Finite Group Databasebeta

← #6640632

Subgroups — #6640632

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic, complemented
3#32978C212yesmaximal, complemented
4#32278C214yesmaximal, complemented
5#32046C214yesmaximal, complemented
6#31930C214yesmaximal, complemented
7#31834C214yesmaximal, complemented
8#29592C212yesmaximal, complemented
9#29103C211yesmaximal, characteristic, complemented
10#28301C211yesmaximal, characteristic, complemented
11#28114C212yesmaximal, complemented
12#27960C211yesmaximal, characteristic, complemented
13#27909C211yesmaximal, characteristic, complemented
14#27644C212yesmaximal, complemented
15#21902C211yesmaximal, characteristic, complemented
16C2×#538—21no—
17C22×#123—21no—
18C22×#123—21no—
19C22×#121C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#480C2212yes—
22C2×#480C2212yes—
23C2×#480C2212yes—
24C2×#447C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2847C2212yes—
28#2639—21no—
29#2617—21no—
30#2559—22no—
31#2557—22no—
32#2501—22no—
33#2501—22no—
34#2501C2214yes—
35#2501C2214yes—
36#2496C2212yescomplemented
37#2491C2212yescomplemented
38#2428C2212yescomplemented
39#2422C2212yescomplemented
40C2×#399—21no—
41C2×#399—21no—
42#2325—21no—
43C4×#121C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2099C2214yescomplemented
50#2093C2214yescomplemented
51#2093C2212yescomplemented
52#2083C2214yescomplemented
53#2071C2214yescomplemented
54#2071C2212yescomplemented
55#2067C2214yescomplemented
56#2054C2214yescomplemented
57#2033C2214yes—
58#2033C2214yes—
59#2023C2212yescomplemented
60#2011C2212yescomplemented
61#1999C2214yes—
62#1999C2214yes—
63#1990C2212yescomplemented
64#1970C2214yescomplemented
65#1957C2212yescomplemented
66#1947C2214yescomplemented
67#1934C2214yescomplemented
68#1925C2214yescomplemented
69#1923C2214yescomplemented
70#1923C2214yescomplemented
71#1879C2214yescomplemented
72#1874C2214yescomplemented
73#1874C2214yescomplemented
74#1852C2214yescomplemented
75#1852C2214yescomplemented
76#1845C2212yescomplemented
77#1834C2212yescomplemented
78#1823C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1802C2214yescomplemented
82#1797C2212yes—
83#1786C2212yes—
84#1779C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×D8C2314yes—
92#538—48no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#124—22no—
97C2×#123—24no—
98C2×#123—24no—
99C2×#123—24no—
100C2×#123—24no—
101C2×#123—22no—
102C2×#123—48no—
103C2×#123—48no—
104C2×C22⋊Q8—21no—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121—22no—
108C2×#121—22no—
109C2×#121—44no—
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×#121C2314yes—
114C2×#120—22no—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—44no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2314yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#480—44no—
138#480—44no—
139#480—44no—
140#447—44no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#399—24no—
159#399—24no—
160#399—24no—
161#399—24no—
162#397—22no—
163#397—22no—
164#394—22no—
165#394—22no—
166#394—22no—
167#393—22no—
168#392—22no—
169#391—21no—
170#390—21no—
171#390—21no—
172#390—22no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2314yes—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#389—21no—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#385—22no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384C2314yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#380—21no—
201#380C2312yes—
202#380C2312yes—
203#380C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4C2312yes—
209C4×C22⋊C4C2312yes—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2312yes—
221C25—21no—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×D4—22no—
225C22×D4—24no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—24no—
232C22×D4—42no—
233C22×D4—42no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—24no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—22no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2412yes—
257C23×C4C2411yescharacteristic
258C2×D8—44no—
259C2×D8—44no—
260C2×D8—44no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263C4⋊Q8—42no—
264#127—42no—
265#126—48no—
266#124—48no—
267#123—416no—
268#123—416no—
269#123—416no—
270#123—416no—
271#123—48no—
272C22⋊Q8—44no—
