Finite Group Databasebeta

← #6657877

Subgroups — #6657877

Order 512.

1653 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#34979C211yesmaximal, characteristic
3#33153C212yesmaximal, complemented
4#32693C212yesmaximal, complemented
5#32644C212yesmaximal, complemented
6#32258C212yesmaximal, complemented
7#32027C212yesmaximal, complemented
8#31793C214yesmaximal, complemented
9#31333C214yesmaximal, complemented
10#29691C212yesmaximal, complemented
11#27631C212yesmaximal, complemented
12#24837C212yesmaximal, complemented
13#24711C212yesmaximal, complemented
14#24302C212yesmaximal, complemented
15#21851C211yesmaximal, characteristic
16C2×#536—21no—
17C2×#533C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#477C2212yes—
21C2×#477C2212yes—
22C2×#432C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2621—21no—
27#2621—21no—
28#2611—21no—
29#2611—21no—
30#2568C2211yescharacteristic
31#2547—21no—
32#2547C2211yescharacteristic
33#2547C2212yes—
34#2546—21no—
35#2546C2212yes—
36#2546C2211yescharacteristic
37#2542C2211yescharacteristic
38#2499—21no—
39#2486—21no—
40#2486—21no—
41#2486C2212yes—
42#2486C2212yes—
43#2432—21no—
44#2415C2212yes—
45#2415C2212yes—
46#2395—21no—
47#2395—21no—
48#2310—21no—
49C4×C4⋊Q8—21no—
50C4×#124—21no—
51C4×C22⋊Q8C2212yes—
52C4×C22⋊Q8C2212yes—
53#2285C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2087C2214yes—
56#2075C2214yes—
57#2075C2214yes—
58#2068C2214yes—
59#2057C2214yes—
60#2057C2214yes—
61#2043C2214yes—
62#2036C2214yes—
63#2036C2214yes—
64#2035C2214yes—
65#2035C2214yes—
66#2024C2214yes—
67#1987C2212yes—
68#1987C2212yes—
69#1985C2212yes—
70#1955C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1944C2214yes—
74#1935C2214yes—
75#1928C2214yes—
76#1919C2214yes—
77#1919C2214yes—
78#1918C2214yes—
79#1892C2214yes—
80#1872C2214yes—
81#1869C2212yes—
82#1869C2212yes—
83#1854C2214yes—
84#1852C2214yes—
85#1825C2214yes—
86#1808C2214yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2212yes—
90#1797C2212yes—
91C22×#41D411yescharacteristic
92C23×Q8—21no—
93C23×C8C2312yes—
94#536—48no—
95#533—42no—
96#533—42no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8—42no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#125—21no—
108C2×#124—22no—
109C2×#124—21no—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C4×#41—42no—
119C4×#41—42no—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C2×#117—21no—
132C2×#117D411yescharacteristic
133C2×#117D411yescharacteristic
134C2×#117D411yescharacteristic
135C2×#117C2312yes—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#477—44no—
142#477—44no—
143#432—42no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8—42no—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2312yes—
159C2×#97C2312yes—
160#400—21no—
161#400—21no—
162#399—21no—
163#399—21no—
164#398—21no—
165#398—21no—
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#397—21no—
171#397—21no—
172#397—22no—
173#397—22no—
174#397—22no—
175#397—22no—
176#397—22no—
177#397—22no—
178#394—22no—
179#394—22no—
180#391—21no—
181#391—21no—
182#391—22no—
183#391—22no—
184#391—21no—
185#391—22no—
186#391—22no—
187#391C2312yes—
188#391C2312yes—
189#390—21no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389—22no—
194#389C2312yes—
195#389C2314yes—
196#389C2314yes—
197#389C2312yes—
198#388—22no—
199#388—22no—
200#387—22no—
201#387—22no—
202#385—22no—
203#385—22no—
204#384—21no—
205#384C2312yes—
206#382—21no—
207#382—21no—
208#381—22no—
209#381—22no—
210#380—22no—
211#380—22no—
212#380—21no—
213#380C2312yes—
214#380C2314yes—
215#380C2312yes—
216#380C2312yes—
217#380C2314yes—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229#376—21no—
230C43—21no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C2×#41—24no—
242C2×#41—42no—
243C2×#41—42no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8—44no—
250C22×Q8C2×D411yescharacteristic
251C22×Q8C2412yes—
252C22×D4C2×D411yescharacteristic
253C22×D4C2×D411yescharacteristic
254C22×D4C2×D411yescharacteristic
