Finite Group Databasebeta

← #6477161

Subgroups — #6477161

Order 512.

1723 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2C4×#536C211yesmaximal, characteristic
3#33154C212yesmaximal, complemented
4#32954C212yesmaximal, complemented
5#30181C214yesmaximal, complemented
6#29311C212yesmaximal, complemented
7#29118C212yesmaximal, complemented
8#27649C212yesmaximal, complemented
9#27609C212yesmaximal
10#24710C212yesmaximal, complemented
11#24703C212yesmaximal, complemented
12#24301C212yesmaximal, complemented
13#24016C212yesmaximal
14#22092C214yesmaximal, complemented
15#20721C211yesmaximal, characteristic
16C2×#536C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#485C2214yes—
21C2×#475C2212yes—
22C2×#475C2212yes—
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2628C2212yes—
28#2628C2212yes—
29#2568—21no—
30#2567—21no—
31#2563—21no—
32#2542—21no—
33#2487—21no—
34#2487—21no—
35#2409—21no—
36#2409—21no—
37#2327—21no—
38#2326—21no—
39#2323C2212yes—
40#2323C2212yes—
41#2322—21no—
42#2322—21no—
43#2317C2211yescharacteristic
44#2317C2212yes—
45#2311C2211yescharacteristic
46#2311C2212yes—
47#2310C2211yescharacteristic
48C4×C4⋊Q8—21no—
49C4×#124—21no—
50C4×C22⋊Q8C2212yes—
51C4×C22⋊Q8C2212yes—
52C4×#123—21no—
53C4×#123—21no—
54#2285C2211yescharacteristic
55C4×#117C2211yescharacteristic
56#2085C2214yes—
57#2077C2214yes—
58#2077C2214yes—
59#2074C2214yes—
60#2058C2214yes—
61#2036C2214yes—
62#2035C2214yes—
63#2024C2214yes—
64#1988C2212yes—
65#1988C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1943C2214yes—
69#1941C2214yes—
70#1935C2214yes—
71#1933C2214yes—
72#1921C2214yes—
73#1920C2214yes—
74#1872C2214yes—
75#1865C2214yes—
76#1865C2214yes—
77#1854C2214yes—
78#1853C2214yes—
79#1853C2214yes—
80#1825C2214yes—
81#1825C2214yes—
82#1818C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1797C2212yes—
86#1797C2212yes—
87#1787C2214yes—
88#1780C2212yes—
89#1750C2212yes—
90#1750C2212yes—
91C22×#41C2311yescharacteristic
92C23×Q8—21no—
93C23×Q8C2311yescharacteristic
94#536—28no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#124—21no—
103C2×#123—21no—
104C2×#123—21no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C4×#41—42no—
110C4×#41—42no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—42no—
114C2×C4×Q8—42no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#485—216no—
140#475—44no—
141#475—44no—
142#432—42no—
143#431—42no—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160#400—22no—
161#400—22no—
162#399—22no—
163#399—22no—
164#395—22no—
165#395—22no—
166#393—22no—
167#393—22no—
168#391—21no—
169#391—22no—
170#391—21no—
171#391—21no—
172#391—22no—
173#391—22no—
174#391C2312yes—
175#391C2312yes—
176#390—22no—
177#390—21no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2312yes—
184#389C2314yes—
185#389C2314yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#386—21no—
195#386—21no—
196#386—22no—
197#386—22no—
198#385—22no—
199#385—22no—
200#385—21no—
201#385—21no—
202#385—22no—
203#385—22no—
204#384—21no—
205#384—22no—
206#384C2312yes—
207#382—22no—
208#382—21no—
209#382C2312yes—
210#381—21no—
211#381—21no—
212#380—22no—
213#380—22no—
214#380—21no—
215#380—21no—
216#380C2312yes—
217#380C2312yes—
218#380C2314yes—
219#380C2314yes—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230#376—21no—
231C43—21no—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#336C2314yes—
241#336C2314yes—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—41no—
245C2×#41—41no—
246C22×Q8—24no—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—24no—
251C22×Q8—41no—
252C22×Q8—41no—
253C22×Q8—44no—
254C22×Q8—84no—
255C22×Q8C2×D411yescharacteristic
256C22×Q8C2412yes—
257C22×D4C2×D411yescharacteristic
258C22×D4C2×D411yescharacteristic
259C22×D4C2×D411yescharacteristic