273#121—24no—
274#121—24no—
275#121—48no—
276#121—48no—
277#121—44no—
278#121—44no—
279#121—48no—
280#121—44no—
281#121—44no—
282#120—48no—
283C4×D4—44no—
284C4×D4—48no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4D818yescomplemented
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—22no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—44no—
357#107—44no—
358#107—44no—
359C4⋊C8—44no—
360#102—44no—
361#102—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365#102—44no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368C22⋊C8—44no—
369C22⋊C8—44no—
370C4×C8—42no—
371C4×C8—42no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24C2×D814yes—
398C24C22×D412yes—
399C24C22×D411yescharacteristic
400C2×Q8—42no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—48no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—88no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—816no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—416no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—48no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—816no—
477C22×C4C2×D814yes—
478C22×C4C2×D814yes—
479C22×C4AGL(1,ℤ/8)14yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4H(5,𝔽2)14yes—
489C22×C4C2511yescharacteristic
490D8—88no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4C2×D818yes—
542C22⋊C4C2×D818yes—
543C42—22no—
544C42—22no—
545C42—24no—
546C42—41no—
547C42—42no—
548C42—41no—
549C42—44no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—28no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23#53512yes—
589C23C22×D814yes—
590C23C22×D812yes—
591C23C2×AGL(1,ℤ/8)12yes—
592C23C23×D411yescharacteristic
593Q8—162no—
594D4—44no—
595D4—44no—
596D4—44no—
597D4—44no—
598D4—84no—
599D4—816no—
600D4—88no—
601D4—162no—
602D4—162no—
603D4—164no—
604D4—164no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—28no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—816no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—164no—
654C2×C4#53512yes—
655C2×C4C22×D814yes—
656C2×C4C22×D812yes—
657C2×C4C22×D812yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×H(5,𝔽2)12yes—
661C2×C4#58512yes—
662C2×C4#58512yes—
663C8—44no—
664C8—44no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—416no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—168no—
694C22#320514yes—
695C22C2×#53511yescharacteristic
696C22#348511yescharacteristic
697C22#348511yescharacteristic
698C22C23×D811yescharacteristic
699C22C22×AGL(1,ℤ/8)11yescharacteristic
700C22#358611yescharacteristic
701C22#358611yescharacteristic
702C4—22no—
703C4—22no—
704C4—22no—
705C4—24no—
706C4—22no—
707C4—22no—
708C4—42no—
709C4—42no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4#353314yes—
717C2—24no—
718C2—44no—
719C2—162no—
720C2—162no—
721C2#3695611yescharacteristic
722C2C2×#320511yescharacteristic
723C2#6080711yescharacteristic
724C2#6169411yescharacteristic
725C2#6169411yescharacteristic
726C2#6170611yescharacteristic
727C2#6171711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35005C2semidirect
2. C2×#2501C2semidirect
3. #32978C2semidirect
4. #32278C2semidirect
5. #32046C2semidirect
6. #31930C2semidirect
7. #31834C2semidirect
8. #29592C2semidirect
9. #29103C2semidirect
10. #28301C2semidirect
11. #28114C2semidirect
12. #27960C2semidirect
13. #27909C2semidirect
14. #27644C2semidirect
15. #21902C2semidirect
36. #2496C22semidirect
37. #2491C22semidirect
38. #2428C22semidirect
39. #2422C22semidirect
49. #2099C22semidirect
50. #2093C22semidirect
51. #2093C22semidirect
52. #2083C22semidirect
53. #2071C22semidirect
54. #2071C22semidirect
55. #2067C22semidirect
56. #2054C22semidirect
59. #2023C22semidirect
60. #2011C22semidirect
63. #1990C22semidirect
64. #1970C22semidirect
65. #1957C22semidirect
66. #1947C22semidirect
67. #1934C22semidirect
68. #1925C22semidirect
69. #1923C22semidirect
70. #1923C22semidirect
71. #1879C22semidirect
72. #1874C22semidirect
73. #1874C22semidirect
74. #1852C22semidirect
75. #1852C22semidirect
76. #1845C22semidirect
77. #1834C22semidirect
81. #1802C22semidirect
342. C2×C22⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#25012
#329782
#322782
#320462
#319302
#318342
#295922
#291032
#283012
#281142
#279602
#279092
#276442
#219022