Finite Group Databasebeta

← #6433525

Subgroups — #6433525

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic, complemented
2#34269C211yesmaximal, characteristic
3#30179C214yesmaximal, complemented
4#30177C214yesmaximal, complemented
5#30173C212yesmaximal, complemented
6#30172C212yesmaximal, complemented
7#30166C212yesmaximal, complemented
8#30165C212yesmaximal, complemented
9#23656C211yesmaximal, characteristic, complemented
10#23643C212yesmaximal, complemented
11#23628C211yesmaximal, characteristic, complemented
12#23552C211yesmaximal, characteristic, complemented
13#23486C212yesmaximal
14C2×#2023C211yesmaximal, characteristic, complemented
15#21039C212yesmaximal, complemented
16#20876C212yesmaximal, complemented
17#20684C211yesmaximal, characteristic, complemented
18C2×#535C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#518C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#418C2212yes—
25C22×#102C2212yes—
26C2×#408C2212yes—
27#2857C2212yescomplemented
28#2857C2212yescomplemented
29#2628C2214yescomplemented
30#2568—21no—
31#2567—21no—
32#2563—21no—
33#2542—21no—
34#2487—22no—
35#2409—22no—
36#2367—21no—
37#2367—21no—
38#2366—21no—
39#2366—21no—
40#2358—22no—
41#2358—22no—
42#2324C2212yes—
43#2324C2212yes—
44#2317C2212yes—
45#2311C2211yescharacteristic
46#2311C2212yes—
47C4×#121C2212yes—
48C4×#121C2212yes—
49#2296C2211yescharacteristic
50C2×#390C2211yescharacteristic
51#2285C2211yescharacteristic
52#2274C2211yescharacteristic
53#2085C2212yescomplemented
54#2085C2212yescomplemented
55#2084C2211yescharacteristic, complemented
56#2084C2211yescharacteristic, complemented
57#2083C2211yescharacteristic, complemented
58#2083C2211yescharacteristic, complemented
59#2077C2214yescomplemented
60#2077C2214yescomplemented
61#2076C2214yescomplemented
62#2075C2214yescomplemented
63#2073C2214yes—
64#2071C2212yes—
65#2071C2212yes—
66#2058C2212yescomplemented
67#2058C2212yescomplemented
68#2057C2212yescomplemented
69#2057C2212yescomplemented
70#2034C2214yes—
71#2034C2214yes—
72#2033C2214yes—
73#2033C2214yes—
74#2023C2212yes—
75#2023C2212yes—
76#2023C2214yes—
77#1921C2214yescomplemented
78#1920C2214yescomplemented
79#1919C2214yescomplemented
80#1918C2214yescomplemented
81#1864C2214yes—
82#1864C2214yes—
83#1862C2214yes—
84#1862C2214yes—
85#1818C2214yescomplemented
86#1817C2212yescomplemented
87#1817C2212yescomplemented
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×D4C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#535—24no—
101#535—24no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#127—21no—
108C2×#124—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×#120—22no—
114#518—44no—
115C4×#41—44no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D412yes—
124C2×C4×D4D412yes—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2312yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—22no—
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141#428—44no—
142#427—28no—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×#107C2314yes—
146C2×#106C2312yes—
147C2×#106C2312yes—
148C2×#106C2314yes—
149#418—28no—
150C2×#102—44no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155#408—44no—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158#400—24no—
159#399—24no—
160#395—24no—
161#393—24no—
162#391—22no—
163#391—21no—
164#391—22no—
165#391—22no—
166#390—22no—
167#390—21no—
168#390—22no—
169#390—22no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#389—22no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#389C2314yes—
181#389C2314yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#386—22no—
187#386—24no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—21no—
193#385—21no—
194#385—22no—
195#385—22no—
196#385—24no—
197#384—22no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382—22no—
201#382C2312yes—
202#380—22no—
203#380—22no—
204#380—21no—
205#380—21no—
206#380C2312yes—
207#380C2312yes—
208#380C2314yes—
209#380C2314yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C25—21no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—24no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—44no—
246C22×Q8—82no—
247C22×Q8—82no—
248C22×Q8C2×D411yescharacteristic
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—24no—
254C22×D4—24no—
255C22×D4C2×D412yes—
256C22×D4C2×D411yescharacteristic
257C22×D4#4114yes—
258C22×D4C2412yes—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2×D412yes—
267C23×C4C2412yes—
268C23×C4C2411yescharacteristic
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282#127—28no—
283#124—28no—
284C22⋊Q8—84no—
285C22⋊Q8—84no—
