Finite Group Databasebeta

← #6489473

Subgroups — #6489473

Order 512.

1609 conjugacy classes of subgroups.

The subgroup lattice has 773 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2#36638C211yesmaximal, characteristic
3#33217C212yesmaximal, complemented
4#33051C212yesmaximal
5#30383C212yesmaximal, complemented
6#30382C212yesmaximal, complemented
7#29225C211yesmaximal, characteristic, complemented
8#29181C212yesmaximal, complemented
9#28726C211yesmaximal, characteristic, complemented
10#28694C211yesmaximal, characteristic, complemented
11#28658C212yesmaximal, complemented
12#28520C211yesmaximal, characteristic, complemented
13#27858C212yesmaximal, complemented
14#27857C212yesmaximal
15#24838C212yesmaximal, complemented
16#24835C212yesmaximal
17#22118C212yesmaximal, complemented
18#22116C212yesmaximal, complemented
19C2×#1767C211yesmaximal, characteristic, complemented
20C2×#533—21no—
21C2×#501C2212yes—
22C2×#489C2212yes—
23C2×#475C2212yes—
24C2×#474C2212yes—
25C22×C4⋊C8C2212yes—
26C22×#97C2212yes—
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2866—21no—
30#2865—21no—
31#2864C2211yescharacteristic
32#2863C2211yescharacteristic
33#2862—21no—
34#2861C2211yescharacteristic
35#2860C2211yescharacteristic
36#2859—21no—
37#2859C2212yes—
38#2857—21no—
39#2857—21no—
40#2857C2212yes—
41#2857C2212yes—
42#2855C2211yescharacteristic
43#2854C2211yescharacteristic
44#2853—21no—
45#2852—21no—
46#2851C2211yescharacteristic
47#2850C2211yescharacteristic
48C4×C4⋊Q8—21no—
49C4×C4⋊Q8—21no—
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2212yes—
52C4×#125C2211yescharacteristic
53C4×#126—21no—
54C4×#124C2211yescharacteristic
55C4×C22⋊Q8C2211yescharacteristic
56C4×#123—21no—
57C4×#121C2211yescharacteristic
58#2298—21no—
59#2295C2211yescharacteristic
60#2293C2211yescharacteristic
61#2290—21no—
62#2285—21no—
63#2285C2212yes—
64C2×C43C2211yescharacteristic
65#2100C2212yes—
66#2100C2212yes—
67#2095C2212yes—
68#2094C2212yes—
69#2088C2214yes—
70#2086C2214yes—
71#2084C2212yes—
72#2083C2212yes—
73#2072C2212yes—
74#2071C2212yes—
75#2067C2214yes—
76#2066C2214yes—
77#1993C2212yes—
78#1993C2212yes—
79#1992C2212yes—
80#1988C2212yes—
81#1988C2212yes—
82#1959C2212yes—
83#1948C2212yes—
84#1947C2212yes—
85#1944C2214yes—
86#1943C2214yes—
87#1936C2214yes—
88#1935C2214yes—
89#1928C2214yes—
90#1926C2214yes—
91#1921C2214yes—
92#1919C2214yes—
93#1893C2214yes—
94#1892C2214yes—
95#1847C2212yes—
96#1836C2212yes—
97#1808C2214yes—
98#1807C2214yes—
99#1803C2212yes—
100#1802C2212yes—
101#1767C2211yescharacteristic
102#1767C2211yescharacteristic
103#1767C2212yes—
104#1767C2212yes—
105#1767C2212yes—
106C2×#336C2212yes—
107C23×C8C2314yes—
108#533—48no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#126—21no—
116C2×#125—22no—
117C2×#125C2311yescharacteristic
118C2×#124C2311yescharacteristic
119C2×#123—21no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8C2311yescharacteristic
123C2×#121C2311yescharacteristic
124C2×C4×Q8—22no—
125C2×C4×Q8—22no—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C2×#117—21no—
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#501—44no—
139#489—44no—
140#475—28no—
141#474—28no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8—28no—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#97—44no—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×#97C2314yes—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2311yescharacteristic
163#391C2311yescharacteristic
164#391C2311yescharacteristic
165#391C2311yescharacteristic
166#390C2311yescharacteristic
167#390C2311yescharacteristic
