Finite Group Databasebeta

← #6469846

Subgroups — #6469846

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29582C214yesmaximal, complemented
4#29220C211yesmaximal, characteristic, complemented
5#28718C211yesmaximal, characteristic, complemented
6#28688C211yesmaximal, characteristic, complemented
7#28514C211yesmaximal, characteristic, complemented
8#27877C211yesmaximal, characteristic, complemented
9#27875C211yesmaximal, characteristic, complemented
10#25673C211yesmaximal, characteristic
11#25671C211yesmaximal, characteristic
12#24706C214yesmaximal, complemented
13#23764C214yesmaximal, complemented
14#21727C211yesmaximal, characteristic, complemented
15#21726C211yesmaximal, characteristic, complemented
16#21646C211yesmaximal, characteristic, complemented
17#21646C211yesmaximal, characteristic, complemented
18#21617C214yesmaximal, complemented
19C4×#432C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×C4×#41—21no—
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#478C2214yes—
27C2×#432C2212yes—
28#2864C2212yes—
29#2863—21no—
30#2863—21no—
31#2862C2211yescharacteristic
32#2859C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2855—21no—
36#2855—21no—
37#2851C2212yes—
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#125C2212yes—
45C4×#124—21no—
46C4×#124—21no—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8—21no—
49C4×#121—21no—
50C4×#121—21no—
51C4×#121C2212yes—
52C4×#121C2212yes—
53#2297—21no—
54#2295—21no—
55#2295—21no—
56#2293—21no—
57#2293C2212yes—
58#2285C2211yescharacteristic
59#2283C2212yes—
60C42×D4—21no—
61C42×D4C2212yes—
62C2×C43C2211yescharacteristic
63#2101C2214yes—
64#2092C2214yes—
65#2085C2214yes—
66#2077C2214yes—
67#2075C2214yes—
68#2073C2214yes—
69#2034C2214yes—
70#2033C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1989C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1949C2214yes—
81#1944C2214yes—
82#1943C2214yes—
83#1936C2214yes—
84#1936C2214yes—
85#1920C2214yes—
86#1918C2214yes—
87#1864C2214yes—
88#1862C2214yes—
89#1845C2211yescharacteristic
90#1844C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1804C2214yes—
94#1776C2214yes—
95#1776C2214yes—
96#1767C2212yes—
97C4×C4⋊C8C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99#1762C2214yes—
100#1762C2214yes—
101#1749C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1745C2212yes—
105C4×#97C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×#41—21no—
108#532—24no—
109#532—24no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#125C2312yes—
115C2×#124—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×#121—21no—
119C2×#121C2312yes—
120C4×#41—48no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2312yes—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C2×#117—22no—
137C22×C42—22no—
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42D411yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#500—24no—
146#498—24no—
147#486—24no—
148#486—24no—
149#478—216no—
150#432—28no—
151C2×#107C2314yes—
152C2×#107C2314yes—
153C2×#106C2314yes—
154C2×#106C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×#97C2312yes—
166C2×#97C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390C2312yes—
186#390C2312yes—
187#389—21no—
188#389—21no—
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#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#386—22no—
216#386—22no—
217#385—22no—
218#385—22no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#383—22no—
224#382—21no—
225#382C2312yes—
226#382C2312yes—
227#382C2312yes—
228#381—22no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380C2312yes—
233#380C2312yes—
234#380C2312yes—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
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—
253C4×C22⋊C4—22no—
254#376—22no—
255C43D411yescharacteristic
256C43D411yescharacteristic
257C43D411yescharacteristic
258C43D411yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C2×#41—28no—
276C2×#41—44no—
277C2×#41—44no—
278C22×Q8—21no—
279C22×Q8—21no—
280C22×Q8—21no—
281C22×D4—22no—
282C22×D4—21no—
283C22×D4—21no—
284C22×D4C2412yes—
285C23×C4—22no—
286C23×C4—22no—
287C23×C4—22no—
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2×D411yescharacteristic
292C23×C4C2×D411yescharacteristic
293C23×C4C2×D411yescharacteristic
294C23×C4C2411yescharacteristic
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8#4114yes—
305#127—22no—
306#127#4114yes—
307#125—28no—
308#124—28no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311#121—28no—
312#121—24no—
313#121—24no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×D4—28no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—28no—
322C4×D4—28no—
