Finite Group Databasebeta

← #6436025

Subgroups — #6436025

Order 512.

1639 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#36645C211yesmaximal, characteristic, complemented
2#36638C211yesmaximal, characteristic
3#30345C212yesmaximal, complemented
4#30344C212yesmaximal, complemented
5#30326C212yesmaximal, complemented
6#30325C212yesmaximal
7#30309C211yesmaximal, characteristic, complemented
8#30267C211yesmaximal, characteristic, complemented
9#30226C211yesmaximal, characteristic, complemented
10#30216C211yesmaximal, characteristic, complemented
11#29852C212yesmaximal, complemented
12#29815C212yesmaximal
13#23679C212yesmaximal, complemented
14#23677C212yesmaximal, complemented
15#23571C212yesmaximal, complemented
16#23568C212yesmaximal
17C4×#497C212yesmaximal, complemented
18C4×#491C212yesmaximal, complemented
19C4×#432C211yesmaximal, characteristic, complemented
20C2×#533C2212yes—
21C2×#497C2212yes—
22C2×#491C2212yes—
23C2×C4×Q16C2214yes—
24C2×C4×SD16C2214yes—
25C2×#432C2212yes—
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2866—21no—
29#2865—21no—
30#2864C2211yescharacteristic
31#2863C2211yescharacteristic
32#2862C2212yes—
33#2861C2211yescharacteristic
34#2860C2211yescharacteristic
35#2859—21no—
36#2859—21no—
37#2857—21no—
38#2857—21no—
39#2857C2212yes—
40#2857C2212yes—
41#2855C2211yescharacteristic
42#2854C2211yescharacteristic
43#2853—21no—
44#2852—21no—
45#2851C2211yescharacteristic
46#2850C2211yescharacteristic
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8—21no—
49C4×C4⋊Q8C2212yes—
50C4×C4⋊Q8C2212yes—
51C4×#125C2211yescharacteristic
52C4×#126—21no—
53C4×#124C2211yescharacteristic
54C4×C22⋊Q8C2211yescharacteristic
55C4×#123—21no—
56C4×#121C2211yescharacteristic
57#2298—21no—
58#2295C2211yescharacteristic
59#2293C2211yescharacteristic
60#2290—21no—
61#2285—21no—
62#2285—21no—
63C2×C43C2211yescharacteristic
64#2084C2212yes—
65#2083C2212yes—
66#2072C2212yes—
67#2071C2212yes—
68#2068C2214yes—
69#2066C2214yes—
70#2058C2212yes—
71#2057C2212yes—
72#2056C2214yes—
73#2055C2214yes—
74#2043C2212yes—
75#2043C2212yes—
76#2043C2212yes—
77#2043C2212yes—
78#2036C2214yes—
79#2035C2214yes—
80#2024C2212yes—
81#2023C2212yes—
82#2016C2212yes—
83#2016C2212yes—
84#2013C2212yes—
85#2013C2212yes—
86#2001C2214yes—
87#2000C2214yes—
88#1992C2212yes—
89#1973C2214yes—
90#1971C2214yes—
91#1960C2212yes—
92#1960C2212yes—
93#1959C2212yes—
94#1935C2214yes—
95#1935C2214yes—
96#1847C2212yes—
97#1836C2212yes—
98#1767C2212yes—
99C4×C4⋊C8C2212yes—
100C4×#103C2214yes—
101C4×#103C2214yes—
102#1749C2212yes—
103C4×C22⋊C8C2212yes—
104#1745C2211yescharacteristic
105#1745C2211yescharacteristic
106C4×#97C2212yes—
107C22×Q16C2314yes—
108C22×SD16C2314yes—
109#533—48no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×#126—21no—
117C2×#125—22no—
118C2×#125C2311yescharacteristic
119C2×#124C2311yescharacteristic
120C2×#123—21no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8C2311yescharacteristic
123C2×C22⋊Q8C2314yes—
124C2×#121C2311yescharacteristic
125C2×C4×Q8—22no—
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C2×#117—21no—
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#497—28no—
140#491—28no—
141C4×Q16—48no—
142C4×SD16—48no—
143#432—28no—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2314yes—
156C2×#97C2312yes—
157C2×#97C2314yes—
158C2×#97C2312yes—
159C2×C4×C8C2314yes—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#391C2311yescharacteristic
165#391C2311yescharacteristic
166#391C2311yescharacteristic
167#391C2311yescharacteristic
168#390C2311yescharacteristic
169#390C2311yescharacteristic
170#390C2311yescharacteristic
171#390C2311yescharacteristic
