Finite Group Databasebeta

← #6586746

Subgroups — #6586746

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#33310C211yesmaximal, characteristic, complemented
4#33197C211yesmaximal, characteristic, complemented
5#33195C211yesmaximal, characteristic, complemented
6#28836C212yesmaximal, complemented
7#28541C212yesmaximal
8#28273C212yesmaximal
9#26091C212yesmaximal, complemented
10#26090C212yesmaximal, complemented
11#25452C212yesmaximal, complemented
12#25451C212yesmaximal, complemented
13#25059C212yesmaximal, complemented
14#25058C212yesmaximal, complemented
15#23163C212yesmaximal, complemented
16#22813C212yesmaximal
17#22729C212yesmaximal, complemented
18#21712C211yesmaximal, characteristic, complemented
19C42×D8C211yesmaximal, characteristic
20C2×#533—21no—
21C2×#498C2212yes—
22C2×#498C2212yes—
23C2×#476C2212yes—
24C2×#474C2212yes—
25C2×#446C2212yes—
26C2×C4×D8C2211yescharacteristic
27C2×C4×D8C2211yescharacteristic
28C2×C4×D8C2211yescharacteristic
29#2868C2211yescharacteristic
30#2868C2211yescharacteristic
31#2857—21no—
32#2857—21no—
33#2571C2212yes—
34#2563C2212yes—
35#2552C2212yes—
36#2542C2212yes—
37#2538C2212yes—
38#2538C2212yes—
39#2497—22no—
40#2492—22no—
41#2441—22no—
42#2439—22no—
43#2429—22no—
44#2423—22no—
45#2369C2212yes—
46#2365C2212yes—
47#2355C2212yes—
48#2354C2212yes—
49#2354C2212yes—
50Q8×C4⋊C4C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52#2285—21no—
53C42×D4C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2094C2214yes—
56#2092C2214yes—
57#2085C2214yes—
58#2084C2214yes—
59#2065C2214yes—
60#2065C2214yes—
61#1988C2212yes—
62#1987C2212yes—
63#1984C2212yes—
64#1972C2214yes—
65#1949C2214yes—
66#1948C2214yes—
67#1948C2212yes—
68#1948C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1944C2214yes—
72#1939C2212yes—
73#1939C2212yes—
74#1926C2214yes—
75#1926C2214yes—
76#1920C2214yes—
77#1919C2214yes—
78#1847C2214yes—
79#1844C2212yes—
80#1844C2212yes—
81#1844C2212yes—
82#1844C2212yes—
83#1842C2211yescharacteristic
84#1802C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1776C2214yes—
89C4×#107C2211yescharacteristic
90C4×#107C2211yescharacteristic
91C4×#107C2211yescharacteristic
92#1770C2212yes—
93#1770C2212yes—
94C4×#102C2211yescharacteristic
95C4×#102C2211yescharacteristic
96C4×#102C2211yescharacteristic
97C4×#102C2211yescharacteristic
98C4×#102C2211yescharacteristic
99C4×#102C2211yescharacteristic
100C42×C8C2211yescharacteristic
101C22×D8C2311yescharacteristic
102#533—48no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8D411yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#125—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117—22no—
126C22×C42—21no—
127C22×C42—21no—
128#498—28no—
129#498—44no—
130#476—44no—
131#474—44no—
132#446—28no—
133C4×D8—24no—
134C4×D8—24no—
135C4×D8Q818yescomplemented
136C2×#107C2311yescharacteristic
137C2×#107C2311yescharacteristic
138C2×#107C2314yes—
139C2×#107C2311yescharacteristic
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×#102C2311yescharacteristic
150C2×#102C2311yescharacteristic
151C2×#102C2311yescharacteristic
152C2×#102C2311yescharacteristic
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2311yescharacteristic
156C2×#102C2311yescharacteristic
157C2×#97C2314yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164#400—24no—
165#399—24no—
166#398—24no—
167#396—24no—
168#396—24no—
169#393—24no—
170#391—22no—
171#391—22no—
172#391—24no—
173#391D412yes—
174#390C2314yes—
175#390C2314yes—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—24no—
181#389D412yes—
182#389D412yes—
183#389C2314yes—
184#389C2314yes—
185#388—24no—
186#388—24no—
187#387—24no—
188#387—24no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384D411yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#384C2314yes—
199#382—22no—
200#382C2314yes—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2311yescharacteristic
231#340C2311yescharacteristic
232#340C2311yescharacteristic
233#340C2311yescharacteristic
234#340C2311yescharacteristic
235#340C2311yescharacteristic
236#336C2314yes—
237#336C2314yes—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8C2×D412yes—