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C22×C8—44no—
266C22×C8—42no—
267C22×C8C2414yes—
268C22×C8C2414yes—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273C4⋊Q8—41no—
274C4⋊Q8—41no—
275C4⋊Q8—41no—
276C4⋊Q8—41no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—48no—
281#125—44no—
282#124—48no—
283#124—44no—
284#123—48no—
285#123—48no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—42no—
289C22⋊Q8—42no—
290C22⋊Q8—42no—
291C22⋊Q8—42no—
292C22⋊Q8—44no—
293C22⋊Q8—48no—
294C22⋊Q8—48no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—42no—
298C4×Q8—41no—
299C4×Q8—41no—
300C4×Q8—44no—
301C4×D4—42no—
302C4×D4—42no—
303C4×D4—42no—
304C4×D4—41no—
305C4×D4—41no—
306C4×D4—41no—
307C4×D4—41no—
308C4×D4—41no—
309C4×D4—41no—
310#117—24no—
311#117—42no—
312#117—42no—
313#117—42no—
314#117—42no—
315#117—44no—
316#117—44no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C22⋊C4C2×D411yescharacteristic
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—22no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—44no—
385#107—44no—
386#107—44no—
387#106—28no—
388C4⋊C8—24no—
389C4⋊C8—24no—
390C4⋊C8—42no—
391#103—44no—
392#103—44no—
393#103—44no—
394#103—44no—
395C22⋊C8—44no—
396C22⋊C8—44no—
397C22⋊C8—42no—
398#97—42no—
399#95—22no—
400#95—21no—
401#95—21no—
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—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416C24C22×D411yescharacteristic
417C24C22×D411yescharacteristic
418C24C22×D411yescharacteristic
419#41—416no—
420C2×Q8—22no—
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—42no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—816no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4AGL(1,ℤ/8)14yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
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×C4C2511yescharacteristic
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—84no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C42−1+414yes—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—44no—
563C42—44no—
564C42—42no—
565C42—44no—
566C42—42no—
567C42—44no—
568C42H(5,𝔽2)12yes—
569C42H(5,𝔽2)12yes—
570C23—21no—
571C23—21no—
572C23—22no—
573C23—22no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—22no—
578C23—21no—
579C23—22no—
580C23—21no—
581C23—21no—
582C23—22no—
583C23C2×AGL(1,ℤ/8)12yes—
584C23C2×AGL(1,ℤ/8)12yes—
585C23C23×D411yescharacteristic
586Q8—44no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—84no—
592Q8—88no—
593Q8—88no—
594D4—44no—
595D4—44no—
596D4—44no—
597C2×C4—28no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—88no—
652C2×C4#53412yes—
653C2×C4#53412yes—
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4#57512yes—
656C2×C4#57814yes—
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×2−1+412yes—
660C2×C4#58512yes—
661C8—44no—
662C8—44no—
663C22—22no—
664C22—22no—
665C22—24no—
666C22—24no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—42no—
672C22—42no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—42no—
677C22—42no—
678C22C2×#53411yescharacteristic
679C22#348912yes—
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C2×#57511yescharacteristic
682C22C2×#57812yes—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—44no—
694C4—82no—
695C4—84no—
696C4—82no—
697C4—82no—
698C4—84no—
699C4#320214yes—
700C4#354712yes—
701C4#355011yescharacteristic
702C4#355111yescharacteristic
703C2—42no—
704C2—42no—
705C2—81no—
706C2#3737611yescharacteristic
707C2C2×#320211yescharacteristic
708C2#6075011yescharacteristic
709C2#6213212yes—
710C2#6213312yes—
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. #33153C2semidirect
4. #32693C2semidirect
5. #32644C2semidirect
6. #32258C2semidirect
7. #32027C2semidirect
8. #31793C2semidirect
9. #31333C2semidirect
10. #29691C2semidirect
11. #27631C2semidirect
12. #24837C2semidirect
13. #24711C2semidirect
14. #24302C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#349792
#331532
#326932
#326442
#322582
#320272
#317932
#313332
#296912
#276312
#248372
#247112
#243022
#218512