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2311yescharacteristic
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#389C2311yescharacteristic
184#389C2311yescharacteristic
185#389C2311yescharacteristic
186#389C2311yescharacteristic
187#389C2311yescharacteristic
188#388—21no—
189#388—21no—
190#388—21no—
191#388—21no—
192#388—21no—
193#388—21no—
194#388—21no—
195#388—21no—
196#388—21no—
197#388—21no—
198#388—21no—
199#388—21no—
200#388—21no—
201#388—21no—
202#388—21no—
203#388—21no—
204#387—21no—
205#387—21no—
206#387—21no—
207#387—21no—
208#387—21no—
209#387—21no—
210#387—21no—
211#387—21no—
212#387—21no—
213#387—21no—
214#387—21no—
215#387—21no—
216#387—21no—
217#387—21no—
218#387—21no—
219#387—21no—
220#386—21no—
221#386—21no—
222#386—21no—
223#386—21no—
224#385—21no—
225#385—21no—
226#385—21no—
227#385—21no—
228#384—21no—
229#384—21no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384C2312yes—
235#384C2312yes—
236#383—21no—
237#382—22no—
238#382—22no—
239#382C2311yescharacteristic
240#382C2311yescharacteristic
241#381—21no—
242#380—22no—
243#380—22no—
244#380—22no—
245#380—22no—
246#380C2311yescharacteristic
247#380C2311yescharacteristic
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—22no—
253C4×C4⋊C4—22no—
254C4×C4⋊C4—22no—
255C4×C4⋊C4—22no—
256C4×C4⋊C4C2311yescharacteristic
257C4×C4⋊C4C2311yescharacteristic
258C4×C22⋊C4—21no—
259C4×C22⋊C4—21no—
260C4×C22⋊C4—21no—
261C4×C22⋊C4—21no—
262C4×C22⋊C4—21no—
263#376—21no—
264C43D411yescharacteristic
265C43D411yescharacteristic
266C43D411yescharacteristic
267C43D411yescharacteristic
268C43C2311yescharacteristic
269C43C2311yescharacteristic
270C43C2311yescharacteristic
271C43C2311yescharacteristic
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#340C2314yes—
276#336C2312yes—
277#336C2312yes—
278#336C2312yes—
279#336C2312yes—
280C22×Q8—22no—
281C22×Q8C2411yescharacteristic
282C22×Q8C2414yes—
283C22×D4—21no—
284C22×D4C2411yescharacteristic
285C23×C4—21no—
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2×D411yescharacteristic
292C23×C4C2411yescharacteristic
293C2×Q16—28no—
294C2×Q16—44no—
295C2×Q16—44no—
296C2×SD16—28no—
297C2×SD16—44no—
298C2×SD16—44no—
299C22×C8C2414yes—
300C22×C8C2414yes—
301C4⋊Q8—24no—
302C4⋊Q8—24no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308#126—44no—
309#125—24no—
310#125—48no—
311#124—24no—
312#123—44no—
313C22⋊Q8—42no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C22⋊Q8—48no—
317#121—42no—
318C4×Q8—28no—
319C4×Q8—28no—
320C4×Q8—24no—
321C4×Q8—48no—
322C4×D4—24no—
323C4×D4—44no—
324#117—48no—
325#117—44no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
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—22no—
344C2×C4⋊C4—22no—
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⋊C4C2411yescharacteristic
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C22⋊C4—24no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C42—24no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42—24no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—42no—
383C2×C42—42no—
384C2×C42—42no—
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×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413#107—44no—
414#106—44no—
415C4⋊C8—28no—
416#103—28no—
417#103—28no—
418#103—44no—
419#103—44no—
420#103—44no—
421#103—44no—
422#103—44no—
423#102—44no—
424C22⋊C8—28no—
425#97—24no—
426#97—44no—
427#97#4118yes—
428C4×C8—28no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—24no—
444#95—24no—
445#95—24no—
446#95—24no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—21no—
457C24—21no—
458C24C22×D411yescharacteristic
459C2×Q8—24no—
