Finite Group Databasebeta

← #6458516

Subgroups — #6458516

Order 512.

1985 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35308C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#27985C214yesmaximal, complemented
4#27984C214yesmaximal, complemented
5#27980C212yesmaximal, complemented
6#27978C212yesmaximal, complemented
7#27976C212yesmaximal, complemented
8#27972C212yesmaximal, complemented
9#23630C211yesmaximal, characteristic, complemented
10#23629C211yesmaximal, characteristic, complemented
11#23627C211yesmaximal, characteristic, complemented
12C2×#2072C211yesmaximal, characteristic, complemented
13#21205C212yesmaximal, complemented
14#20915C212yesmaximal
15#20913C212yesmaximal, complemented
16#20763C212yesmaximal, complemented
17#20378C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C2×#536C2211yescharacteristic
20C2×#536C2211yescharacteristic
21C2×#520C2212yes—
22C2×C4×#41C2211yescharacteristic
23C2×#417C2212yes—
24C2×#419C2212yes—
25C2×#416C2212yes—
26C22×#103C2212yes—
27C2×#408C2212yes—
28C2×#405C2212yes—
29#2857C2212yes—
30#2850C2212yes—
31#2621C2212yes—
32#2620C2212yes—
33#2429—21no—
34#2429—21no—
35#2427—21no—
36#2427—21no—
37#2425—22no—
38#2425—22no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—21no—
41D4×C4⋊C4—22no—
42D4×C4⋊C4—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45#2317C2211yescharacteristic
46#2317C2212yes—
47#2317C2214yes—
48#2297C2211yescharacteristic
49C2×#391C2211yescharacteristic
50#2285C2212yes—
51#2285C2211yescharacteristic
52C2×#384C2212yes—
53C2×#384C2211yescharacteristic
54#2074C2212yes—
55#2074C2214yes—
56#2073C2212yes—
57#2072C2211yescharacteristic
58#2072C2212yes—
59#2072C2212yes—
60#2072C2212yes—
61#2072C2212yes—
62#2072C2212yes—
63#2072C2214yes—
64#2072C2211yescharacteristic
65#2072C2212yes—
66#2071C2211yescharacteristic
67#2071C2212yes—
68#2071C2211yescharacteristic
69#2071C2212yes—
70#1960C2214yes—
71#1959C2214yes—
72#1958C2214yes—
73#1956C2214yes—
74#1875C2214yes—
75#1874C2214yes—
76#1873C2214yes—
77#1873C2214yes—
78#1873C2214yes—
79#1872C2214yes—
80#1872C2214yes—
81#1872C2214yes—
82#1836C2214yes—
83#1836C2214yes—
84#1835C2214yes—
85#1834C2214yes—
86#1750C2214yes—
87#1749C2212yes—
88C4×C22⋊C8C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#537—28no—
101#536—28no—
102#536—28no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×#124—21no—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×#121—22no—
116C2×#121—21no—
117C2×#121C2312yes—
118#520—44no—
119#520—44no—
120C4×#41—28no—
121C2×C4×Q8—22no—
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8D411yescharacteristic
124C2×C4×D4—24no—
125C2×C4×D4—22no—
126C2×C4×D4D412yes—
127C2×C4×D4D412yes—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2314yes—
133C2×#117C2312yes—
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2314yes—
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144#419—44no—
145#417—44no—
146#416—44no—
147C2×#103—44no—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158#408—28no—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161#405—28no—
162C2×#97C2314yes—
163C2×C4×C8C2314yes—
164#398—22no—
165#398—22no—
166#397—22no—
167#397—22no—
168#396—22no—
169#396—22no—
170#394—22no—
171#394—22no—
172#391—21no—
173#391—21no—
174#391C2312yes—
175#391C2311yescharacteristic
176#391C2312yes—
177#391C2311yescharacteristic
178#390C2311yescharacteristic
179#390C2311yescharacteristic
180#389—24no—
181#389—24no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—21no—
186#389—22no—
187#389—22no—
188#388—22no—
189#388—22no—
190#386—22no—
191#386—22no—
192#386—21no—
193#386—21no—
194#385—24no—
195#385—24no—
196#385—24no—
197#385—22no—
198#384—21no—
199#384—21no—
200#384—22no—
201#384—22no—
202#384C2312yes—
203#384C2314yes—
204#384C2312yes—
205#384C2312yes—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#382C2312yes—
209#382C2314yes—
210#382C2312yes—
211#380—22no—
212#380—22no—
213#380—21no—
214#380—21no—
215#380—22no—
216#380—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C2×#95—21no—
224C2×#95—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2312yes—
239#336C2312yes—
240C2×#41—24no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41#4118yes—
248C22×Q8—21no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8C2×D411yescharacteristic
254C22×Q8C2412yes—
255C22×Q8C2411yescharacteristic
256C22×D4—21no—
257C22×D4—22no—
258C22×D4C2×D412yes—
259C22×D4C2×D411yescharacteristic
