Finite Group Databasebeta

← #6856086

Subgroups — #6856086

Order 512.

1667 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36580C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#33107C211yesmaximal, characteristic
4#32638C214yesmaximal, complemented
5#32580C214yesmaximal, complemented
6#29761C212yesmaximal, complemented
7#29336C212yesmaximal, complemented
8#29234C212yesmaximal, complemented
9#28645C212yesmaximal, complemented
10#26277C212yesmaximal, complemented
11#25903C211yesmaximal, characteristic
12#25648C212yesmaximal, complemented
13#25480C212yesmaximal, complemented
14#24302C211yesmaximal, characteristic, complemented
15#22510C211yesmaximal, characteristic, complemented
16#22082C212yesmaximal, complemented
17C4×#484C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#486C2212yes—
22C2×#484C2211yescharacteristic
23C2×#481C2212yes—
24C2×#477C2212yes—
25C2×#446C2212yes—
26#2856C2211yescharacteristic
27#2850C2212yes—
28#2592C2212yes—
29#2582C2211yescharacteristic
30#2580C2212yes—
31#2567—21no—
32#2567—21no—
33#2567—22no—
34#2567C2212yes—
35#2567C2214yes—
36#2567C2212yes—
37#2548C2211yescharacteristic
38#2458—22no—
39#2448—21no—
40#2443—22no—
41#2436—21no—
42C2×#395—22no—
43#2367—21no—
44#2361—21no—
45#2355C2211yescharacteristic
46C4×C22⋊Q8C2211yescharacteristic
47C4×#121C2212yes—
48C2×#391—21no—
49C2×#389C2212yes—
50C2×#389C2212yes—
51C2×#384—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2086C2214yes—
55#2085C2214yes—
56#2067C2214yes—
57#2067C2214yes—
58#2066C2214yes—
59#2057C2212yes—
60#2042C2214yes—
61#2042C2214yes—
62#2024C2212yes—
63#2023C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1985C2214yes—
67#1984C2212yes—
68#1982C2212yes—
69#1948C2212yes—
70#1944C2214yes—
71#1936C2214yes—
72#1936C2214yes—
73#1935C2214yes—
74#1928C2212yes—
75#1926C2214yes—
76#1921C2212yes—
77#1919C2214yes—
78#1918C2214yes—
79#1874C2212yes—
80#1874C2212yes—
81#1874C2212yes—
82#1872C2212yes—
83#1872C2211yescharacteristic
84#1862C2214yes—
85#1862C2214yes—
86#1852C2211yescharacteristic
87#1852C2212yes—
88#1824C2211yescharacteristic
89#1797C2212yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97#559—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#124—21no—
105C2×#124—21no—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8C2311yescharacteristic
109C2×#121—21no—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×C4×Q8—44no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D412yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D412yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#486—44no—
135#484—24no—
136#481—28no—
137#477—44no—
138#446—44no—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#399—22no—
158#398—22no—
159#397—22no—
160#395—24no—
161#395—24no—
162#395—24no—
163#395—24no—
164#394—22no—
165#393—22no—
166#393—22no—
167#391—22no—
168#391—22no—
169#391—24no—
170#391—21no—
171#391—21no—
172#391C2312yes—
173#391C2311yescharacteristic
174#390—21no—
175#390—21no—
176#390—21no—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2314yes—
186#389C2314yes—
187#389C2314yes—
188#389C2314yes—
189#389C2311yescharacteristic
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—22no—
197#387—21no—
198#386—22no—
199#386—22no—
200#386—22no—
201#385—21no—
202#385—22no—
203#385—22no—
204#384—22no—
205#384—22no—
206#384—24no—
207#384C2311yescharacteristic
208#382C2312yes—
209#380C2311yescharacteristic
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95—22no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—42no—
240C22×Q8—42no—
241C22×Q8—44no—
242C22×Q8—82no—
243C22×Q8—82no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D412yes—
247C22×Q8C2411yescharacteristic
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269#127—24no—
270#124—44no—
271#124—44no—
272#124—44no—
273#123—48no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276#121—24no—
277#121—24no—
278#121—44no—
279#121—82no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—42no—
286C4×Q8—42no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—84no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4#4114yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
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—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C42—24no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—24no—
358#107—44no—
359#106—28no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—24no—
363#103—24no—
364#103—44no—
365#102—28no—
366#102—28no—
367#102—44no—
368#102—44no—
369C22⋊C8—24no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372#97—44no—
373C4×C8—24no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391C24—22no—
392C24C22×D411yescharacteristic
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—416no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—816no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—81no—
426C2×D4—81no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4#13714yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C42−1+412yes—
472C22×C4C2511yescharacteristic
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—82no—
513C4⋊C4—81no—
514C4⋊C4C2×#4114yes—
515C4⋊C42−1+414yes—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—22no—
532C42—24no—
533C42—42no—
534C42—42no—
535C42—44no—
536C42—42no—
537C42—44no—
538C42—42no—
539C42—82no—
540C42H(5,𝔽2)14yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23#54812yes—
553C23C2×#13712yes—
554C23#57612yes—
555C23C23×D411yescharacteristic
556Q8—48no—
557Q8—44no—
558Q8—44no—
559Q8—42no—
560Q8—42no—
561Q8—88no—
562Q8—88no—
563D4—44no—
564D4—44no—
565D4—88no—
566D4—84no—
567D4—162no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4—164no—
623C2×C4#53412yes—
624C2×C4#53412yes—
625C2×C4#54812yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4#57512yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4C2×2−1+412yes—
632C2×C4C2×2−1+411yescharacteristic
633C8—44no—
634C8—42no—
635C8—42no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22#337514yes—
650C22C2×#53411yescharacteristic
651C22C2×#54811yescharacteristic
652C22#348611yescharacteristic
653C22C22×#13711yescharacteristic
654C22C2×#57511yescharacteristic
655C22C2×#57611yescharacteristic
656C22#358511yescharacteristic
657C4—22no—
658C4—22no—
659C4—22no—
660C4—42no—
661C4—42no—
662C4—41no—
663C4—42no—
664C4—42no—
665C4—44no—
666C4—44no—
667C4—81no—
668C4—84no—
669C4—81no—
670C4—82no—
671C4—81no—
672C4—81no—
673C4#319614yes—
674C4#351112yes—
675C4#355212yes—
676C2—24no—
677C2—162no—
678C2#3972111yescharacteristic
679C2C2×#319611yescharacteristic
680C2#6075611yescharacteristic
681C2C2×#337511yescharacteristic
682C2#6131811yescharacteristic
683C2#6183011yescharacteristic
684C2#6183511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
4. #32638C2semidirect
5. #32580C2semidirect
6. #29761C2semidirect
7. #29336C2semidirect
8. #29234C2semidirect
9. #28645C2semidirect
10. #26277C2semidirect
12. #25648C2semidirect
13. #25480C2semidirect
14. #24302C2semidirect
15. #22510C2semidirect
16. #22082C2semidirect
17. C4×#484C2semidirect

Maximal subgroups

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

subgroupindex
#365802
C2×#25672
#331072
#326382
#325802
#297612
#293362
#292342
#286452
#262772
#259032
#256482
#254802
#243022
#225102
#220822
C4×#4842