Finite Group Databasebeta

← #6480331

Subgroups — #6480331

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#33067C212yesmaximal, complemented
4#33025C212yesmaximal, complemented
5#30149C212yesmaximal, complemented
6#30129C212yesmaximal, complemented
7#29534C214yesmaximal, complemented
8#29276C211yesmaximal, characteristic, complemented
9#29258C211yesmaximal, characteristic, complemented
10#28205C211yesmaximal, characteristic, complemented
11#28160C211yesmaximal, characteristic, complemented
12#27857C212yesmaximal, complemented
13#25497C212yesmaximal, complemented
14#24838C212yesmaximal, complemented
15#24753C212yesmaximal
16#22180C214yesmaximal, complemented
17C2×#1831C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529—21no—
20C2×#475C2212yes—
21C2×#465C2214yes—
22C2×#460C2212yes—
23C22×C4⋊C8C2212yes—
24#2861—21no—
25#2860—21no—
26#2860C2212yes—
27#2859C2211yescharacteristic
28#2858C2211yescharacteristic
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2854—21no—
32#2854C2212yes—
33#2850—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×#127C2211yescharacteristic
40C4×#125—21no—
41C4×#125—21no—
42C4×#124—21no—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8C2212yes—
46C4×C22⋊Q8C2212yes—
47C4×#121—21no—
48C4×#121C2212yes—
49#2296—21no—
50#2295—21no—
51#2295C2212yes—
52#2293—21no—
53#2293C2212yes—
54#2285C2211yescharacteristic
55C42×Q8—21no—
56C42×Q8C2212yes—
57C4×#117—21no—
58C2×C43C2211yescharacteristic
59#2100C2212yes—
60#2094C2212yes—
61#2088C2214yes—
62#2084C2212yes—
63#2071C2212yes—
64#2066C2214yes—
65#2065C2214yes—
66#2058C2212yes—
67#2055C2214yes—
68#2043C2212yes—
69#2023C2212yes—
70#2013C2212yes—
71#1988C2212yes—
72#1982C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1927C2214yes—
76#1926C2214yes—
77#1926C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1919C2214yes—
81#1893C2214yes—
82#1892C2214yes—
83#1880C2214yes—
84#1880C2214yes—
85#1877C2214yes—
86#1865C2214yes—
87#1865C2214yes—
88#1864C2214yes—
89#1831C2211yescharacteristic
90#1831C2212yes—
91#1831C2211yescharacteristic
92#1831C2212yes—
93#1831C2212yes—
94#1807C2214yes—
95#1803C2212yes—
96#1803C2212yes—
97C2×#336C2212yes—
98C2×#336C2212yes—
99C22×SD16C2314yes—
100C23×C8C2314yes—
101#533—28no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106#529—48no—
107C2×#126—22no—
108C2×#125—21no—
109C2×#125—21no—
110C2×#124—21no—
111C2×#124—21no—
112C2×#123—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117—21no—
132C2×#117—22no—
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#475—44no—
141#465—216no—
142#460—44no—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8—44no—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391C2312yes—
161#391C2312yes—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
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#385—22no—
203#385—22no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#382—21no—
209#382—21no—
210#380—21no—
211#380—21no—
212#380C2312yes—
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
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×C4⋊C4C2312yes—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236#376—22no—
237C43—22no—
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#336C2312yes—
255#336C2314yes—
256#336C2314yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8C2412yes—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4C2412yes—
265C23×C4—22no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2411yescharacteristic
273C2×SD16—28no—
274C2×SD16—44no—
275C2×SD16—44no—
276C22×C8—28no—
277C22×C8—44no—
278C22×C8—44no—
279C22×C8C2414yes—
280C22×C8C2414yes—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285#127—22no—
286#127—22no—
287#126—48no—
288#125—28no—
289#125—44no—
290#124—44no—
291#124—44no—
292#123—48no—
293C22⋊Q8—28no—
294C22⋊Q8—24no—
295C22⋊Q8—44no—
296C22⋊Q8#4118yes—
297#121—28no—
298#121—44no—
299#121—44no—
300C4×Q8—28no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—42no—
306C4×Q8—42no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—48no—