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—21no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—28no—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281#124—28no—
282#123—28no—
283#123—28no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—84no—
288C22⋊Q8—84no—
289C22⋊Q8#4118yes—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×D4—42no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—42no—
303C4×D4—42no—
304#117—24no—
305#117—28no—
306#117—24no—
307#117—41no—
308#117—41no—
309#117—41no—
310#117—41no—
311#117—41no—
312#117—41no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—41no—
328C2×C4⋊C4—41no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
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⋊C4C2414yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C22⋊C4C2×D411yescharacteristic
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C22⋊C4C2×D411yescharacteristic
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—44no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—28no—
384#106—28no—
385#106—44no—
386#106—44no—
387C4⋊C8—24no—
388C4⋊C8—24no—
389#103—28no—
390#103—28no—
391#103—44no—
392#103—44no—
393C22⋊C8—28no—
394C22⋊C8—28no—
395C22⋊C8—42no—
396C22⋊C8—42no—
397#97—42no—
398C4×C8—42no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415C24C22×D411yescharacteristic
416C24C22×D411yescharacteristic
417C24C22×D411yescharacteristic
418#41—48no—
419#41—48no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—41no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—84no—
438C2×Q8—816no—
439C2×Q8C2×#4114yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—82no—
483C22×C4—84no—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4114yes—
496C22×C42−1+414yes—
497C22×C4C2511yescharacteristic
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4C2×#4114yes—
533C22⋊C4—24no—
534C22⋊C4—28no—
535C22⋊C4—28no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—44no—
562C42—44no—
563C42—44no—
564C42—42no—
565C42—44no—
566C42—82no—
567C42H(5,𝔽2)12yes—
568C42H(5,𝔽2)12yes—
569C23—22no—
570C23—22no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—22no—
582C23#52512yes—
583C23#53612yes—
584C23C23×D411yescharacteristic
585Q8—24no—
586Q8—24no—
587Q8—44no—
588Q8—48no—
589Q8—42no—
590Q8—42no—
591Q8—88no—
592Q8—88no—
593D4—44no—
594D4—44no—
595D4—44no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4#52512yes—
650C2×C4#53412yes—
651C2×C4#53412yes—
652C2×C4#53612yes—
653C2×C4#57714yes—
654C2×C4C22×#4112yes—
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×2−1+412yes—
658C8—44no—
659C8—44no—
660C22—22no—
661C22—22no—
662C22—24no—
663C22—24no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—42no—
669C22—42no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—42no—
674C22—42no—
675C22C2×#52511yescharacteristic
676C22C2×#53411yescharacteristic
677C22C2×#53611yescharacteristic
678C22C2×#57712yes—
679C22#358612yes—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—41no—
684C4—44no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—44no—
692C4—84no—
693C4—82no—
694C4—84no—
695C4—82no—
696C4#320014yes—
697C4#352012yes—
698C4#352012yes—
699C2—42no—
700C2—42no—
701C2—81no—
702C2#3508411yescharacteristic
703C2#6059912yes—
704C2C2×#320011yescharacteristic
705C2#6075411yescharacteristic
706C2#6088912yes—
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. #33154C2semidirect
4. #32954C2semidirect
5. #30181C2semidirect
6. #29311C2semidirect
7. #29118C2semidirect
8. #27649C2semidirect
10. #24710C2semidirect
11. #24703C2semidirect
12. #24301C2semidirect
14. #22092C2semidirect

Maximal subgroups

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

subgroupindex
#352792
C4×#5362
#331542
#329542
#301812
#293112
#291182
#276492
#276092
#247102
#247032
#243012
#240162
#220922
#207212