Finite Group Databasebeta

← #6475425

Subgroups — #6475425

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35333C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#33218C212yesmaximal, complemented
4#33217C212yesmaximal, complemented
5#30299C214yesmaximal, complemented
6#29311C212yesmaximal, complemented
7#29310C212yesmaximal, complemented
8#29309C214yesmaximal, complemented
9#27955C212yesmaximal, complemented
10#27954C212yesmaximal, complemented
11#25819C212yesmaximal, complemented
12#25816C212yesmaximal, complemented
13#21947C212yesmaximal
14#21946C212yesmaximal
15#20375C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#501C2212yes—
20C2×#501C2212yes—
21C2×#408C2212yes—
22C2×C4×M16C2211yescharacteristic
23C22×#97C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2851C2212yes—
27#2851C2212yes—
28#2629C2212yes—
29#2629C2212yes—
30#2628C2212yes—
31#2628C2212yes—
32#2579—21no—
33#2579—21no—
34#2579—21no—
35#2579—21no—
36#2568—21no—
37#2567—21no—
38#2566—21no—
39#2566—21no—
40#2565—21no—
41#2565—21no—
42#2563—21no—
43#2542—21no—
44#2487—21no—
45#2487—21no—
46#2409—21no—
47#2409—21no—
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50#2317C2212yes—
51#2311C2211yescharacteristic
52#2311C2211yescharacteristic
53#2311C2212yes—
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56#2085C2214yes—
57#2085C2214yes—
58#2084C2214yes—
59#2083C2214yes—
60#2059C2214yes—
61#2059C2214yes—
62#2058C2214yes—
63#2057C2214yes—
64#1993C2212yes—
65#1993C2212yes—
66#1992C2212yes—
67#1992C2212yes—
68#1944C2214yes—
69#1943C2214yes—
70#1942C2214yes—
71#1941C2214yes—
72#1854C2214yes—
73#1854C2214yes—
74#1854C2214yes—
75#1854C2214yes—
76#1853C2214yes—
77#1853C2214yes—
78#1852C2214yes—
79#1852C2214yes—
80#1847C2212yes—
81#1847C2212yes—
82#1809C2214yes—
83#1809C2214yes—
84#1808C2214yes—
85#1807C2214yes—
86#1750C2212yes—
87#1750C2212yes—
88#1749C2212yes—
89#1749C2212yes—
90#1729C2212yes—
91C22×#41C2311yescharacteristic
92C23×Q8—21no—
93C22×M16C2312yes—
94C23×C8C2312yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#125—21no—
102C2×#125—21no—
103C2×#125—42no—
104C2×#125—42no—
105C2×#125C2312yes—
106C2×#125C2312yes—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C4×#41—42no—
110C4×#41—42no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#501—44no—
135#501—44no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144#408Q8116yescomplemented
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C4×M16—42no—
148C2×#97—42no—
149C2×#97C2312yes—
150C2×#97C2312yes—
151C2×#97C2312yes—
152C2×C4×C8C2312yes—
153#400—22no—
154#400—22no—
155#399—22no—
156#399—22no—
157#398—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#396—22no—
162#396—22no—
163#396—22no—
164#396—22no—
165#395—22no—
166#395—22no—
167#393—22no—
168#393—22no—
169#391—21no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391—22no—
174#390—22no—
175#390—21no—
176#390—21no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#389C2314yes—
182#389C2312yes—
183#389C2312yes—
184#389C2314yes—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#386—21no—
190#386—21no—
191#386—21no—
192#386—21no—
193#386—22no—
194#386—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—21no—
200#385—21no—
201#385—21no—
202#385—21no—
203#385—22no—
204#385—22no—
205#384—21no—
206#384—21no—
207#384—22no—
208#384C2312yes—
209#384C2312yes—
210#382—22no—
211#382—21no—
212#382—21no—
213#382C2312yes—
214#382C2312yes—
215#380—22no—
216#380—22no—
217#380—21no—
218#380—21no—
219#380—21no—
220#380—21no—
221#380C2312yes—
222#380C2314yes—
223#380C2312yes—
224#380C2312yes—
225#380C2312yes—
226#380C2314yes—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C2×#41—41no—
242C2×#41—41no—
243C2×#41—42no—