168#390C2311yescharacteristic
169#390C2311yescharacteristic
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#389C2314yes—
179#389C2314yes—
180#389C2311yescharacteristic
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#389C2311yescharacteristic
184#389C2311yescharacteristic
185#389C2311yescharacteristic
186#388—21no—
187#388—21no—
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#387—21no—
203#387—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#386—21no—
219#386—21no—
220#386—21no—
221#386—21no—
222#385—21no—
223#385—21no—
224#385—21no—
225#385—21no—
226#384—21no—
227#384—21no—
228#384—21no—
229#384—21no—
230#384C2312yes—
231#384C2312yes—
232#384C2312yes—
233#384C2312yes—
234#383—21no—
235#382—22no—
236#382—22no—
237#382C2311yescharacteristic
238#382C2311yescharacteristic
239#381—21no—
240#380—22no—
241#380—22no—
242#380—22no—
243#380C2311yescharacteristic
244#380C2314yes—
245#380C2311yescharacteristic
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—22no—
251C4×C4⋊C4—22no—
252C4×C4⋊C4—22no—
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2314yes—
256C4×C22⋊C4—21no—
257C4×C22⋊C4—21no—
258C4×C22⋊C4—21no—
259C4×C22⋊C4—21no—
260C4×C22⋊C4—21no—
261#376—21no—
262C43D411yescharacteristic
263C43D411yescharacteristic
264C43D411yescharacteristic
265C43D411yescharacteristic
266C43C2311yescharacteristic
267C43C2311yescharacteristic
268C43C2311yescharacteristic
269C43C2311yescharacteristic
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#340C2314yes—
276#340C2314yes—
277#340C2314yes—
278#336C2312yes—
279#336C2312yes—
280#336C2314yes—
281C22×Q8—22no—
282C22×Q8—22no—
283C22×Q8C2411yescharacteristic
284C22×D4—21no—
285C22×D4C2411yescharacteristic
286C23×C4—21no—
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2×D411yescharacteristic
292C23×C4C2×D411yescharacteristic
293C23×C4C2411yescharacteristic
294C22×C8—28no—
295C22×C8—28no—
296C22×C8—44no—
297C22×C8C2414yes—
298C22×C8C2414yes—
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301C4⋊Q8—24no—
302C4⋊Q8—24no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306C4⋊Q8—44no—
307#126—44no—
308#125—22no—
309#125—22no—
310#125—48no—
311#124—22no—
312#124—22no—
313#123—44no—
314C22⋊Q8—24no—
315C22⋊Q8—48no—
316C22⋊Q8—48no—
317#121—24no—
318C4×Q8—24no—
319C4×Q8—48no—
320C4×Q8—48no—
321C4×D4—24no—
322C4×D4—44no—
323#117—44no—
324#117—48no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
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⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2414yes—
356C2×C4⋊C4C2414yes—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—21no—
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×C42—24no—
374C2×C42—24no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—24no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
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—28no—
414#107—44no—
415#106—28no—
416#106—44no—
417C4⋊C8—28no—
418C4⋊C8—24no—
419C4⋊C8#4118yes—
420#103—28no—
421#103—44no—
422#102—28no—
423#102—44no—
424#97—24no—
425#97—24no—
426#97—44no—
427#95—21no—
428#95—21no—
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—24no—
442#95—24no—
443#95—24no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—24no—
453#95—21no—
454#95—21no—
455C24—21no—
456C24C22×D411yescharacteristic
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—44no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—21no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
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—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4AGL(1,ℤ/8)14yes—