313C4×D4—44no—
314#117—28no—
315#117—44no—
316#117—48no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4C2412yes—
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—22no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—42no—
369C2×C42—42no—
370C2×C42—42no—
371C2×C42—42no—
372C2×C42SD1618yescomplemented
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×C42C2×D411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401#106—28no—
402#106—44no—
403#106—44no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—44no—
407#103—28no—
408#103—44no—
409#103—44no—
410#102—28no—
411#102—44no—
412#102—44no—
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—
429C24—22no—
430C24C22×D411yescharacteristic
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8C2×#4114yes—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
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—22no—
474C22×C4—22no—
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—44no—
484C22×C4C2×SD1614yes—
485C22×C4C2×SD1614yes—
486C22×C4C2×SD1614yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
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×C4C2×#4114yes—
501C22×C4C2511yescharacteristic
502SD16—88no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—88no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—28no—
513C4⋊C4—24no—
514C4⋊C4—28no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—44no—
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—28no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—21no—
567C42—21no—
568C42—22no—
569C42—22no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—44no—
581C42—44no—
582C42—44no—
583C42—44no—
584C42—44no—
585C42—44no—
586C42C2×SD1614yes—
587C42C2×SD1614yes—
588C42C2×SD1614yes—
589C42C2×SD1614yes—
590C42H(5,𝔽2)12yes—
591C422−1+412yes—
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—24no—
608Q8—24no—
609Q8—48no—
610Q8—42no—
611Q8—42no—
612Q8—84no—
613D4—44no—
614D4—44no—
615D4—84no—
616D4—88no—
617D4—84no—
618C2×C4—28no—
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—28no—
636C2×C4—24no—
637C2×C4—24no—
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—21no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
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—88no—
676C2×C4#52512yes—
677C2×C4#53712yes—
678C2×C4#53712yes—
679C2×C4C22×SD1612yes—
680C2×C4C22×SD1612yes—
681C2×C4C22×SD1612yes—
682C2×C4C22×SD1612yes—
683C2×C4C22×SD1612yes—
684C2×C4C22×SD1612yes—
685C2×C4#57512yes—
686C2×C4#57512yes—
687C2×C4C22×#4112yes—
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×2−1+411yescharacteristic
690C2×C4#58512yes—
691C8—44no—
692C8—44no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22#310314yes—
707C22C2×#52511yescharacteristic
708C22C2×#53711yescharacteristic
709C22#344911yescharacteristic
710C22C23×SD1611yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57511yescharacteristic
713C22C2×#57511yescharacteristic
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—41no—
723C4—41no—
724C4—44no—
725C4—41no—
726C4—44no—
727C4—41no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4#324614yes—
734C4#351412yes—
735C4#351512yes—
736C2—24no—
737C2—162no—
738C2#3513011yescharacteristic
739C2C2×#310311yescharacteristic
740C2#6054611yescharacteristic
741C2C2×#324611yescharacteristic
742C2#6091711yescharacteristic
743C2#6094111yescharacteristic
744C2#6096111yescharacteristic
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. C4×#533C2semidirect
3. #33067C2semidirect
4. #33025C2semidirect
5. #30149C2semidirect
6. #30129C2semidirect
7. #29534C2semidirect
8. #29276C2semidirect
9. #29258C2semidirect
10. #28205C2semidirect
11. #28160C2semidirect
12. #27857C2semidirect
13. #25497C2semidirect
14. #24838C2semidirect
16. #22180C2semidirect
17. C2×#1831C2semidirect
372. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#5332
#330672
#330252
#301492
#301292
#295342
#292762
#292582
#282052
#281602
#278572
#254972
#248382
#247532
#221802
C2×#18312