286#121—24no—
287#121—24no—
288#121—24no—
289#121—24no—
290#120—216no—
291C4×Q8—42no—
292C4×Q8—42no—
293C4×D4—42no—
294C4×D4—42no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—84no—
302C4×D4—84no—
303#117—22no—
304#117—24no—
305#117—24no—
306#117—24no—
307#117—24no—
308#117—44no—
309#117#4114yes—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—48no—
356C2×C22⋊C4—82no—
357C2×C22⋊C4—82no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—21no—
362C2×C42—24no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—44no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—24no—
376#107—24no—
377#107—44no—
378#106—24no—
379#106—24no—
380#106—44no—
381#102—28no—
382#102—28no—
383#102—44no—
384#102—44no—
385C22⋊C8—44no—
386C22⋊C8—44no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401C24—22no—
402C24—24no—
403C24—42no—
404C24—42no—
405C24—81no—
406C24—81no—
407C24—81no—
408C24—81no—
409C24C22×D412yes—
410C24C22×D411yescharacteristic
411#41—48no—
412#41—48no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—82no—
421C2×Q8—82no—
422C2×Q8—816no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—216no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—48no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4C2×#4114yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C42−1+414yes—
492C22×C4C2511yescharacteristic
493M16—48no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—48no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—81no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4C2×#4112yes—
534C4⋊C4C2×#4112yes—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—24no—
559C42—42no—
560C42—42no—
561C42—44no—
562C42—42no—
563C42—82no—
564C42—82no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—48no—
584C23—84no—
585C23—84no—
586C23—162no—
587C23—162no—
588C23—162no—
589C23—162no—
590C23#52512yes—
591C23#53412yes—
592C23#53612yes—
593C23C23×D411yescharacteristic
594Q8—44no—
595Q8—88no—
596Q8—88no—
597D4—28no—
598D4—48no—
599D4—48no—
600D4—44no—
601D4—44no—
602D4—88no—
603D4—88no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—88no—
654C2×C4—164no—
655C2×C4—164no—
656C2×C4—162no—
657C2×C4#52512yes—
658C2×C4#53412yes—
659C2×C4#53612yes—
660C2×C4#53918yes—
661C2×C4#54218yes—
662C2×C4#57612yes—
663C2×C4#57712yes—
664C2×C4C22×#4111yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C2×H(5,𝔽2)12yes—
667C2×C4C2×2−1+412yes—
668C8—44no—
669C8—42no—
670C8—42no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—42no—
678C22—44no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—44no—
683C22—42no—
684C22—44no—
685C22—42no—
686C22—42no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—168no—
692C22#263814yes—
693C22C2×#52511yescharacteristic
694C22C2×#53411yescharacteristic
695C22C2×#53611yescharacteristic
696C22C2×#57611yescharacteristic
697C22C2×#57711yescharacteristic
698C22#358511yescharacteristic
699C22#358611yescharacteristic
700C4—22no—
701C4—22no—
702C4—42no—
703C4—41no—
704C4—42no—
705C4—41no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—84no—
710C4—81no—
711C4—81no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4#263814yes—
716C4C2×#53912yes—
717C4C2×#54212yes—
718C2—24no—
719C2—82no—
720C2—84no—
721C2C2×#263811yescharacteristic
722C2#6059611yescharacteristic
723C2#6059711yescharacteristic
724C2#6075611yescharacteristic
725C2#6075811yescharacteristic
726C2#6087711yescharacteristic
727C2#6088511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
1. #35279C2semidirect
3. #30179C2semidirect
4. #30177C2semidirect
5. #30173C2semidirect
6. #30172C2semidirect
7. #30166C2semidirect
8. #30165C2semidirect
9. #23656C2semidirect
10. #23643C2semidirect
11. #23628C2semidirect
12. #23552C2semidirect
14. C2×#2023C2semidirect
15. #21039C2semidirect
16. #20876C2semidirect
17. #20684C2semidirect
27. #2857C22semidirect
28. #2857C22semidirect
29. #2628C22semidirect
53. #2085C22semidirect
54. #2085C22semidirect
55. #2084C22semidirect
56. #2084C22semidirect
57. #2083C22semidirect
58. #2083C22semidirect
59. #2077C22semidirect
60. #2077C22semidirect
61. #2076C22semidirect
62. #2075C22semidirect
66. #2058C22semidirect
67. #2058C22semidirect
68. #2057C22semidirect
69. #2057C22semidirect
77. #1921C22semidirect
78. #1920C22semidirect
79. #1919C22semidirect
80. #1918C22semidirect
85. #1818C22semidirect
86. #1817C22semidirect
87. #1817C22semidirect

Maximal subgroups

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

subgroupindex
#352792
#342692
#301792
#301772
#301732
#301722
#301662
#301652
#236562
#236432
#236282
#235522
#234862
C2×#20232
#210392
#208762
#206842