Finite Group Databasebeta

← #6657560

Subgroups — #6657560

Order 512.

1683 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#34952C211yesmaximal, characteristic
3#32954C212yesmaximal, complemented
4#32591C214yesmaximal, complemented
5#32468C212yesmaximal, complemented
6#32269C212yesmaximal, complemented
7#32016C214yesmaximal, complemented
8#31683C212yesmaximal, complemented
9#31273C212yesmaximal, complemented
10#29936C212yesmaximal, complemented
11#27629C212yesmaximal, complemented
12#24845C212yesmaximal, complemented
13#24710C212yesmaximal, complemented
14#24301C212yesmaximal, complemented
15#21853C211yesmaximal, characteristic
16C2×#536—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#475C2212yes—
21C2×#475C2212yes—
22C2×#432C2211yescharacteristic
23C2×#423C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2621—21no—
27#2621—21no—
28#2582C2212yes—
29#2581—21no—
30#2580—21no—
31#2579—21no—
32#2579—21no—
33#2547—21no—
34#2547C2212yes—
35#2546—21no—
36#2546C2212yes—
37#2538C2212yes—
38#2487C2211yescharacteristic
39#2487C2211yescharacteristic
40#2486—21no—
41#2486—21no—
42#2486C2212yes—
43#2486C2212yes—
44#2409C2211yescharacteristic
45#2409C2211yescharacteristic
46#2396—21no—
47#2396—21no—
48#2310—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2212yes—
51#2297—21no—
52#2296—21no—
53#2285C2211yescharacteristic
54#2285C2211yescharacteristic
55#2099C2214yes—
56#2088C2214yes—
57#2083C2214yes—
58#2077C2214yes—
59#2076C2214yes—
60#2075C2214yes—
61#2075C2214yes—
62#2068C2214yes—
63#2058C2214yes—
64#2036C2214yes—
65#2036C2214yes—
66#2024C2214yes—
67#1988C2212yes—
68#1988C2212yes—
69#1985C2212yes—
70#1955C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1943C2214yes—
74#1935C2214yes—
75#1928C2214yes—
76#1921C2214yes—
77#1894C2214yes—
78#1872C2214yes—
79#1870C2212yes—
80#1870C2212yes—
81#1865C2214yes—
82#1865C2214yes—
83#1853C2214yes—
84#1853C2214yes—
85#1809C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2212yes—
90#1787C2214yes—
91C22×#41D411yescharacteristic
92C23×Q8—21no—
93C23×Q8C2311yescharacteristic
94C22×M16C2312yes—
95#536—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#124—22no—
104C2×#123—22no—
105C2×#123—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C4×#41—42no—
111C4×#41—42no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—42no—
115C2×C4×Q8—42no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—21no—
127C2×#117—21no—
128C2×#117D411yescharacteristic
129C2×#117D411yescharacteristic
130C2×#117D411yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#475—44no—
140#475—44no—
141#432—42no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146#423—42no—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2312yes—
158#400—21no—
159#400—21no—
160#399—21no—
161#399—21no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#397—22no—
169#397—22no—
170#397—22no—
171#397—22no—
172#396—22no—
173#396—22no—
174#395—21no—
175#395—21no—
176#393—21no—
177#393—21no—
178#391—22no—
179#391—22no—
180#391—22no—
181#391—22no—
182#391—22no—
183#391C2312yes—
184#391C2312yes—
185#390—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389C2312yes—
191#389C2314yes—
192#389C2312yes—
193#389C2314yes—
194#387—22no—
195#387—22no—
196#386—21no—
197#386—21no—
198#386—21no—
199#386—21no—
200#385—22no—
201#385—22no—
202#385—21no—
203#385—21no—
204#385—21no—
205#385—21no—
206#384C2312yes—
207#384C2312yes—
208#382—21no—
209#382—21no—
210#381—22no—
211#381—22no—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—21no—
216#380C2312yes—
217#380C2314yes—
218#380C2311yescharacteristic
219#380C2311yescharacteristic
220#380C2314yes—
221#380C2312yes—
222#380C2312yes—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
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—24no—
243C2×#41—42no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—24no—
248C22×Q8—41no—
249C22×Q8—41no—
250C22×Q8—42no—
251C22×Q8—44no—
252C22×Q8—44no—
253C22×Q8—44no—