244C2×#41C2×Q818yes—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—44no—
248C22×Q8—84no—
249C22×Q8C2×D411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D411yescharacteristic
252C22×D4C2×D411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×M16—42no—
266C2×M16—42no—
267C2×M16C2×Q818yescomplemented
268C22×C8—42no—
269C22×C8—42no—
270C22×C8C2×Q818yescomplemented
271C22×C8C2414yes—
272C22×C8C2414yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—42no—
276C4⋊Q8—42no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279#125—24no—
280#125—24no—
281#125—42no—
282#125—42no—
283#125—44no—
284#125—44no—
285C22⋊Q8—84no—
286C22⋊Q8—84no—
287C4×Q8—42no—
288C4×Q8—42no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—42no—
295#117—41no—
296#117—41no—
297#117—41no—
298#117—41no—
299#117—41no—
300#117—41no—
301#117—42no—
302#117—42no—
303#117—42no—
304#117—42no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—41no—
325C2×C4⋊C4—41no—
326C2×C4⋊C4—41no—
327C2×C4⋊C4—41no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C22⋊C4C2×D411yescharacteristic
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C22⋊C4C2×D411yescharacteristic
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—44no—
380#107—44no—
381#107—44no—
382#107—44no—
383#106—44no—
384#106—44no—
385#106—44no—
386#106—44no—
387C22⋊C8C2×Q8116yes—
388C22⋊C8C2×Q8116yescomplemented
389#97—24no—
390#97—24no—
391#97—42no—
392C4×C8—42no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411C24C22×D411yescharacteristic
412#41—48no—
413#41—48no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—84no—
421C2×Q8—816no—
422C2×Q8C4⋊Q814yes—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4C4⋊Q814yes—
430C2×D4C4⋊Q814yes—
431C2×D4C4⋊Q814yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—84no—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×Q814yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q814yes—
479C22×C4C2511yescharacteristic
480M16—48no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—48no—
486C2×C8C22×Q818yes—
487C2×C8C22×Q818yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—44no—
551C42—42no—
552C42—42no—
553C42H(5,𝔽2)12yes—
554C42H(5,𝔽2)12yes—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C2×C4⋊Q812yes—
568C23C2×C4⋊Q812yes—
569C23C2×C4⋊Q812yes—
570C23C23×D411yescharacteristic
571Q8—44no—
572Q8—88no—
573Q8—88no—
574D4—44no—
575D4—44no—
576D4—44no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—88no—
630C2×C4C2×C4⋊Q812yes—
631C2×C4C2×C4⋊Q812yes—
632C2×C4C2×C4⋊Q812yes—
633C2×C4#53412yes—
634C2×C4#53412yes—
635C2×C4C23×Q812yes—
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4#58512yes—
639C8—28no—
640C8—28no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22C22×C4⋊Q811yescharacteristic
657C22C2×#53411yescharacteristic
658C22#344911yescharacteristic
659C22#358512yes—
660C22#358612yes—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—82no—
673C4—84no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4—82no—
678C4#350112yes—
679C4#350112yes—
680C2—42no—
681C2—42no—
682C2—42no—
683C2#3501111yescharacteristic
684C2#6066812yes—
685C2#6075311yescharacteristic
686C2#6075411yescharacteristic
687C2#6102112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33218C2semidirect
4. #33217C2semidirect
5. #30299C2semidirect
6. #29311C2semidirect
7. #29310C2semidirect
8. #29309C2semidirect
9. #27955C2semidirect
10. #27954C2semidirect
11. #25819C2semidirect
12. #25816C2semidirect
144. #408Q8semidirect
267. C2×M16C2×Q8semidirect
270. C22×C8C2×Q8semidirect
388. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#353332
#352792
#332182
#332172
#302992
#293112
#293102
#293092
#279552
#279542
#258192
#258162
#219472
#219462
#203752