Finite Group Databasebeta

← #6856201

Subgroups — #6856201

Order 512.

1635 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#36587C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3#32903C211yesmaximal, characteristic
4#32441C214yesmaximal, complemented
5#31649C214yesmaximal, complemented
6#29719C212yesmaximal, complemented
7#28402C212yesmaximal, complemented
8#27977C212yesmaximal, complemented
9#27730C212yesmaximal, complemented
10#27684C212yesmaximal, complemented
11#25903C211yesmaximal, characteristic
12#25325C212yesmaximal, complemented
13#24683C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22041C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#486C2212yes—
21C2×#475C2212yes—
22C2×#460C2212yes—
23C2×#451C2211yescharacteristic
24C2×#448C2212yes—
25C2×C4×Q16C2212yes—
26#2856C2211yescharacteristic
27#2850C2212yes—
28#2597—22no—
29#2571C2211yescharacteristic
30#2568—22no—
31#2568—21no—
32#2568—21no—
33#2568C2212yes—
34#2568C2212yes—
35#2568C2214yes—
36#2548C2211yescharacteristic
37#2513—21no—
38#2513C2212yes—
39#2504—21no—
40#2447C2212yes—
41#2444—22no—
42C2×#400—22no—
43#2366C2211yescharacteristic
44#2361—21no—
45#2357—21no—
46#2355C2211yescharacteristic
47C4×C22⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49C2×#391—21no—
50C2×#391—21no—
51C2×#391C2212yes—
52C2×#382C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2092C2214yes—
56#2074C2214yes—
57#2074C2214yes—
58#2073C2214yes—
59#2058C2212yes—
60#2056C2214yes—
61#2055C2214yes—
62#2024C2212yes—
63#2023C2214yes—
64#2015C2214yes—
65#1990C2212yes—
66#1988C2212yes—
67#1968C2214yes—
68#1958C2214yes—
69#1949C2214yes—
70#1939C2212yes—
71#1933C2214yes—
72#1927C2214yes—
73#1916C2214yes—
74#1880C2214yes—
75#1878C2212yes—
76#1877C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1863C2212yes—
83#1853C2212yes—
84#1852C2211yescharacteristic
85#1835C2214yes—
86#1804C2214yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#124C2312yes—
106C2×C22⋊Q8C2311yescharacteristic
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—44no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#486—44no—
129#475—44no—
130#460—44no—
131#451—24no—
132#448—28no—
133C4×Q16—44no—
134C2×#107C2312yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#400—24no—
153#400—24no—
154#400—24no—
155#400—24no—
156#400—22no—
157#397—22no—
158#397—22no—
159#397—22no—
160#397—22no—
161#394—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—24no—
167#391—24no—
168#391—22no—
169#391—22no—
170#391C2314yes—
171#391C2312yes—
172#391C2314yes—
173#391C2311yescharacteristic
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
182#389—21no—
183#389C2311yescharacteristic
184#389C2312yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—21no—
191#387—22no—
192#386—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#385—21no—
197#385—22no—
198#384C2311yescharacteristic
199#382—21no—
200#382—21no—
201#382—22no—
202#382—22no—
203#382C2314yes—
204#382C2314yes—
205#382C2312yes—
206#381—22no—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8#4114yes—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×Q16—24no—
262C2×Q16—42no—
263C2×Q16—42no—
264C2×SD16—28no—
265C2×SD16—44no—
266C2×SD16—44no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275#127—24no—
276#124—44no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279#121—24no—
280#121—24no—
281#121—44no—
282#121—82no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—84no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
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×C42—24no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351#107—42no—
352#106—44no—
353C4⋊C8—24no—
354C4⋊C8—44no—
355#103—28no—
356#103—24no—
357#103—42no—
358#103—42no—
359#103—42no—
360#103—44no—
361#102—44no—
362#102—44no—
363#102—44no—
364C22⋊C8—24no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C4×C8—24no—
368C4×C8—44no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—22no—
393C24C22×D411yescharacteristic
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—48no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—48no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—816no—
415C2×Q8C2×#4114yes—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—81no—
428C2×D4—81no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4#13514yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4112yes—
473C22×C42−1+412yes—
474C22×C4C2511yescharacteristic
475Q16—48no—
476SD16—88no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—81no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—24no—
531C42—42no—
532C42—42no—
533C42—44no—
534C42—44no—
535C42—42no—
536C42—42no—
537C42—82no—
538C42H(5,𝔽2)14yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23#54812yes—
551C23C2×#13512yes—
552C23#57712yes—
553C23C23×D411yescharacteristic
554Q8—24no—
555Q8—24no—
556Q8—48no—
557Q8—42no—
558Q8—42no—
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—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4—164no—
623C2×C4#53512yes—
624C2×C4#53512yes—
625C2×C4#54812yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13512yes—
628C2×C4#57512yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4C2×2−1+411yescharacteristic
632C2×C4#58512yes—
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#336514yes—
650C22C2×#53511yescharacteristic
651C22C2×#54811yescharacteristic
652C22#348511yescharacteristic
653C22C22×#13511yescharacteristic
654C22C2×#57511yescharacteristic
655C22C2×#57711yescharacteristic
656C22#358611yescharacteristic
657C4—22no—
658C4—22no—
659C4—22no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—81no—
667C4—81no—
668C4—84no—
669C4—82no—
670C4—81no—
671C4—81no—
672C4—82no—
673C4#321114yes—
674C4#350512yes—
675C4#354112yes—
676C2—24no—
677C2—162no—
678C2#3972211yescharacteristic
679C2C2×#321111yescharacteristic
680C2#6081311yescharacteristic
681C2C2×#336511yescharacteristic
682C2#6132011yescharacteristic
683C2#6170111yescharacteristic
684C2#6173911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2568C2semidirect
4. #32441C2semidirect
5. #31649C2semidirect
6. #29719C2semidirect
7. #28402C2semidirect
8. #27977C2semidirect
9. #27730C2semidirect
10. #27684C2semidirect
12. #25325C2semidirect
13. #24683C2semidirect
14. #24301C2semidirect
15. #22625C2semidirect
16. #22041C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#25682
#329032
#324412
#316492
#297192
#284022
#279772
#277302
#276842
#259032
#253252
#246832
#243012
#226252
#220412
C4×#4512