Finite Group Databasebeta

← #6498221

Subgroups — #6498221

Order 512.

1561 conjugacy classes of subgroups.

The subgroup lattice has 751 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#30309C211yesmaximal, characteristic, complemented
6#30285C211yesmaximal, characteristic, complemented
7#30233C211yesmaximal, characteristic, complemented
8#30216C211yesmaximal, characteristic, complemented
9#29580C212yesmaximal, complemented
10#29579C212yesmaximal, complemented
11#29520C212yesmaximal, complemented
12#29518C212yesmaximal, complemented
13#29159C212yesmaximal, complemented
14#29157C212yesmaximal
15#27858C212yesmaximal, complemented
16#27857C212yesmaximal
17#22250C212yesmaximal, complemented
18#22223C212yesmaximal, complemented
19C2×#1745C211yesmaximal, characteristic, complemented
20C2×#533—21no—
21C2×#501C2212yes—
22C2×#489C2212yes—
23C22×#97C2212yes—
24#2868C2211yescharacteristic
25#2868C2211yescharacteristic
26#2866—21no—
27#2865—21no—
28#2864C2211yescharacteristic
29#2863C2211yescharacteristic
30#2862—21no—
31#2861C2211yescharacteristic
32#2860C2211yescharacteristic
33#2859—21no—
34#2859C2212yes—
35#2857—21no—
36#2857—21no—
37#2857C2212yes—
38#2857C2212yes—
39#2855C2211yescharacteristic
40#2854C2211yescharacteristic
41#2853—21no—
42#2852—21no—
43#2851C2211yescharacteristic
44#2850C2211yescharacteristic
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2212yes—
49C4×#125C2211yescharacteristic
50C4×#126—21no—
51C4×#124C2211yescharacteristic
52C4×C22⋊Q8C2211yescharacteristic
53C4×#123—21no—
54C4×#121C2211yescharacteristic
55#2298—21no—
56#2295C2211yescharacteristic
57#2293C2211yescharacteristic
58#2290—21no—
59#2285—21no—
60#2285C2212yes—
61C2×C43C2211yescharacteristic
62#2100C2212yes—
63#2100C2212yes—
64#2099C2212yes—
65#2099C2212yes—
66#2096C2212yes—
67#2095C2212yes—
68#2094C2212yes—
69#2093C2212yes—
70#2084C2212yes—
71#2083C2212yes—
72#2072C2212yes—
73#2071C2212yes—
74#2058C2212yes—
75#2057C2212yes—
76#2024C2212yes—
77#2023C2212yes—
78#1993C2212yes—
79#1993C2212yes—
80#1992C2212yes—
81#1959C2212yes—
82#1944C2214yes—
83#1943C2214yes—
84#1936C2214yes—
85#1935C2214yes—
86#1928C2214yes—
87#1926C2214yes—
88#1921C2214yes—
89#1919C2214yes—
90#1919C2214yes—
91#1918C2214yes—
92#1893C2214yes—
93#1892C2214yes—
94#1865C2214yes—
95#1862C2214yes—
96#1847C2212yes—
97#1836C2212yes—
98#1808C2214yes—
99#1807C2214yes—
100#1745C2211yescharacteristic
101#1745C2211yescharacteristic
102#1745C2212yes—
103#1745C2212yes—
104#1745C2212yes—
105C2×#336C2212yes—
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—28no—
139#489—28no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#97—28no—
149C2×#97C2314yes—
150C2×#97C2312yes—
151C2×#97C2312yes—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2311yescharacteristic
157#391C2311yescharacteristic
158#391C2311yescharacteristic
159#391C2311yescharacteristic
160#390C2311yescharacteristic
161#390C2311yescharacteristic
162#390C2311yescharacteristic
163#390C2311yescharacteristic
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2314yes—
173#389C2314yes—
174#389C2311yescharacteristic
175#389C2311yescharacteristic
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#388—21no—
181#388—21no—
182#388—21no—
183#388—21no—
184#388—21no—
185#388—21no—
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#387—21no—
197#387—21no—
198#387—21no—
199#387—21no—
200#387—21no—
201#387—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#386—21no—
213#386—21no—
214#386—21no—
215#386—21no—
216#385—21no—
217#385—21no—
218#385—21no—
219#385—21no—
220#384—21no—
221#384—21no—
222#384—21no—
223#384—21no—
224#384C2312yes—
225#384C2312yes—
226#384C2312yes—
227#384C2312yes—
228#383—21no—
229#382—22no—
230#382—22no—
231#382C2311yescharacteristic
232#382C2311yescharacteristic
233#381—21no—
234#380—22no—
235#380—22no—
236#380—22no—
237#380C2311yescharacteristic
238#380C2314yes—
239#380C2311yescharacteristic
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—22no—
245C4×C4⋊C4—22no—
246C4×C4⋊C4—22no—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2314yes—
250C4×C22⋊C4—21no—
251C4×C22⋊C4—21no—
252C4×C22⋊C4—21no—
253C4×C22⋊C4—21no—
254C4×C22⋊C4—21no—