254C22×Q8C2×D411yescharacteristic
255C22×Q8C2412yes—
256C22×D4C2×D411yescharacteristic
257C22×D4C2×D411yescharacteristic
258C22×D4C2×D411yescharacteristic
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C2×M16—44no—
272C2×M16—42no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—48no—
282#124—48no—
283#123—48no—
284#123—48no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—42no—
288C22⋊Q8—42no—
289C22⋊Q8—48no—
290C22⋊Q8—48no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—42no—
294C4×Q8—41no—
295C4×Q8—41no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—42no—
303C4×D4—41no—
304C4×D4—41no—
305C4×D4—41no—
306C4×D4—41no—
307C4×D4—41no—
308C4×D4—41no—
309#117—42no—
310#117—42no—
311#117—42no—
312#117—44no—
313#117—44no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4—41no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—44no—
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⋊C4C2412yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2414yes—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
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—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—44no—
384#107—44no—
385#106—44no—
386#106—44no—
387C4⋊C8—24no—
388C4⋊C8—24no—
389C4⋊C8—42no—
390#103—28no—
391#103—44no—
392#103—44no—
393#103—44no—
394C22⋊C8—44no—
395C22⋊C8—44no—
396C22⋊C8—42no—
397#97—42no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—21no—
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—416no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—41no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—816no—
439C2×Q82−1+414yes—
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—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—82no—
487C22×C4AGL(1,ℤ/8)14yes—
488C22×C4#13714yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C2511yescharacteristic
501M16—84no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—48no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—44no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42H(5,𝔽2)12yes—
572C42H(5,𝔽2)12yes—
573C23—22no—
574C23—22no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—22no—
579C23—21no—
580C23—21no—
581C23—22no—
582C23—21no—
583C23—22no—
584C23—21no—
585C23—21no—
586C23C2×AGL(1,ℤ/8)12yes—
587C23C2×#13712yes—
588C23C23×D411yescharacteristic
589Q8—24no—
590Q8—24no—
591Q8—44no—
592Q8—48no—
593Q8—42no—
594Q8—42no—
595Q8—88no—
596Q8—88no—
597D4—44no—
598D4—44no—
599D4—44no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—88no—
655C2×C4—84no—
656C2×C4#53412yes—
657C2×C4#53412yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×#13712yes—
661C2×C4C2×#13712yes—
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×2−1+412yes—
665C2×C4#58512yes—
666C8—44no—
667C8—44no—
668C22—22no—
669C22—22no—
670C22—24no—
671C22—24no—
672C22—41no—
673C22—41no—
674C22—42no—
675C22—42no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—42no—
681C22—42no—
682C22—41no—
683C22C2×#53411yescharacteristic
684C22#348912yes—
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C22×#13711yescharacteristic
687C22#358612yes—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—44no—
699C4—84no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—84no—
704C4#319314yes—
705C4#353811yescharacteristic
706C4#354011yescharacteristic
707C4#354112yes—
708C2—42no—
709C2—42no—
710C2—81no—
711C2#3737411yescharacteristic
712C2C2×#319311yescharacteristic
713C2#6075411yescharacteristic
714C2#6211112yes—
715C2#6211312yes—
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. #32954C2semidirect
4. #32591C2semidirect
5. #32468C2semidirect
6. #32269C2semidirect
7. #32016C2semidirect
8. #31683C2semidirect
9. #31273C2semidirect
10. #29936C2semidirect
11. #27629C2semidirect
12. #24845C2semidirect
13. #24710C2semidirect
14. #24301C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#349522
#329542
#325912
#324682
#322692
#320162
#316832
#312732
#299362
#276292
#248452
#247102
#243012
#218532