—
424#102—44no—
425#97—24no—
426#97—24no—
427C4×C8—22no—
428C4×C8—22no—
429C4×C8—22no—
430C4×C8—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#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—
449C24—22no—
450C24C22×D411yescharacteristic
451#41—432no—
452C2×Q8—24no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×D4—28no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4C22×Q814yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—28no—
476C22×C4—22no—
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—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4C22⋊Q814yes—
503C22×C4C22⋊Q814yes—
504C22×C4C22×D411yescharacteristic
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×Q812yes—
518C22×C4C2×#4112yes—
519C22×C4C2511yescharacteristic
520M16—48no—
521M16—48no—
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×C8C2×#4114yes—
533C2×C8C2×#4114yes—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—44no—
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⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4C22×Q814yes—
565C22⋊C4—28no—
566C22⋊C4—28no—
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—22no—
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—41no—
591C42—41no—
592C42—41no—
593C42—41no—
594C42—41no—
595C42—41no—
596C42—41no—
597C42—41no—
598C42—44no—
599C42—44no—
600C42—44no—
601C42—44no—
602C42—44no—
603C42C22⋊Q814yes—
604C42C22⋊Q814yes—
605C42AGL(1,ℤ/8)14yes—
606C42C22×Q812yes—
607C42C22×Q812yes—
608C42C2×#4112yes—
609C42C2×#4112yes—
610C42C2×#4114yes—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—24no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23C2×C22⋊Q812yes—
625C23C23×D411yescharacteristic
626Q8—48no—
627D4—28no—
628D4—416no—
629D4—48no—
630D4—48no—
631D4—44no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
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—28no—
647C2×C4—28no—
648C2×C4—24no—
649C2×C4—24no—
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—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—416no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
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—88no—
689C2×C4—88no—
690C2×C4C2×C22⋊Q812yes—
691C2×C4C2×C22⋊Q812yes—
692C2×C4C2×C22⋊Q812yes—
693C2×C4C2×C22⋊Q812yes—
694C2×C4C2×C22⋊Q812yes—
695C2×C4#53212yes—
696C2×C4#53212yes—
697C2×C4C2×AGL(1,ℤ/8)12yes—
698C2×C4C2×AGL(1,ℤ/8)12yes—
699C2×C4C2×AGL(1,ℤ/8)12yes—
700C2×C4#57512yes—
701C2×C4#57512yes—
702C2×C4#57512yes—
703C2×C4C23×Q811yescharacteristic
704C2×C4C22×#4111yescharacteristic
705C2×C4C22×#4112yes—
706C2×C4#58512yes—
707C8—24no—
708C8—24no—
709C8—42no—
710C8—42no—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22C22×C22⋊Q811yescharacteristic
726C22C2×#53211yescharacteristic
727C22#344911yescharacteristic
728C22C22×AGL(1,ℤ/8)11yescharacteristic
729C22C2×#57511yescharacteristic
730C22C2×#57511yescharacteristic
731C22C2×#57511yescharacteristic
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—44no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4#307714yes—
751C4#308014yes—
752C4#347512yes—
753C4Q8×#4112yes—
754C2—42no—
755C2—84no—
756C2#3494711yescharacteristic
757C2C2×#307711yescharacteristic
758C2C2×#308011yescharacteristic
759C2#6069311yescharacteristic
760C2#6070011yescharacteristic
761C2#6096511yescharacteristic
762C2#6097211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
1. #36640C2semidirect
3. #29220C2semidirect
4. #29207C2semidirect
5. #29192C2semidirect
6. #29190C2semidirect
7. #29178C2semidirect
8. #29176C2semidirect
9. #29167C2semidirect
10. #29143C2semidirect
11. #28514C2semidirect
12. #24841C2semidirect
13. #23784C2semidirect
14. #21617C2semidirect
17. #20824C2semidirect
18. #20824C2semidirect
19. C4×#423C2semidirect
420. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#366402
C4×#5322
#292202
#292072
#291922
#291902
#291782
#291762
#291672
#291432
#285142
#248412
#237842
#216172
#214102
#208242
C4×#4232