Finite Group Databasebeta

← #6817427

Subgroups — #6817427

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C212yesmaximal, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#36115C211yesmaximal, characteristic
4#33230C211yesmaximal, characteristic, complemented
5#31138C214yesmaximal, complemented
6#31134C214yesmaximal, complemented
7#29854C212yesmaximal, complemented
8#28754C211yesmaximal, characteristic, complemented
9#28694C211yesmaximal, characteristic, complemented
10#26462C212yesmaximal, complemented
11#26460C212yesmaximal, complemented
12#26236C212yesmaximal, complemented
13#26232C212yesmaximal
14#25908C212yesmaximal, complemented
15#25678C212yesmaximal, complemented
16C4×#501C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#529C2212yes—
21#3239—216no—
22C2×#501C2212yes—
23C2×#501C2211yescharacteristic
24C2×#489C2212yes—
25C2×#489C2212yes—
26C2×#489C2212yes—
27C2×#484C2214yes—
28C2×#481C2214yes—
29C2×#475C2214yes—
30C2×#474C2214yes—
31C2×#432C2212yes—
32#2868C2211yescharacteristic
33#2858C2211yescharacteristic
34#2857C2212yes—
35#2850C2212yes—
36#2551—21no—
37#2551C2212yes—
38#2546—21no—
39#2542C2211yescharacteristic
40#2541C2211yescharacteristic
41#2512—21no—
42#2471—22no—
43#2467—22no—
44#2445—21no—
45#2440—22no—
46#2440C2214yes—
47#2434—22no—
48#2427C2212yes—
49#2426C2212yes—
50#2366—21no—
51#2366C2211yescharacteristic
52#2365C2211yescharacteristic
53#2354—21no—
54D4×C4⋊C4—21no—
55D4×C4⋊C4C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×#125C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2084C2212yes—
60#2083C2212yes—
61#2067C2214yes—
62#2066C2214yes—
63#2042C2214yes—
64#2036C2214yes—
65#2036C2214yes—
66#2034C2214yes—
67#2034C2214yes—
68#2024C2214yes—
69#2023C2214yes—
70#1993C2212yes—
71#1993C2211yescharacteristic
72#1992C2211yescharacteristic
73#1982C2212yes—
74#1981C2212yes—
75#1959C2212yes—
76#1959C2214yes—
77#1957C2212yes—
78#1956C2212yes—
79#1954C2212yes—
80#1948C2212yes—
81#1947C2212yes—
82#1944C2212yes—
83#1943C2212yes—
84#1939C2212yes—
85#1924C2214yes—
86#1874C2214yes—
87#1872C2214yes—
88#1865C2214yes—
89#1862C2214yes—
90#1847C2211yescharacteristic
91C4×#107C2212yes—
92C4×#106C2212yes—
93#1767C2211yescharacteristic
94#1749C2212yes—
95C4×#97C2211yescharacteristic
96C23×Q8—22no—
97D4×Q8—48no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127—21no—
110C2×#127C2311yescharacteristic
111#529—28no—
112#529—44no—
113C2×#126—22no—
114C2×#125C2311yescharacteristic
115C2×#124C2312yes—
116C2×#124C2312yes—
117C2×#123—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×#121C2312yes—
123C2×#121C2312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8C2312yes—
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2312yes—
132C2×C4×D4C2311yescharacteristic
133C2×#117—22no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2314yes—
136C22×C42C2312yes—
137#501—28no—
138#501D418yescomplemented
139#489—28no—
140#489—28no—
141#489—44no—
142#484—216no—
143#481—216no—
144#475—216no—
145#474—216no—
146#432—28no—
147C2×#107C2314yes—
148C2×#107C2312yes—
149C2×#107C2312yes—
150C2×#106C2314yes—
151C2×#106C2312yes—
152C2×#106C2312yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×#97C2311yescharacteristic
166C2×#97C2312yes—
167C2×#97C2312yes—
168C2×#97C2311yescharacteristic
169C2×C4×C8C2312yes—
170#399—22no—
171#398—24no—
172#397—22no—
173#396—22no—
174#395—24no—
175#395—24no—
176#394—22no—
177#394—22no—
178#392—22no—
179#391—21no—
180#391—22no—
181#391C2312yes—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—22no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390C2312yes—
191#389—22no—
192#389—22no—
193#389—21no—
194#389—21no—
195#389—22no—
196#389C2311yescharacteristic
197#389C2311yescharacteristic
198#389C2312yes—
199#389C2314yes—
200#388—22no—
201#388—22no—
202#388—22no—
203#387—21no—
204#387—21no—
205#387—22no—
206#386—22no—
207#386—22no—
208#386—22no—
209#386—22no—
210#385—24no—
211#385—24no—
212#384—21no—
213#384—21no—
214#384—21no—
215#384C2311yescharacteristic
216#384C2312yes—
217#384C2312yes—
218#382C2311yescharacteristic
219#380—22no—
220#380—22no—
221#380C2314yes—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2311yescharacteristic
232C4×C22⋊C4D412yes—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4C2312yes—
235C43C2311yescharacteristic
