Finite Group Databasebeta

← #6479936

Subgroups — #6479936

Order 512.

1732 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic, complemented
3#33095C212yesmaximal, complemented
4#33005C212yesmaximal, complemented
5#29967C211yesmaximal, characteristic, complemented
6#29951C211yesmaximal, characteristic, complemented
7#29755C211yesmaximal, characteristic
8#29713C211yesmaximal, characteristic
9#29575C212yesmaximal, complemented
10#29555C212yesmaximal, complemented
11#29534C214yesmaximal, complemented
12#29434C214yesmaximal, complemented
13#29390C212yesmaximal, complemented
14#29366C212yesmaximal, complemented
15#22076C212yesmaximal, complemented
16#22041C212yesmaximal, complemented
17#20777C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C22×C4×Q8C2211yescharacteristic
20C2×#481C2212yes—
21C2×#448C2212yes—
22#2861—21no—
23#2860—21no—
24#2860C2212yes—
25#2859C2211yescharacteristic
26#2858C2211yescharacteristic
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2854—21no—
30#2854C2212yes—
31#2850—21no—
32C4×C4⋊Q8C2211yescharacteristic
33C4×#127C2211yescharacteristic
34C4×#124—21no—
35C4×#121—21no—
36C4×#121C2212yes—
37#2295—21no—
38#2295—21no—
39#2295—21no—
40#2295—21no—
41#2295C2212yes—
42#2295C2212yes—
43#2295C2212yes—
44#2295C2212yes—
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47#2293—21no—
48#2293C2212yes—
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#387—21no—
52C2×#387—21no—
53C2×C4×C4⋊C4—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×#376—21no—
56C2×C43C2211yescharacteristic
57#2099C2212yes—
58#2096C2212yes—
59#2088C2214yes—
60#2084C2212yes—
61#2072C2212yes—
62#2066C2214yes—
63#2065C2214yes—
64#2058C2212yes—
65#2055C2214yes—
66#2041C2212yes—
67#2024C2212yes—
68#2014C2212yes—
69#1968C2214yes—
70#1941C2214yes—
71#1934C2214yes—
72#1933C2214yes—
73#1927C2214yes—
74#1926C2214yes—
75#1926C2214yes—
76#1920C2214yes—
77#1919C2214yes—
78#1919C2214yes—
79#1880C2214yes—
80#1880C2214yes—
81#1877C2214yes—
82#1874C2212yes—
83#1874C2212yes—
84#1865C2214yes—
85#1865C2214yes—
86#1864C2214yes—
87#1853C2212yes—
88#1853C2212yes—
89#1839C2212yes—
90#1831C2212yes—
91#1824C2212yes—
92#1817C2212yes—
93#1797C2211yescharacteristic
94#1797C2211yescharacteristic
95#1787C2212yes—
96C4×C4⋊C8C2212yes—
97C4×C22⋊C8C2212yes—
98#1744C2212yes—
99C23×Q8C2311yescharacteristic
100C22×SD16C2314yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103#529—48no—
104C2×#126—22no—
105C2×#125—21no—
106C2×#124—21no—
107C2×#124—21no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42—21no—
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#481—28no—
141#448—28no—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C4×C8C2314yes—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#386—22no—
209#386—22no—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#382—21no—
213#380—21no—
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227#376—22no—
228#376—22no—
229#376—22no—
230#376—22no—
231C2×#95—21no—
232C2×#95—21no—
233C2×#95—21no—
234C2×#95—21no—
235C2×#95—21no—
236C2×#95—21no—
237C2×#95—21no—
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#336C2312yes—
253#336C2312yes—
254#336C2314yes—
255#336C2314yes—
256#336C2314yes—
257C22×Q8—21no—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—42no—
261C22×Q8—42no—
262C22×Q8#4114yes—
263C22×Q8C2412yes—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4C2412yes—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—22no—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2411yescharacteristic
280C2×SD16—28no—
281C2×SD16—44no—
282C2×SD16—44no—
283C22×C8C2414yes—
284C22×C8C2414yes—
285C4⋊Q8—42no—
286#127—42no—
287#126—48no—
288#125—44no—
289#124—44no—
290#124—44no—
291#123—48no—
292C22⋊Q8—44no—
293#121—24no—
294#121—24no—
295#121—44no—
296#121—44no—
297C4×Q8—28no—
298C4×Q8—28no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—44no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—48no—
307C4×D4—44no—
308#117—48no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4#4114yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—22no—
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×C42—28no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—42no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42—42no—
363C2×C42—42no—
364C2×C42—44no—
365C2×C42SD1618yescomplemented
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394#106—28no—
395#106—44no—
396#106—44no—
397C4⋊C8—42no—
398C4⋊C8—42no—
399#103—28no—
400#103—44no—
401#103—44no—
402#102—28no—
403#102—44no—
404#102—44no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C4×C8—44no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428C24—22no—
429C24C22×D411yescharacteristic
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—48no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—48no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8C2×#4114yes—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—42no—
454C2×D4—42no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
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—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4C2×SD1614yes—
495C22×C4C2×SD1614yes—
496C22×C4C2×SD1614yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2511yescharacteristic
513SD16—88no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4C2×#4114yes—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C42—28no—
558C42—28no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—44no—
586C42—44no—
587C42—44no—
588C42C2×SD1614yes—
589C42C2×SD1614yes—
590C42C2×SD1614yes—
591C42C2×SD1614yes—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23#52512yes—
604C23C22×SD1612yes—
605C23#57512yes—
606C23C23×D411yescharacteristic
607Q8—28no—
608Q8—48no—
609Q8—44no—
610Q8—84no—
611D4—44no—
612D4—44no—
613D4—84no—
614D4—84no—
615D4—88no—
616C2×C4—28no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—48no—
657C2×C4—48no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—88no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4#52512yes—
681C2×C4C22×SD1612yes—
682C2×C4C22×SD1612yes—
683C2×C4C22×SD1612yes—
684C2×C4C22×SD1612yes—
685C2×C4C22×SD1612yes—
686C2×C4C22×SD1612yes—
687C2×C4#57512yes—
688C2×C4#57512yes—
689C2×C4C22×#4111yescharacteristic
690C2×C4C22×#4111yescharacteristic
691C2×C4#58512yes—
692C2×C4#58512yes—
693C8—44no—
694C8—44no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22#310314yes—
709C22C2×#52511yescharacteristic
710C22#344911yescharacteristic
711C22#344911yescharacteristic
712C22C23×SD1611yescharacteristic
713C22C2×#57511yescharacteristic
714C22C2×#57511yescharacteristic
715C22C2×#57511yescharacteristic
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—41no—
726C4—41no—
727C4—44no—
728C4—41no—
729C4—41no—
730C4—44no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4#349112yes—
737C4#349312yes—
738C2—24no—
739C2—162no—
740C2#3511811yescharacteristic
741C2C2×#310311yescharacteristic
742C2#6055911yescharacteristic
743C2#6094111yescharacteristic
744C2#6094111yescharacteristic
745C2#6095911yescharacteristic
746C2#6096411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2295C2semidirect
3. #33095C2semidirect
4. #33005C2semidirect
5. #29967C2semidirect
6. #29951C2semidirect
9. #29575C2semidirect
10. #29555C2semidirect
11. #29534C2semidirect
12. #29434C2semidirect
13. #29390C2semidirect
14. #29366C2semidirect
15. #22076C2semidirect
16. #22041C2semidirect
17. #20777C2semidirect
365. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×#22952
#330952
#330052
#299672
#299512
#297552
#297132
#295752
#295552
#295342
#294342
#293902
#293662
#220762
#220412
#207772