506C22×C4#13714yes—
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×C4C22×D411yescharacteristic
520C22×C42−1+414yes—
521C22×C4C2511yescharacteristic
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—416no—
528C2×C8C2×#4118yes—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—24no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—48no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—48no—
560C4⋊C4—44no—
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—42no—
570C22⋊C4—48no—
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—21no—
601C42—22no—
602C42—22no—
603C42—21no—
604C42—41no—
605C42—41no—
606C42—41no—
607C42—41no—
608C42—48no—
609C42—42no—
610C42—42no—
611C42—42no—
612C42C2×#4114yes—
613C42H(5,𝔽2)12yes—
614C422−1+412yes—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23—41no—
620C23—41no—
621C23—41no—
622C23—41no—
623C23—41no—
624C23—41no—
625C23—41no—
626C23#53612yes—
627C23C2×AGL(1,ℤ/8)12yes—
628C23C2×#13712yes—
629C23C23×D411yescharacteristic
630Q8—42no—
631Q8—42no—
632Q8—88no—
633Q8—88no—
634D4—42no—
635D4—42no—
636D4—84no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—28no—
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—21no—
653C2×C4—21no—
654C2×C4—28no—
655C2×C4—24no—
656C2×C4—24no—
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—41no—
674C2×C4—41no—
675C2×C4—41no—
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—44no—
686C2×C4—44no—
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—41no—
697C2×C4—41no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—84no—
701C2×C4#53212yes—
702C2×C4#53212yes—
703C2×C4#53612yes—
704C2×C4#53712yes—
705C2×C4#53712yes—
706C2×C4C2×AGL(1,ℤ/8)12yes—
707C2×C4C2×AGL(1,ℤ/8)12yes—
708C2×C4C2×#13712yes—
709C2×C4C2×#13712yes—
710C2×C4#57512yes—
711C2×C4#57512yes—
712C2×C4C22×#4112yes—
713C2×C4C2×H(5,𝔽2)11yescharacteristic
714C2×C4C2×2−1+412yes—
715C2×C4C2×2−1+411yescharacteristic
716C8—28no—
717C8—44no—
718C22—22no—
719C22—22no—
720C22—22no—
721C22—22no—
722C22—22no—
723C22—22no—
724C22—81no—
725C22—81no—
726C22—81no—
727C22—81no—
728C22—81no—
729C22—81no—
730C22—81no—
731C22#323914yes—
732C22C2×#53211yescharacteristic
733C22C2×#53611yescharacteristic
734C22C2×#53711yescharacteristic
735C22C22×AGL(1,ℤ/8)11yescharacteristic
736C22C22×#13711yescharacteristic
737C22C2×#57511yescharacteristic
738C22C2×#57511yescharacteristic
739C4—22no—
740C4—22no—
741C4—22no—
742C4—22no—
743C4—22no—
744C4—22no—
745C4—22no—
746C4—22no—
747C4—41no—
748C4—41no—
749C4—41no—
750C4—41no—
751C4—81no—
752C4—84no—
753C4—81no—
754C4—81no—
755C4—81no—
756C4—81no—
757C4—81no—
758C4—84no—
759C4—81no—
760C4#318014yes—
761C4#352012yes—
762C4#352312yes—
763C2—24no—
764C2—161no—
765C2#3524111yescharacteristic
766C2C2×#318011yescharacteristic
767C2#6070111yescharacteristic
768C2C2×#323911yescharacteristic
769C2#6085411yescharacteristic
770C2#6091611yescharacteristic
771C2#6091811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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. #36645C2semidirect
3. #33217C2semidirect
5. #30383C2semidirect
6. #30382C2semidirect
7. #29225C2semidirect
8. #29181C2semidirect
9. #28726C2semidirect
10. #28694C2semidirect
11. #28658C2semidirect
12. #28520C2semidirect
13. #27858C2semidirect
15. #24838C2semidirect
17. #22118C2semidirect
18. #22116C2semidirect
19. C2×#1767C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366382
#332172
#330512
#303832
#303822
#292252
#291812
#287262
#286942
#286582
#285202
#278582
#278572
#248382
#248352
#221182
#221162
C2×#17672