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2312yes—
240#340C2312yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—48no—
248C22×Q8—48no—
249C22×Q8—84no—
250C22×Q8C2412yes—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4C2×D411yescharacteristic
256C22×D4C2412yes—
257C23×C4—24no—
258C23×C4—22no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—28no—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272C4⋊Q8—48no—
273C4⋊Q8—44no—
274C4⋊Q8—48no—
275C4⋊Q8—82no—
276C4⋊Q8C2×D414yes—
277#127—24no—
278#127—44no—
279#126—48no—
280#125—22no—
281#125C2×D414yes—
282#124—24no—
283#124—24no—
284#124—44no—
285#123—48no—
286C22⋊Q8—24no—
287C22⋊Q8—416no—
288C22⋊Q8—42no—
289C22⋊Q8—48no—
290C22⋊Q8—44no—
291#121—24no—
292#121—42no—
293#121—44no—
294C4×Q8—24no—
295C4×Q8—42no—
296C4×Q8—44no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—42no—
302C4×D4—48no—
303C4×D4—48no—
304C4×D4—44no—
305#117—48no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
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—22no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—28no—
372#107—42no—
373#107C2×D418yes—
374#106—28no—
375#106—42no—
376#106C2×D418yes—
377C4⋊C8—28no—
378C4⋊C8—24no—
379#103—28no—
380#103—28no—
381#103—44no—
382#103—44no—
383#102—28no—
384#102—28no—
385#102—44no—
386#102—44no—
387C22⋊C8—28no—
388C22⋊C8—44no—
389#97—24no—
390#97—22no—
391#97—42no—
392#97C2×D414yes—
393C4×C8—24no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—21no—
402#95—21no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—22no—
410C24C22×D412yes—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—48no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—48no—
423C2×Q8—84no—
424C2×Q8—832no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4#13714yes—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—82no—
467C22×C4—84no—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C2511yescharacteristic
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8C22×D414yes—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4AGL(1,ℤ/8)14yes—
520C4⋊C4C22×D414yes—
521C4⋊C4C22×D412yes—
522C4⋊C4C22×D412yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—41no—
546C42—41no—
547C42—44no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42C22×D412yes—
552C42H(5,𝔽2)12yes—
553C422−1+414yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23C2×#13714yes—
568C23C23×D411yescharacteristic
569Q8—48no—
570Q8—44no—
571Q8—44no—
572Q8—816no—
573Q8—88no—
574D4—28no—
575D4—44no—
576D4—44no—
577D4—88no—
578D4—84no—
579D4—162no—
580D4—164no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—88no—
630C2×C4—816no—
631C2×C4#53612yes—
632C2×C4#53612yes—
633C2×C4D4212yes—
634C2×C4D4212yes—
635C2×C4C2×AGL(1,ℤ/8)14yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4C2×#13712yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+412yes—
641C2×C4#58512yes—
642C8—24no—
643C8—44no—
644C8—42no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22C2×#53611yescharacteristic
660C22C2×D4211yescharacteristic
661C22#350011yescharacteristic
662C22C22×AGL(1,ℤ/8)11yescharacteristic
663C22C22×#13711yescharacteristic
664C22#358511yescharacteristic
665C22#358611yescharacteristic
666C4—22no—
667C4—24no—
668C4—22no—
669C4—22no—
670C4—41no—
671C4—41no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—81no—
678C4—81no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4#323914yescomplemented
683C4#349712yes—
684C4#355412yes—
685C2—44no—
686C2—162no—
687C2#3936311yescharacteristic
688C2C2×#323911yescharacteristic
689C2#6088511yescharacteristic
690C2D4×AGL(1,ℤ/8)11yescharacteristic
691C2D4×#13711yescharacteristic
692C2#6258211yescharacteristic
693C2#6259511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3239C2semidirect
2. D4×C4⋊Q8C2semidirect
4. #33230C2semidirect
5. #31138C2semidirect
6. #31134C2semidirect
7. #29854C2semidirect
8. #28754C2semidirect
9. #28694C2semidirect
10. #26462C2semidirect
11. #26460C2semidirect
12. #26236C2semidirect
14. #25908C2semidirect
15. #25678C2semidirect
16. C4×#501C2semidirect
17. #21270C2semidirect
138. #501D4semidirect
682. C4#3239semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
D4×C4⋊Q82
#361152
#332302
#311382
#311342
#298542
#287542
#286942
#264622
#264602
#262362
#262322
#259082
#256782
C4×#5012
#212702