Finite Group Databasebeta

← #6563973

Subgroups — #6563973

Order 512.

2035 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36288C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#30039C212yesmaximal, complemented
4#29922C211yesmaximal, characteristic, complemented
5#29366C212yesmaximal, complemented
6#28370C212yesmaximal, complemented
7#27224C214yesmaximal, complemented
8#26464C212yesmaximal, complemented
9#25717C211yesmaximal, characteristic
10#25368C212yesmaximal, complemented
11#25156C214yesmaximal, complemented
12#24211C211yesmaximal, characteristic
13#23756C211yesmaximal, characteristic, complemented
14#23678C212yesmaximal, complemented
15#23182C212yesmaximal, complemented
16#23095C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518—21no—
20C2×#492C2212yes—
21C2×#476C2214yescomplemented
22C2×#460C2212yes—
23C2×#461C2214yescomplemented
24C2×#450C2212yes—
25C2×#450C2214yes—
26C2×C4×SD16C2214yescomplemented
27#2862C2212yescomplemented
28#2857C2211yescharacteristic
29#2612—22no—
30#2574—22no—
31#2551C2211yescharacteristic, complemented
32#2546C2211yescharacteristic, complemented
33#2508—21no—
34#2492C2212yes—
35#2439—21no—
36#2429C2212yes—
37#2369—21no—
38#2366—21no—
39#2366—21no—
40#2366C2212yes—
41#2366C2212yes—
42#2365C2211yescharacteristic, complemented
43#2355C2211yescharacteristic, complemented
44#2346—21no—
45D4×C22⋊C4—22no—
46D4×C22⋊C4C2214yes—
47#2328—22no—
48#2309—21no—
49C4×#127C2212yes—
50C4×#127C2212yes—
51C4×C22⋊Q8C2211yescharacteristic
52#2295C2212yescomplemented
53#2285C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2099C2211yescharacteristic, complemented
56#2096C2212yescomplemented
57#2096C2211yescharacteristic, complemented
58#2084C2211yescharacteristic, complemented
59#2077C2212yes—
60#2072C2211yescharacteristic, complemented
61#2072C2212yescomplemented
62#2065C2214yescomplemented
63#2055C2214yescomplemented
64#2041C2212yes—
65#2035C2212yes—
66#2035C2214yes—
67#2023C2214yescomplemented
68#2014C2212yes—
69#2014C2212yes—
70#2014C2212yes—
71#2013C2214yescomplemented
72#1982C2212yes—
73#1981C2212yescomplemented
74#1968C2214yescomplemented
75#1968C2214yescomplemented
76#1957C2212yes—
77#1948C2212yescomplemented
78#1939C2212yescomplemented
79#1926C2214yescomplemented
80#1923C2214yes—
81#1880C2214yescomplemented
82#1877C2214yescomplemented
83#1877C2214yescomplemented
84#1877C2214yescomplemented
85#1864C2214yescomplemented
86#1839C2212yescomplemented
87#1797C2211yescharacteristic
88#1786C2212yes—
89C4×#106C2212yescomplemented
90C8×C4⋊C4C2212yescomplemented
91C4×C4⋊C8C2212yescomplemented
92C4×#103C2212yescomplemented
93C4×#102C2214yescomplemented
94C4×C22⋊C8C2211yescharacteristic
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C22×SD16C2314yes—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#127—28no—
106C2×#127—21no—
107C2×#127—21no—
108C2×#127C2312yes—
109C2×#127C2312yes—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C22⋊Q8C2312yes—
112#518—48no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4C2314yes—
122C2×C4×D4C2314yes—
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C22×C22⋊C4—24no—
127C22×C42C2311yescharacteristic
128#492—28no—
129#476—48no—
130#461—48no—
131#460—28no—
132#450—48no—
133#450—44no—
134C4×SD16—48no—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—22no—
154#398—22no—
155#398—22no—
156#396—22no—
157#395—22no—
158#393—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#391C2311yescharacteristic
165#391C2312yes—
166#391C2312yes—
167#390—22no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2314yes—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389C2312yes—
183#389C2311yescharacteristic
184#388—22no—
185#388—22no—
186#387—22no—
187#387—21no—
188#387—22no—
189#387—22no—
190#387—22no—
191#386—24no—
192#386—24no—
193#386—24no—
194#386—24no—
195#385—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#382—21no—
200#382—22no—
201#382—21no—
202#382—22no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380C2311yescharacteristic
207#380C2312yes—
208#379—24no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—21no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224#376—22no—
225#376D412yes—
226C43C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C25—22no—
238C22×Q8—22no—
239C22×Q8C2411yescharacteristic
240C22×Q8C2412yes—
241C22×D4—28no—
242C22×D4—28no—
243C22×D4—28no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—42no—
250C22×D4—44no—
251C22×D4—42no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4—82no—
255C22×D4—82no—
256C22×D4C2414yes—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×SD16—28no—
263C2×SD16—28no—
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—44no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×SD16—44no—
270C2×SD16—44no—
271C22×C8C2412yes—
272C22×C8C2414yes—
273C22×C8C2412yes—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#127—28no—