460C2×Q8—22no—
461C2×Q8—24no—
462C2×Q8—22no—
463C2×Q8—24no—
464C2×Q8—24no—
465C2×Q8—24no—
466C2×Q8—24no—
467C2×Q8—22no—
468C2×Q8—44no—
469C2×Q8—44no—
470C2×Q8—44no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
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—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—41no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4#13714yes—
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×C4C2511yescharacteristic
524Q16—416no—
525SD16—416no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8C2×#4118yes—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—48no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—48no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—44no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—42no—
568C22⋊C4—42no—
569C22⋊C4—48no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—42no—
577C22⋊C4—42no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—42no—
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—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—41no—
601C42—41no—
602C42—41no—
603C42—41no—
604C42—41no—
605C42—41no—
606C42—48no—
607C42—42no—
608C42—42no—
609C42—42no—
610C42#13714yes—
611C42C2×#4114yes—
612C42H(5,𝔽2)14yes—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23—41no—
620C23—41no—
621C23—41no—
622C23—41no—
623C23—41no—
624C23—41no—
625C23—41no—
626C23C2×#13712yes—
627C23C23×D411yescharacteristic
628Q8—44no—
629Q8—44no—
630Q8—44no—
631Q8—44no—
632Q8—44no—
633Q8—88no—
634D4—44no—
635D4—84no—
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—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—28no—
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—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—44no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—84no—
699C2×C4#53212yes—
700C2×C4#53212yes—
701C2×C4#53512yes—
702C2×C4#53512yes—
703C2×C4C2×#13712yes—
704C2×C4C2×#13712yes—
705C2×C4C2×#13712yes—
706C2×C4C2×#13712yes—
707C2×C4C2×#13712yes—
708C2×C4#57512yes—
709C2×C4#57512yes—
710C2×C4C22×#4112yes—
711C2×C4C2×H(5,𝔽2)12yes—
712C2×C4#58512yes—
713C8—28no—
714C8—44no—
715C22—22no—
716C22—22no—
717C22—22no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—81no—
723C22—81no—
724C22—81no—
725C22—81no—
726C22—81no—
727C22—81no—
728C22—81no—
729C22C2×#53211yescharacteristic
730C22C2×#53511yescharacteristic
731C22#344911yescharacteristic
732C22C22×#13711yescharacteristic
733C22C22×#13711yescharacteristic
734C22C2×#57511yescharacteristic
735C22C2×#57511yescharacteristic
736C4—22no—
737C4—22no—
738C4—22no—
739C4—22no—
740C4—22no—
741C4—22no—
742C4—22no—
743C4—22no—
744C4—41no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—81no—
749C4—81no—
750C4—81no—
751C4—81no—
752C4—81no—
753C4—84no—
754C4—81no—
755C4—81no—
756C4—84no—
757C4#321814yes—
758C4#346014yes—
759C2—42no—
760C2—161no—
761C2#3462411yescharacteristic
762C2#6070111yescharacteristic
763C2#6070111yescharacteristic
764C2C2×#321811yescharacteristic
765C2#6079011yescharacteristic
766C2#6095011yescharacteristic
767C2#6095211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #30345C2semidirect
4. #30344C2semidirect
5. #30326C2semidirect
7. #30309C2semidirect
8. #30267C2semidirect
9. #30226C2semidirect
10. #30216C2semidirect
11. #29852C2semidirect
13. #23679C2semidirect
14. #23677C2semidirect
15. #23571C2semidirect
17. C4×#497C2semidirect
18. C4×#491C2semidirect
19. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366382
#303452
#303442
#303262
#303252
#303092
#302672
#302262
#302162
#298522
#298152
#236792
#236772
#235712
#235682
C4×#4972
C4×#4912
C4×#4322