Finite Group Databasebeta

← #6686262

Subgroups — #6686262

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3360C211yesmaximal, characteristic, complemented
2C2×#3341C211yesmaximal, characteristic
3#39489C211yesmaximal, characteristic, complemented
4#36185C211yesmaximal, characteristic, complemented
5#33197C211yesmaximal, characteristic, complemented
6#30640C214yesmaximal, complemented
7#30595C214yesmaximal, complemented
8#30279C211yesmaximal, characteristic, complemented
9#29828C211yesmaximal, characteristic, complemented
10#28676C211yesmaximal, characteristic, complemented
11#28598C211yesmaximal, characteristic, complemented
12#26552C214yesmaximal, complemented
13#26349C214yesmaximal, complemented
14#25839C211yesmaximal, characteristic, complemented
15#25704C211yesmaximal, characteristic, complemented
16#25646C211yesmaximal, characteristic, complemented
17#25568C211yesmaximal, characteristic, complemented
18#22208C211yesmaximal, characteristic, complemented
19#21113C211yesmaximal, characteristic, complemented
20C2×#556C2211yescharacteristic
21C2×#555C2211yescharacteristic
22C2×#555C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3360—28no—
25#3341—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#488C2212yes—
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#482C2212yes—
32C2×#482C2212yes—
33C2×#479C2212yes—
34C2×#476C2211yescharacteristic
35C2×#476C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#466C2212yes—
38C2×#463C2212yes—
39C2×#461C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×SD16C2212yes—
42C2×C4×D8C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#427C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2850—21no—
48#2674—24no—
49#2620—21no—
50#2567C2211yescharacteristic
51#2567C2211yescharacteristic
52#2563C2211yescharacteristic
53#2542C2211yescharacteristic
54#2521—24no—
55#2501—21no—
56#2500—21no—
57#2499—21no—
58#2499—21no—
59#2441C2214yes—
60#2427C2214yes—
61#2422C2214yes—
62#2362C2214yes—
63#2317—21no—
64C4×#124C2211yescharacteristic
65#2293C2211yescharacteristic
66#2285—21no—
67C42×D4C2211yescharacteristic
68#2095C2214yes—
69#2095C2214yes—
70#2087C2214yes—
71#2066C2214yes—
72#2059C2214yes—
73#2055C2214yes—
74#2042C2214yes—
75#2036C2214yes—
76#2033C2214yes—
77#2000C2214yes—
78#1987C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1948C2214yes—
85#1942C2214yes—
86#1925C2212yes—
87#1923C2214yes—
88#1918C2212yes—
89#1892C2214yes—
90#1879C2212yes—
91#1873C2214yes—
92#1865C2214yes—
93#1862C2212yes—
94#1862C2214yes—
95#1854C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1802C2214yes—
99C4×#107C2211yescharacteristic
100C4×#102C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×SD16C2312yes—
104C22×D8C2311yescharacteristic
105C23×C8C2312yes—
106#556—24no—
107#556—42no—
108#555—24no—
109#555—24no—
110#555—42no—
111#555—42no—
112#554—24no—
113#554—42no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×#126—22no—
120C2×#125C2312yes—
121C2×#124—22no—
122C2×#124—22no—
123C2×#124D411yescharacteristic
124C2×#124C2312yes—
125C2×#123—24no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8C2314yes—
130C2×#121—21no—
131C2×#121—21no—
132C2×#121C2312yes—
133C2×#121C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2311yescharacteristic
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2311yescharacteristic
139C2×C4×D4C2311yescharacteristic
140C2×C4×D4C2311yescharacteristic
141C2×#117—21no—
142C2×#117—21no—
143C2×#117—21no—
144C2×#117—21no—
145C2×#117C2312yes—
146C2×#117C2312yes—
147C22×C4⋊C4—21no—
148C22×C42—21no—
149#500—24no—
150#500—42no—
151#488—28no—
152#487—24no—
153#487—42no—
154#482—28no—
155#482—28no—
156#479—28no—
157#476—24no—
158#476—24no—
159#476—24no—
160#466—28no—
161#463—28no—
162#461—24no—
163C8×Q8—24no—
164C4×SD16—28no—
165C4×D8D418yes—
166C8×D4—24no—
167#427—28no—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2314yes—
172C2×#107C2311yescharacteristic
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2314yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2311yescharacteristic
191C2×#102C2311yescharacteristic
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2314yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2312yes—
196C2×C4×C8C2311yescharacteristic
197#399—24no—
198#399—22no—
199#397—24no—
200#397—24no—
201#396—24no—
202#395—22no—
203#395—22no—
204#394—24no—
205#394—24no—
206#393—22no—
207#391—21no—
208#391—21no—
209#391—22no—
210#391D411yescharacteristic
211#391C2314yes—
212#390—21no—
213#390—21no—
214#390D411yescharacteristic
215#390C2312yes—
216#389—24no—
217#389C2314yes—
218#389C2314yes—
219#389C2312yes—
220#388—24no—
221#388—24no—
222#388—22no—
223#388D414yes—
224#387—24no—
225#387—24no—
226#386—24no—
227#385—24no—
228#384—21no—
229#384—21no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384C2314yes—
234#382—21no—
235#382—21no—
236#382—21no—
237#382D411yescharacteristic
238#380—24no—
239#380C2314yes—
240C4×C4⋊C4—24no—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4—21no—