323C4×D4—44no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—48no—
328C4×D4—48no—
329#117—48no—
330#117—44no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
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⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—22no—
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×C42—24no—
369C2×C42—22no—
370C2×C42—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—48no—
382C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410#107—28no—
411#107—28no—
412#106—28no—
413#106—28no—
414C4⋊C8—24no—
415C4⋊C8#4118yescomplemented
416#102—28no—
417#102—28no—
418#102—28no—
419#102—28no—
420C22⋊C8—24no—
421C22⋊C8—24no—
422#97—24no—
423#97—24no—
424#97—24no—
425#97—24no—
426C4×C8—22no—
427C4×C8—22no—
428C4×C8—22no—
429C4×C8—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#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
452#41—432no—
453C2×Q8—24no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×D4—28no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—24no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4C2×#4114yes—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—28no—
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—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4AGL(1,ℤ/8)14yes—
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×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C2511yescharacteristic
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—
537C2×C8C2×#4114yes—
538C2×C8C2×#4114yes—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—24no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—44no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4C2×#4114yes—
568C22⋊C4—28no—
569C22⋊C4—28no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
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—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—21no—
597C42—21no—
598C42—22no—
599C42—41no—
600C42—41no—
601C42—41no—
602C42—41no—
603C42—41no—
604C42—41no—
605C42—44no—
606C42—44no—
607C42—44no—
608C42—44no—
609C42—44no—
610C42C2×#4112yes—
611C42C2×#4112yes—
612C42C2×#4114yes—
613C422−1+414yes—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23—21no—
620C23—24no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23C2×AGL(1,ℤ/8)12yes—
628C23C23×D411yescharacteristic
629Q8—48no—
630Q8—84no—
631Q8—84no—
632D4—28no—
633D4—416no—
634D4—48no—
635D4—48no—
636D4—44no—
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—28no—
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—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—416no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
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—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—88no—
696C2×C4—88no—
697C2×C4#52512yes—
698C2×C4#52512yes—
699C2×C4#53212yes—
700C2×C4#53212yes—
701C2×C4#53612yes—
702C2×C4#53612yes—
703C2×C4C2×AGL(1,ℤ/8)12yes—
704C2×C4C2×AGL(1,ℤ/8)12yes—
705C2×C4#57512yes—
706C2×C4#57512yes—
707C2×C4#57512yes—
708C2×C4C22×#4111yescharacteristic
709C2×C4C22×#4111yescharacteristic
710C2×C4C22×#4112yes—
711C2×C4C2×2−1+412yes—
712C8—24no—
713C8—24no—
714C8—42no—
715C8—42no—
716C22—22no—
717C22—22no—
718C22—22no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—44no—
724C22—44no—
725C22—44no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22C2×#52511yescharacteristic
731C22C2×#53211yescharacteristic
732C22C2×#53611yescharacteristic
733C22C22×AGL(1,ℤ/8)11yescharacteristic
734C22C2×#57511yescharacteristic
735C22C2×#57511yescharacteristic
736C22C2×#57511yescharacteristic
737C4—22no—
738C4—22no—
739C4—22no—
740C4—22no—
741C4—22no—
742C4—41no—
743C4—41no—
744C4—44no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—41no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4#311114yes—
756C4#311414yes—
757C4#348212yes—
758C4#348212yes—
759C2—42no—
760C2—84no—
761C2#3497911yescharacteristic
762C2C2×#311411yescharacteristic
763C2C2×#311111yescharacteristic
764C2#6069311yescharacteristic
765C2#6070011yescharacteristic
766C2#6086211yescharacteristic
767C2#6086411yescharacteristic
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. #36635C2semidirect
3. #29582C2semidirect
4. #29220C2semidirect
5. #28718C2semidirect
6. #28688C2semidirect
7. #28514C2semidirect
8. #27877C2semidirect
9. #27875C2semidirect
12. #24706C2semidirect
13. #23764C2semidirect
14. #21727C2semidirect
15. #21726C2semidirect
16. #21646C2semidirect
17. #21646C2semidirect
18. #21617C2semidirect
19. C4×#432C2semidirect
415. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#366352
C4×#5322
#295822
#292202
#287182
#286882
#285142
#278772
#278752
#256732
#256712
#247062
#237642
#217272
#217262
#216462
#216172
C4×#4322