Finite Group Databasebeta

← #6464693

Subgroups — #6464693

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#29521C212yesmaximal, complemented
4#29500C212yesmaximal, complemented
5#29467C214yesmaximal, complemented
6#29346C211yesmaximal, characteristic, complemented
7#29312C211yesmaximal, characteristic, complemented
8#29123C211yesmaximal, characteristic, complemented
9#28374C211yesmaximal, characteristic, complemented
10#24710C212yesmaximal, complemented
11#24661C212yesmaximal
12#23760C212yesmaximal, complemented
13#23574C212yesmaximal, complemented
14#21615C212yesmaximal, complemented
15#21505C212yesmaximal, complemented
16#21288C214yesmaximal, complemented
17C4×#432C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529—21no—
20C2×#475C2212yes—
21C2×#460C2212yes—
22C2×#440C2214yes—
23C2×#432C2212yes—
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#2083C2212yes—
62#2077C2214yes—
63#2071C2212yes—
64#2057C2212yes—
65#2043C2212yes—
66#2036C2214yes—
67#2033C2214yes—
68#2023C2212yes—
69#2013C2212yes—
70#2000C2214yes—
71#1988C2212yes—
72#1982C2212yes—
73#1973C2214yes—
74#1968C2214yes—
75#1947C2212yes—
76#1947C2212yes—
77#1943C2214yes—
78#1942C2214yes—
79#1936C2214yes—
80#1935C2214yes—
81#1934C2214yes—
82#1933C2214yes—
83#1925C2214yes—
84#1920C2214yes—
85#1880C2214yes—
86#1863C2214yes—
87#1803C2212yes—
88#1803C2212yes—
89#1776C2214yes—
90#1767C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94#1762C2214yes—
95#1749C2212yes—
96C4×C22⋊C8C2212yes—
97#1745C2212yes—
98C4×#97C2212yes—
99C22×SD16C2314yes—
100#533—28no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105#529—48no—
106C2×#126—22no—
107C2×#125—21no—
108C2×#125—21no—
109C2×#124—21no—
110C2×#124—21no—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2312yes—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C2×#117—21no—
131C2×#117—22no—
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#475—28no—
140#460—28no—
141#440—216no—
142#432—28no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2314yes—
155C2×#97C2314yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2312yes—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391C2312yes—
164#391C2312yes—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
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#389—21no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#386—22no—
204#386—22no—
205#385—22no—
206#385—22no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#382—21no—
212#382—21no—
213#380—21no—
214#380—21no—
215#380C2312yes—
216#380C2312yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2312yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239#376—22no—
240C43—22no—
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43D411yescharacteristic
244C43D411yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247C43C2311yescharacteristic
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#336C2312yes—
256#336C2312yes—
257#336C2314yes—
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—28no—
275C2×SD16—44no—
276C22×C8C2414yes—
277C22×C8C2414yes—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281#127—24no—
282#126—48no—
283#125—28no—
284#125—44no—
285#124—44no—
286#124—44no—
287#123—48no—
288C22⋊Q8—28no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—44no—
292#121—42no—
293#121—42no—
294#121—44no—
295#121—44no—
296C4×Q8—28no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—42no—
301C4×Q8—42no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×Q8#4118yes—
306C4×D4—28no—
307C4×D4—44no—
308C4×D4—44no—
309C4×D4—48no—
310#117—28no—
311#117—48no—
312#117—44no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
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—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
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⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—42no—
366C2×C42—41no—
367C2×C42—41no—
368C2×C42—42no—
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×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—28no—
398#107—44no—
399#106—28no—
400C4⋊C8—24no—
401C4⋊C8#4118yes—
402#103—28no—
403#103—28no—
404#103—44no—
405#102—28no—
406#102—28no—
407#102—44no—
408C22⋊C8—28no—
409#97—28no—
410#97—28no—
411C4×C8—42no—
412C4×C8—42no—
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—24no—
445C2×D4—22no—
446C2×D4—22no—
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—22no—
461C22×C4—22no—
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—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
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×C4C22×D411yescharacteristic
501C22×C4C2511yescharacteristic
502SD16—416no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8C2×#4118yes—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—28no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
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—42no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4C2×#4114yes—
541C22⋊C4—28no—
542C22⋊C4—44no—
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—
554C42—22no—
555C42—22no—
556C42—28no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—44no—
582C42—44no—
583C42—44no—
584C42—44no—
585C42—44no—
586C42—44no—
587C42AGL(1,ℤ/8)14yes—
588C42#13714yes—
589C42C2×#4114yes—
590C42C2×#4114yes—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23#57512yes—
605C23C23×D411yescharacteristic
606Q8—24no—
607Q8—24no—
608Q8—48no—
609Q8—42no—
610Q8—42no—
611Q8—84no—
612D4—44no—
613D4—44no—
614D4—84no—
615D4—88no—
616D4—84no—
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—21no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—21no—
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—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
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—88no—
675C2×C4#52512yes—
676C2×C4#52512yes—
677C2×C4#53212yes—
678C2×C4#53212yes—
679C2×C4C2×AGL(1,ℤ/8)12yes—
680C2×C4C2×AGL(1,ℤ/8)12yes—
681C2×C4C2×AGL(1,ℤ/8)12yes—
682C2×C4C2×#13712yes—
683C2×C4C2×#13712yes—
684C2×C4C2×#13712yes—
685C2×C4#57512yes—
686C2×C4C22×#4112yes—
687C2×C4C22×#4112yes—
688C2×C4#58511yescharacteristic
689C2×C4#58511yescharacteristic
690C8—28no—
691C8—44no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22C2×#52511yescharacteristic
707C22C2×#53211yescharacteristic
708C22#344911yescharacteristic
709C22C22×AGL(1,ℤ/8)11yescharacteristic
710C22C22×#13711yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57511yescharacteristic
713C4—22no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—44no—
726C4—44no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4#318214yes—
733C4#347512yes—
734C4#348012yes—
735C2—42no—
736C2—162no—
737C2#3491111yescharacteristic
738C2#6056011yescharacteristic
739C2#6056111yescharacteristic
740C2C2×#318211yescharacteristic
741C2#6070211yescharacteristic
742C2#6096211yescharacteristic
743C2#6096311yescharacteristic
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. #29521C2semidirect
4. #29500C2semidirect
5. #29467C2semidirect
6. #29346C2semidirect
7. #29312C2semidirect
8. #29123C2semidirect
9. #28374C2semidirect
10. #24710C2semidirect
12. #23760C2semidirect
13. #23574C2semidirect
14. #21615C2semidirect
15. #21505C2semidirect
16. #21288C2semidirect
17. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#5332
#295212
#295002
#294672
#293462
#293122
#291232
#283742
#247102
#246612
#237602
#235742
#216152
#215052
#212882
C4×#4322