245C4×C22⋊C4—21no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4C2314yes—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2412yes—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4C2×D411yescharacteristic
266C22×D4C2411yescharacteristic
267C22×D4C2411yescharacteristic
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4C2412yes—
273C2×SD16—24no—
274C2×SD16—44no—
275C2×D8—42no—
276C2×D8C2×D414yes—
277C22×C8—24no—
278C22×C8—44no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2411yescharacteristic
283C22×C8C2411yescharacteristic
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—84no—
293#126—48no—
294#125—24no—
295#125—42no—
296#124—24no—
297#124—24no—
298#124—42no—
299#124—48no—
300#124—48no—
301#123—88no—
302C22⋊Q8—28no—
303C22⋊Q8—48no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C22⋊Q8—44no—
307#121—24no—
308#121—24no—
309#121—42no—
310#121—42no—
311#121—44no—
312#121—44no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—24no—
316C4×Q8—42no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—42no—
324C4×D4—42no—
325C4×D4C2×D414yes—
326C4×D4C2×D414yes—
327#117—24no—
328#117—24no—
329#117—42no—
330#117—42no—
331#117—44no—
332#117—44no—
333#117—44no—
334#117—44no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D414yes—
376C2×C22⋊C4C2×D414yes—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—44no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D412yes—
386C2×C42C2×D411yescharacteristic
387C2×C42C2412yes—
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390#107—24no—
391#107—24no—
392#107—24no—
393#107—44no—
394#107C2×D414yes—
395#106—24no—
396#106—24no—
397#106—24no—
398C4⋊C8—24no—
399C4⋊C8—22no—
400C4⋊C8—22no—
401C4⋊C8—44no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—44no—
406#103—44no—
407#102—24no—
408#102—24no—
409#102—24no—
410#102—24no—
411#102—42no—
412#102C2×D414yes—
413#102C2×D414yes—
414C22⋊C8—24no—
415C22⋊C8—44no—
416C4×C8—24no—
417C4×C8—22no—
418C4×C8C2×D414yes—
419#95—22no—
420#95—21no—
421#95—22no—
422#95—24no—
423#95—21no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95C2×D414yes—
429#95C2×D414yes—
430C24—21no—
431C24—21no—
432C24C22×D412yes—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—82no—
444C2×Q8—82no—
445C2×Q8—82no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—84no—
458C2×D4—84no—
459C2×D4C22×D412yes—
460C2×D4C22×D412yes—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—82no—
483C22×C4—84no—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C2511yescharacteristic
492SD16—48no—
493D8—44no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—48no—
504C2×C8C22×D412yes—
505C2×C8C22×D412yes—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4C22×D412yes—
535C4⋊C4C22×D412yes—
536C4⋊C42−1+414yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—28no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—84no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—42no—
560C42—44no—
561C42—44no—
562C42—44no—
563C42—44no—
564C42—42no—
565C42—42no—
566C42#13514yes—
567C42C22×D412yes—
568C42H(5,𝔽2)12yes—
569C23—24no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—24no—
574C23—22no—
575C23—22no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23C23×D411yescharacteristic
585Q8—44no—
586Q8—42no—
587Q8—42no—
588Q8—44no—
589Q8—162no—
590Q8—162no—
591Q8—162no—
592D4—22no—
593D4—22no—
594D4—22no—
595D4—22no—
596D4—44no—
597D4—42no—
598D4—42no—
599D4—164no—
600D4—164no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—88no—
640C2×C4—164no—
641C2×C4D4212yes—
642C2×C4D4212yes—
643C2×C4C2×#13514yes—
644C2×C4C2×#13512yes—
645C2×C4#57812yes—
646C2×C4C23×D411yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×2−1+412yes—
649C2×C4#58512yes—
650C8—22no—
651C8—22no—
652C8—42no—
653C8—42no—
654C8—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22C2×D4211yescharacteristic
671C22#348912yes—
672C22C22×#13511yescharacteristic
673C22C2×#57811yescharacteristic
674C22#358512yes—
675C4—24no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#328314yes—
690C4#344911yescharacteristic
691C4#349911yescharacteristic
692C4#355412yes—
693C2—82no—
694C2—82no—
695C2—82no—
696C2#3768011yescharacteristic
697C2C2×#328311yescharacteristic
698C2#6108211yescharacteristic
699C2#6209812yes—
700C2#6218312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3341C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3360C2semidirect
3. #39489C2semidirect
4. #36185C2semidirect
5. #33197C2semidirect
6. #30640C2semidirect
7. #30595C2semidirect
8. #30279C2semidirect
9. #29828C2semidirect
10. #28676C2semidirect
11. #28598C2semidirect
12. #26552C2semidirect
13. #26349C2semidirect
14. #25839C2semidirect
15. #25704C2semidirect
16. #25646C2semidirect
17. #25568C2semidirect
18. #22208C2semidirect
19. #21113C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33602
C2×#33412
#394892
#361852
#331972
#306402
#305952
#302792
#298282
#286762
#285982
#265522
#263492
#258392
#257042
#256462
#255682
#222082
#211132