260C22×D4C2411yescharacteristic
261C23×C4—21no—
262C23×C4—24no—
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—24no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C4⋊Q8—28no—
282C4⋊Q8—44no—
283#124—28no—
284#124—44no—
285#123—48no—
286#123—48no—
287C22⋊Q8—28no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290C22⋊Q8—44no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293#121—28no—
294#121—44no—
295#121—48no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—48no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—28no—
302C4×D4—28no—
303C4×D4—416no—
304C4×D4—48no—
305#117—28no—
306#117—24no—
307#117—24no—
308#117—24no—
309#117—44no—
310#117—44no—
311#117#4118yes—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—84no—
338C2×C4⋊C4—82no—
339C2×C4⋊C4—84no—
340C2×C4⋊C4—82no—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—24no—
368C2×C42—28no—
369C2×C42—24no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2412yes—
379#103—44no—
380#103—44no—
381#103—42no—
382#103—42no—
383#103—44no—
384#103—44no—
385#103—44no—
386#102—42no—
387#102—42no—
388#102—44no—
389C22⋊C8—28no—
390C22⋊C8—28no—
391#97—28no—
392C4×C8—28no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407C24—21no—
408C24C22×D412yes—
409C24C22×D411yescharacteristic
410#41—216no—
411#41—216no—
412#41—48no—
413#41—48no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—48no—
428C2×Q8—48no—
429C2×Q8#12414yes—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4#12418yes—
445C2×D4#12414yes—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—48no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—82no—
479C22×C4—84no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4—81no—
483C22×C4—81no—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C2×#4118yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4C2511yescharacteristic
498M16—48no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—48no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4#12314yes—
541C4⋊C4#12314yes—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—48no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—81no—
556C22⋊C4—81no—
557C22⋊C4#12318yes—
558C42—24no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—48no—
565C42—42no—
566C42—44no—
567C42C2×#4118yes—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23C2×#12312yes—
586C23C2×#12414yes—
587C23C2×#12412yes—
588C23#53512yes—
589C23C23×D411yescharacteristic
590Q8—28no—
591Q8—44no—
592Q8—44no—
593Q8—42no—
594Q8—42no—
595Q8—84no—
596D4—216no—
597D4—28no—
598D4—42no—
599D4—42no—
600D4—84no—
601D4—164no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—164no—
651C2×C4—162no—
652C2×C4—168no—
653C2×C4C2×#12314yes—
654C2×C4C2×#12314yes—
655C2×C4C2×#12312yes—
656C2×C4C2×#12414yes—
657C2×C4C2×#12412yes—
658C2×C4#53512yes—
659C2×C4#55318yes—
660C2×C4#55318yes—
661C2×C4#57712yes—
662C2×C4#57812yes—
663C2×C4C22×#4112yes—
664C2×C4C22×#4112yes—
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C8—42no—
668C8—42no—
669C8—44no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22#250414yes—
691C22C22×#12311yescharacteristic
692C22C22×#12411yescharacteristic
693C22C2×#53511yescharacteristic
694C22C2×#57711yescharacteristic
695C22C2×#57811yescharacteristic
696C22#358511yescharacteristic
697C22#358611yescharacteristic
698C4—22no—
699C4—24no—
700C4—22no—
701C4—24no—
702C4—24no—
703C4—41no—
704C4—41no—
705C4—84no—
706C4—82no—
707C4—82no—
708C4—81no—
709C4—81no—
710C4—82no—
711C4—81no—
712C4—82no—
713C4#250414yes—
714C4C2×#55312yes—
715C4C2×#55312yes—
716C2—24no—
717C2—44no—
718C2—161no—
719C2C2×#250411yescharacteristic
720C2#6052211yescharacteristic
721C2#6052611yescharacteristic
722C2#6062311yescharacteristic
723C2#6062511yescharacteristic
724C2#6083311yescharacteristic
725C2#6083711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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
2. #34183C2semidirect
3. #27985C2semidirect
4. #27984C2semidirect
5. #27980C2semidirect
6. #27978C2semidirect
7. #27976C2semidirect
8. #27972C2semidirect
9. #23630C2semidirect
10. #23629C2semidirect
11. #23627C2semidirect
12. C2×#2072C2semidirect
13. #21205C2semidirect
15. #20913C2semidirect
16. #20763C2semidirect
17. #20378C2semidirect

Maximal subgroups

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

subgroupindex
#353082
#341832
#279852
#279842
#279802
#279782
#279762
#279722
#236302
#236292
#236272
C2×#20722
#212052
#209152
#209132
#207632
#203782