279#127—28no—
280#127—28no—
281#127—24no—
282#127#4118yes—
283C22⋊Q8—22no—
284C22⋊Q8—41no—
285C22⋊Q8—44no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—42no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4—48no—
296C4×D4—48no—
297C4×D4—84no—
298C4×D4—84no—
299#117—42no—
300#117—44no—
301#117SD1618yescomplemented
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—416no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#106—24no—
359#106—24no—
360#106—44no—
361C4⋊C8—28no—
362C4⋊C8—24no—
363#103—24no—
364#103—42no—
365#103—44no—
366#103—44no—
367#102—28no—
368#102—28no—
369#102—28no—
370#102—44no—
371#102—44no—
372C22⋊C8—24no—
373C22⋊C8—42no—
374C4×C8—24no—
375C4×C8—44no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—28no—
398C24—24no—
399C24—44no—
400C24—44no—
401C24—82no—
402C24—82no—
403C24—82no—
404C24—82no—
405C24C22×D411yescharacteristic
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—44no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—432no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—88no—
439C2×D4—88no—
440C2×D4—816no—
441C2×D4C2×#4118yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—41no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4C2×SD1614yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476SD16—416no—
477SD16—88no—
478SD16—88no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8H(5,𝔽2)14yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4C2×SD1614yes—
516C4⋊C4C2×SD1614yes—
517C4⋊C4H(5,𝔽2)12yes—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—81no—
528C22⋊C4—82no—
529C22⋊C4C2×SD1618yes—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—22no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—42no—
541C42—44no—
542C42—42no—
543C42—82no—
544C42C2×SD1618yes—
545C42C2×#4114yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—28no—
553C23—416no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—88no—
561C23—88no—
562C23—164no—
563C23—164no—
564C23—164no—
565C23—164no—
566C23C22×SD1612yes—
567C23C23×D411yescharacteristic
568Q8—44no—
569Q8—42no—
570Q8—44no—
571Q8—42no—
572Q8—164no—
573D4—28no—
574D4—28no—
575D4—416no—
576D4—416no—
577D4—416no—
578D4—44no—
579D4—44no—
580D4—88no—
581D4—88no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—168no—
630C2×C4—168no—
631C2×C4#52512yes—
632C2×C4#52512yes—
633C2×C4C22×SD1612yes—
634C2×C4C22×SD1614yes—
635C2×C4C22×SD1614yes—
636C2×C4C22×SD1612yes—
637C2×C4#57512yes—
638C2×C4#57712yes—
639C2×C4C22×#4112yes—
640C2×C4C2×H(5,𝔽2)12yes—
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4#58511yescharacteristic
643C8—24no—
644C8—42no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—48no—
654C22—48no—
655C22—48no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—1616no—
661C22C2×#52511yescharacteristic
662C22#348811yescharacteristic
663C22#349811yescharacteristic
664C22C23×SD1611yescharacteristic
665C22C2×#57511yescharacteristic
666C22C2×#57711yescharacteristic
667C22#358511yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—24no—
672C4—24no—
673C4—42no—
674C4—42no—
675C4—41no—
676C4—42no—
677C4—81no—
678C4—81no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—81no—
683C4—82no—
684C4#310114yes—
685C4#351412yes—
686C4#351912yes—
687C2—42no—
688C2—88no—
689C2#3600411yescharacteristic
690C2C2×#310111yescharacteristic
691C2#6057911yescharacteristic
692C2#6195111yescharacteristic
693C2#6198611yescharacteristic
694C2#6241311yescharacteristic
695C2#6245511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36288C2semidirect
3. #30039C2semidirect
4. #29922C2semidirect
5. #29366C2semidirect
6. #28370C2semidirect
7. #27224C2semidirect
8. #26464C2semidirect
10. #25368C2semidirect
11. #25156C2semidirect
13. #23756C2semidirect
14. #23678C2semidirect
15. #23182C2semidirect
16. #23095C2semidirect
17. #22639C2semidirect
21. C2×#476C22semidirect
23. C2×#461C22semidirect
26. C2×C4×SD16C22semidirect
27. #2862C22semidirect
31. #2551C22semidirect
32. #2546C22semidirect
42. #2365C22semidirect
43. #2355C22semidirect
52. #2295C22semidirect
55. #2099C22semidirect
56. #2096C22semidirect
57. #2096C22semidirect
58. #2084C22semidirect
60. #2072C22semidirect
61. #2072C22semidirect
62. #2065C22semidirect
63. #2055C22semidirect
67. #2023C22semidirect
71. #2013C22semidirect
73. #1981C22semidirect
74. #1968C22semidirect
75. #1968C22semidirect
77. #1948C22semidirect
78. #1939C22semidirect
79. #1926C22semidirect
81. #1880C22semidirect
82. #1877C22semidirect
83. #1877C22semidirect
84. #1877C22semidirect
85. #1864C22semidirect
86. #1839C22semidirect
89. C4×#106C22semidirect
90. C8×C4⋊C4C22semidirect
91. C4×C4⋊C8C22semidirect
92. C4×#103C22semidirect
93. C4×#102C22semidirect
301. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#362882
#342102
#300392
#299222
#293662
#283702
#272242
#264642
#257172
#253682
#251562
#242112
#237562
#236782
#231822
#230952
#226392