255#376—21no—
256C43D411yescharacteristic
257C43D411yescharacteristic
258C43D411yescharacteristic
259C43D411yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263C43C2311yescharacteristic
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275#336C2312yes—
276#336C2314yes—
277#336C2314yes—
278C22×Q8—22no—
279C22×Q8—22no—
280C22×Q8C2411yescharacteristic
281C22×D4—21no—
282C22×D4C2411yescharacteristic
283C23×C4—21no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2411yescharacteristic
291C22×C8—28no—
292C22×C8—28no—
293C22×C8—44no—
294C22×C8C2414yes—
295C22×C8C2414yes—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303#126—44no—
304#125—24no—
305#125—48no—
306#124—24no—
307#123—44no—
308C22⋊Q8—24no—
309C22⋊Q8—48no—
310C22⋊Q8—48no—
311#121—24no—
312C4×Q8—42no—
313C4×Q8—48no—
314C4×Q8—48no—
315C4×D4—42no—
316C4×D4—44no—
317#117—44no—
318#117—48no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2414yes—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
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×C42—24no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—42no—
376C2×C42—42no—
377C2×C42—42no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#107—28no—
407#107—44no—
408#106—28no—
409#106—44no—
410#103—28no—
411#103—44no—
412#102—28no—
413#102—44no—
414#97—28no—
415#97—24no—
416#97#4118yes—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—24no—
432#95—24no—
433#95—24no—
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—24no—
443#95—21no—
444#95—21no—
445C24—21no—
446C24C22×D411yescharacteristic
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4AGL(1,ℤ/8)14yes—
493C22×C4#13714yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C42−1+414yes—
508C22×C4C2511yescharacteristic
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—416no—
515C2×C8C2×#4118yes—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—48no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—48no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—41no—
591C42—41no—
592C42—41no—
593C42—42no—
594C42—48no—
595C42—42no—
596C42—42no—
597C42C2×#4114yes—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23#53612yes—
610C23C2×AGL(1,ℤ/8)12yes—
611C23C2×#13712yes—
612C23C23×D411yescharacteristic
613Q8—44no—
614Q8—88no—
615Q8—88no—
616D4—44no—
617D4—84no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—28no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—28no—
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—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—44no—
666C2×C4—41no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—84no—
683C2×C4#53212yes—
684C2×C4#53212yes—
685C2×C4#53612yes—
686C2×C4C2×AGL(1,ℤ/8)12yes—
687C2×C4C2×AGL(1,ℤ/8)12yes—
688C2×C4C2×#13712yes—
689C2×C4C2×#13712yes—
690C2×C4#57512yes—
691C2×C4#57512yes—
692C2×C4C22×#4112yes—
693C2×C4C2×2−1+412yes—
694C2×C4#58512yes—
695C8—28no—
696C8—44no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22#323914yes—
711C22C2×#53211yescharacteristic
712C22C2×#53611yescharacteristic
713C22#344911yescharacteristic
714C22C22×AGL(1,ℤ/8)11yescharacteristic
715C22C22×#13711yescharacteristic
716C22C2×#57511yescharacteristic
717C22C2×#57511yescharacteristic
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—81no—
732C4—84no—
733C4—81no—
734C4—81no—
735C4—84no—
736C4—81no—
737C4—81no—
738C4—81no—
739C4—81no—
740C4#349514yes—
741C2—24no—
742C2—161no—
743C2#3532611yescharacteristic
744C2#6070011yescharacteristic
745C2#6070111yescharacteristic
746C2C2×#323911yescharacteristic
747C2#6085411yescharacteristic
748C2#6095811yescharacteristic
749C2#6096311yescharacteristic
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. #30309C2semidirect
6. #30285C2semidirect
7. #30233C2semidirect
8. #30216C2semidirect
9. #29580C2semidirect
10. #29579C2semidirect
11. #29520C2semidirect
12. #29518C2semidirect
13. #29159C2semidirect
15. #27858C2semidirect
17. #22250C2semidirect
18. #22223C2semidirect
19. C2×#1745C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366382
#332172
#330512
#303092
#302852
#302332
#302162
#295802
#295792
#295202
#295182
#291592
#291572
#278582
#278572
#222502
#222232
C2×#17452