Finite Group Databasebeta

← #6435286

Subgroups — #6435286

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#30114C214yesmaximal, complemented
4#30060C211yesmaximal, characteristic, complemented
5#30059C211yesmaximal, characteristic, complemented
6#30042C211yesmaximal, characteristic, complemented
7#30041C211yesmaximal, characteristic, complemented
8#29985C212yesmaximal, complemented
9#29927C212yesmaximal, complemented
10#29801C212yesmaximal, complemented
11#29668C212yesmaximal, complemented
12#23817C212yesmaximal, complemented
13#23797C212yesmaximal, complemented
14#23571C212yesmaximal, complemented
15#23570C212yesmaximal
16C4×#464C214yesmaximal, complemented
17#20777C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#532C2211yescharacteristic
21C2×#464C2214yes—
22C2×C4×Q16C2214yes—
23C2×C4×SD16C2214yes—
24#2862C2211yescharacteristic
25#2862C2211yescharacteristic
26#2861—21no—
27#2861—21no—
28#2858C2211yescharacteristic
29#2858C2211yescharacteristic
30#2850—21no—
31#2850—21no—
32#2850C2212yes—
33#2850C2212yes—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×#125—21no—
39C4×#125—21no—
40C4×#124—21no—
41C4×#124—21no—
42C4×#124C2212yes—
43C4×#124C2212yes—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2212yes—
48#2297—21no—
49#2297—21no—
50#2296—21no—
51#2295—21no—
52#2295C2212yes—
53#2285C2211yescharacteristic
54C42×Q8—21no—
55C42×Q8C2212yes—
56C4×#117—21no—
57C2×C43C2211yescharacteristic
58#2101C2214yes—
59#2097C2214yes—
60#2096C2212yes—
61#2093C2212yes—
62#2072C2212yes—
63#2071C2212yes—
64#2058C2212yes—
65#2057C2212yes—
66#2043C2212yes—
67#2042C2214yes—
68#2041C2212yes—
69#2036C2214yes—
70#2034C2214yes—
71#2024C2212yes—
72#2024C2212yes—
73#2024C2212yes—
74#2023C2212yes—
75#2016C2212yes—
76#2016C2212yes—
77#2015C2214yes—
78#2014C2212yes—
79#2013C2212yes—
80#2001C2214yes—
81#2000C2214yes—
82#1983C2214yes—
83#1949C2214yes—
84#1927C2214yes—
85#1925C2214yes—
86#1880C2214yes—
87#1878C2214yes—
88#1839C2211yescharacteristic
89#1839C2211yescharacteristic
90#1831C2212yes—
91#1817C2212yes—
92#1805C2214yes—
93#1797C2212yes—
94C4×C4⋊C8C2212yes—
95C4×#103C2214yes—
96C4×#102C2214yes—
97C4×C22⋊C8C2212yes—
98#1744C2212yes—
99C22×#41—21no—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102#533—28no—
103#533—44no—
104#532—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#125—21no—
112C2×#125—21no—
113C2×#124—21no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#124C2312yes—
117C2×#124C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×#121—21no—
123C2×#121C2312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117—21no—
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#464—216no—
145C4×Q16—48no—
146C4×SD16—48no—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C4×C8C2314yes—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#390C2312yes—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#386—22no—
201#386—22no—
202#386—22no—
203#386—22no—
204#385—22no—
205#385—22no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#382—21no—
209#382—21no—
210#382—21no—
211#382C2312yes—
212#382C2312yes—
213#380—21no—
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237#376—22no—
238C43—22no—
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2312yes—
254#336C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C2×#41—44no—
259C2×#41—44no—
260C2×#41—44no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8C2412yes—
265C22×Q8C2412yes—
266C22×D4—22no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C2412yes—
270C23×C4—22no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C2×Q16—28no—
279C2×Q16—44no—
280C2×Q16—44no—
281C2×SD16—28no—
282C2×SD16—44no—
283C2×SD16—44no—
284C22×C8C2414yes—
285C22×C8C2414yes—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—42no—
289C4⋊Q8—42no—
290C4⋊Q8—42no—
291#127—42no—
292#125—28no—
293#125—44no—
294#124—28no—
295#124—42no—
296#124—42no—
297#124—44no—
298#124—44no—
