Finite Group Databasebeta

← #6492187

Subgroups — #6492187

Order 512.

1640 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2#34127C211yesmaximal, characteristic
3#33146C212yesmaximal, complemented
4#33109C212yesmaximal, complemented
5#29941C212yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#26221C214yesmaximal, complemented
8#26145C214yesmaximal, complemented
9#25063C212yesmaximal, complemented
10#25016C212yesmaximal, complemented
11#24995C212yesmaximal, complemented
12#24969C212yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21894C211yesmaximal, characteristic, complemented
15#21849C211yesmaximal, characteristic
16#21389C211yesmaximal, characteristic, complemented
17#21171C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#484C2214yes—
21C2×#431C2211yescharacteristic
22C2×#428C2212yes—
23C2×#423C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2629—22no—
27#2582—21no—
28#2582C2212yes—
29#2538—21no—
30#2538C2212yes—
31#2417—22no—
32#2417C2214yes—
33#2355C2211yescharacteristic
34#2355C2211yescharacteristic
35#2352C2212yes—
36#2352C2212yes—
37D4×C4⋊C4C2211yescharacteristic
38D4×C4⋊C4C2211yescharacteristic
39#2347—21no—
40#2343—22no—
41#2339—21no—
42#2316—21no—
43#2311—21no—
44#2295—21no—
45#2290—22no—
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2283C2212yes—
49C42×D4—21no—
50C4×#117C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52#2101C2214yes—
53#2097C2214yes—
54#2085C2214yes—
55#2077C2214yes—
56#2076C2214yes—
57#2074C2214yes—
58#2036C2214yes—
59#2035C2214yes—
60#1988C2211yescharacteristic
61#1988C2212yes—
62#1988C2211yescharacteristic
63#1984C2212yes—
64#1981C2211yescharacteristic
65#1981C2212yes—
66#1981C2211yescharacteristic
67#1954C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1928C2214yes—
75#1928C2214yes—
76#1918C2214yes—
77#1918C2214yes—
78#1894C2214yes—
79#1873C2212yes—
80#1873C2212yes—
81#1872C2214yes—
82#1870C2214yes—
83#1852C2214yes—
84#1852C2214yes—
85#1845C2212yes—
86#1834C2212yes—
87#1823C2214yes—
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91#1797C2212yes—
92C4×#103C2212yes—
93C4×#103C2212yes—
94#1736C2212yes—
95C22×#41D411yescharacteristic
96C22×M16C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2314yes—
106C4×#41—24no—
107C4×#41—24no—
108C2×C4×Q8—21no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4D412yes—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D412yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C42—22no—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#484—48no—
133#431—24no—
134#428—44no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139#423—24no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8C2312yes—
154#398—24no—
155#398—24no—
156#396—24no—
157#396—24no—
158#391—22no—
159#391—21no—
160#391—21no—
161#391—22no—
162#391C2314yes—
163#390—22no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#388—22no—
173#388—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#386—24no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#383—22no—
190#382—21no—
191#382C2312yes—
192#382C2312yes—
193#381—22no—
194#380—22no—
195#380—22no—
196#380—21no—
197#380—22no—
198#380—24no—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202#380C2312yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220#376—21no—
221#376—22no—
222C2×#95—22no—
223C43—21no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41—42no—
239C22×Q8—21no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2412yes—
242C22×D4C2×D412yes—
243C22×D4C2×D411yescharacteristic
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—42no—
255C2×M16—42no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263#125—44no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—84no—
270C4×Q8—24no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—44no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—44no—
283C4×D4—44no—
284#117—24no—
285#117—28no—
286#117—22no—
287#117—22no—
288#117—41no—
289#117—41no—
290#117—44no—
291#117—82no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—84no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42#4114yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—28no—
364#107—44no—
365#107—44no—
366#106—44no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371#103—24no—
372#103—24no—
373#103—44no—
374#103—42no—
375#103—42no—
376#103—44no—
377C22⋊C8—24no—
378C22⋊C8—44no—
379C22⋊C8—44no—
380C4×C8—24no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399C24C22×D411yescharacteristic
400C24C22×D412yes—
401#41—416no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4#13714yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4#12314yes—
459C22×C4#12314yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2511yescharacteristic
471M16—48no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4H(5,𝔽2)12yes—
518C4⋊C42−1+412yes—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—82no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—42no—
542C42—82no—
543C42—82no—
544C42#12314yes—
545C42#12314yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C23—22no—
551C23—21no—
552C23—22no—
553C23—21no—
554C23—22no—
555C23—21no—
556C23—22no—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—22no—
561C23—21no—
562C23C2×#12312yes—
563C23C2×#13714yes—
564C23C23×D411yescharacteristic
565Q8—44no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—84no—
571D4—28no—
572D4—48no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—22no—
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—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—88no—
628C2×C4—88no—
629C2×C4—164no—
630C2×C4C2×#12312yes—
631C2×C4C2×#12312yes—
632C2×C4C2×#12312yes—
633C2×C4C2×#12312yes—
634C2×C4C2×#12312yes—
635C2×C4C2×#13712yes—
636C2×C4#57512yes—
637C2×C4#57612yes—
638C2×C4#57812yes—
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C2×C4#58511yescharacteristic
644C2×C4#58511yescharacteristic
645C8—44no—
646C8—42no—
647C8—42no—
648C22—24no—
649C22—22no—
650C22—24no—
651C22—22no—
652C22—24no—
653C22—22no—
654C22—41no—
655C22—42no—
656C22—41no—
657C22—42no—
658C22—41no—
659C22—42no—
660C22—41no—
661C22—42no—
662C22C22×#12311yescharacteristic
663C22#348811yescharacteristic
664C22#348911yescharacteristic
665C22C22×#13711yescharacteristic
666C22C2×#57511yescharacteristic
667C22C2×#57611yescharacteristic
668C22C2×#57811yescharacteristic
669C4—22no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—81no—
678C4—82no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—81no—
684C4#309514yes—
685C4#309814yes—
686C4#346011yescharacteristic
687C4#346111yescharacteristic
688C4#351111yescharacteristic
689C4#351211yescharacteristic
690C2—44no—
691C2—42no—
692C2#3525511yescharacteristic
693C2C2×#309511yescharacteristic
694C2C2×#309811yescharacteristic
695C2#6198411yescharacteristic
696C2#6199511yescharacteristic
697C2#6212711yescharacteristic
698C2#6214611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
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. #33146C2semidirect
4. #33109C2semidirect
5. #29941C2semidirect
6. #29926C2semidirect
7. #26221C2semidirect
8. #26145C2semidirect
9. #25063C2semidirect
10. #25016C2semidirect
11. #24995C2semidirect
12. #24969C2semidirect
13. #21982C2semidirect
14. #21894C2semidirect
16. #21389C2semidirect
17. #21171C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#341272
#331462
#331092
#299412
#299262
#262212
#261452
#250632
#250162
#249952
#249692
#219822
#218942
#218492
#213892
#211712