241C22×D4C2411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C2×D8—22no—
250C2×D8—22no—
251C2×D8C2×Q814yes—
252C22×C8C2414yes—
253C22×C8C2411yescharacteristic
254C22×C8C2411yescharacteristic
255C22×C8C2411yescharacteristic
256C22×C8C2411yescharacteristic
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264C4⋊Q8—82no—
265C4⋊Q8—82no—
266#125—48no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269C22⋊Q8—84no—
270C22⋊Q8—84no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—48no—
274C4×Q8—48no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4C2×Q814yes—
286C4×D4C2×Q814yes—
287#117—48no—
288#117—84no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
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—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—24no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2414yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—22no—
346#107—22no—
347#107—44no—
348#107C2×Q814yes—
349#106—28no—
350#106—28no—
351#106—44no—
352#106—44no—
353#106—44no—
354C4⋊C8—28no—
355C4⋊C8—24no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358#102—22no—
359#102—22no—
360#102—22no—
361#102—22no—
362#102—44no—
363#102—44no—
364#102C2×Q814yes—
365#102C2×Q814yes—
366#97—28no—
367C4×C8—22no—
368C4×C8—22no—
369C4×C8—22no—
370C4×C8C2×Q814yes—
371C4×C8#4114yes—
372C4×C8#4114yes—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95C2×D414yes—
386#95C2×D414yes—
387C24—21no—
388C24—21no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4C22×Q812yes—
409C2×D4C22×Q812yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445D8—28no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8C22×Q812yes—
456C2×C8C22×Q812yes—
457C2×C8C2×#4112yes—
458C2×C8C2×#4112yes—
459C2×C8C2×#4112yes—
460C2×C8C2×#4112yes—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4C22×Q812yes—
496C4⋊C4C22×Q812yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—48no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—24no—
512C42—48no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—84no—
520C42AGL(1,ℤ/8)14yes—
521C42C22×Q812yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C42H(5,𝔽2)12yes—
525C422−1+412yes—
526C422−1+412yes—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C23×D411yescharacteristic
542Q8—48no—
543Q8—88no—
544Q8—164no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4#54712yes—
596C2×C4#54712yes—
597C2×C4D4×Q812yes—
598C2×C4D4×Q812yes—
599C2×C4#55012yes—
600C2×C4#55012yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4#57712yes—
605C2×C4#57712yes—
606C2×C4#57712yes—
607C2×C4C23×Q811yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C8—22no—
614C8—22no—
615C8—22no—
616C8—44no—
617C8#53314yescomplemented
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22C2×#54711yescharacteristic
633C22C2×D4×Q811yescharacteristic
634C22C2×#55011yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C2×#57711yescharacteristic
637C22C2×#57711yescharacteristic
638C22C2×#57711yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—84no—
651C4—84no—
652C4#335414yes—
653C4#338414yes—
654C4#339714yes—
655C4C2×#53311yescharacteristic
656C4#350311yescharacteristic
657C4#350911yescharacteristic
658C4#351111yescharacteristic
659C2—82no—
660C2—82no—
661C2#3613411yescharacteristic
662C2C2×#335411yescharacteristic
663C2#6128811yescharacteristic
664C2C2×#338411yescharacteristic
665C2#6137411yescharacteristic
666C2C2×#339711yescharacteristic
667C2#6141011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D8C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
2. #36134C2semidirect
3. #33310C2semidirect
4. #33197C2semidirect
5. #33195C2semidirect
6. #28836C2semidirect
9. #26091C2semidirect
10. #26090C2semidirect
11. #25452C2semidirect
12. #25451C2semidirect
13. #25059C2semidirect
14. #25058C2semidirect
15. #23163C2semidirect
17. #22729C2semidirect
18. #21712C2semidirect
135. C4×D8Q8semidirect
617. C8#533semidirect

Maximal subgroups

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

subgroupindex
#361942
#361342
#333102
#331972
#331952
#288362
#285412
#282732
#260912
#260902
#254522
#254512
#250592
#250582
#231632
#228132
#227292
#217122
C42×D82