299#124—44no—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305#121—44no—
306#121—44no—
307C4×Q8—28no—
308C4×Q8—24no—
309C4×Q8—28no—
310C4×Q8—28no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8#4118yescomplemented
316C4×D4—28no—
317C4×D4—44no—
318#117—28no—
319#117—44no—
320#117—44no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
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⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—42no—
370C2×C42—42no—
371C2×C42—42no—
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×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—44no—
401#106—44no—
402C4⋊C8—28no—
403#103—28no—
404#103—44no—
405#103—44no—
406#103—44no—
407#103—44no—
408#102—28no—
409#102—44no—
410#102—44no—
411C22⋊C8—44no—
412C4×C8—28no—
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—
428#95—22no—
429#95—22no—
430#95—22no—
431C24—22no—
432C24C22×D411yescharacteristic
433#41—816no—
434C2×Q8—24no—
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8C2×#4114yes—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
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—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4#13714yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
509Q16—416no—
510SD16—416no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—28no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4C2×#4114yes—
542C22⋊C4—28no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
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—
555C42—22no—
556C42—22no—
557C42—28no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—44no—
582C42—44no—
583C42—44no—
584C42—44no—
585C42—44no—
586C42—44no—
587C42#13514yes—
588C42C2×#4114yes—
589C42H(5,𝔽2)14yes—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23C2×#13712yes—
604C23C23×D411yescharacteristic
605Q8—28no—
606Q8—48no—
607Q8—44no—
608Q8—44no—
609Q8—44no—
610Q8—84no—
611Q8—84no—
612D4—44no—
613D4—44no—
614D4—84no—
615D4—84no—
616D4—88no—
617C2×C4—28no—
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—28no—
635C2×C4—21no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
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—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
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—88no—
674C2×C4#52512yes—
675C2×C4#52512yes—
676C2×C4#53512yes—
677C2×C4#53512yes—
678C2×C4C2×#13512yes—
679C2×C4C2×#13512yes—
680C2×C4C2×#13512yes—
681C2×C4C2×#13712yes—
682C2×C4C2×#13712yes—
683C2×C4#57512yes—
684C2×C4#57512yes—
685C2×C4C22×#4112yes—
686C2×C4C2×H(5,𝔽2)12yes—
687C2×C4#58512yes—
688C8—44no—
689C8—44no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22C2×#52511yescharacteristic
705C22C2×#53511yescharacteristic
706C22#344911yescharacteristic
707C22C22×#13511yescharacteristic
708C22C22×#13711yescharacteristic
709C22C2×#57511yescharacteristic
710C22C2×#57511yescharacteristic
711C4—22no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—44no—
724C4—44no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4#321514yes—
731C4#346014yes—
732C2—42no—
733C2—162no—
734C2#3460911yescharacteristic
735C2#6053611yescharacteristic
736C2#6056011yescharacteristic
737C2#6078211yescharacteristic
738C2C2×#321511yescharacteristic
739C2#6094711yescharacteristic
740C2#6095311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#533C2semidirect
3. #30114C2semidirect
4. #30060C2semidirect
5. #30059C2semidirect
6. #30042C2semidirect
7. #30041C2semidirect
8. #29985C2semidirect
9. #29927C2semidirect
10. #29801C2semidirect
11. #29668C2semidirect
12. #23817C2semidirect
13. #23797C2semidirect
14. #23571C2semidirect
16. C4×#464C2semidirect
17. #20777C2semidirect
315. C4×Q8#41semidirect

Maximal subgroups

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

subgroupindex
#366392
C4×#5332
#301142
#300602
#300592
#300422
#300412
#299852
#299272
#298012
#296682
#238172
#237972
#235712
#235702
C4×#4642
#207772