Finite Group Databasebeta

← #6437339

Subgroups — #6437339

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#30384C212yesmaximal, complemented
4#30378C212yesmaximal, complemented
5#30367C212yesmaximal, complemented
6#30356C212yesmaximal, complemented
7#30345C212yesmaximal, complemented
8#30337C212yesmaximal, complemented
9#30327C212yesmaximal
10#30317C212yesmaximal, complemented
11#23839C211yesmaximal, characteristic, complemented
12#23823C211yesmaximal, characteristic, complemented
13#23822C211yesmaximal, characteristic, complemented
14#23803C211yesmaximal, characteristic, complemented
15#23710C214yesmaximal, complemented
16#21288C214yesmaximal, complemented
17C4×#432C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#440C2214yes—
20C2×#432C2212yes—
21#2864—21no—
22#2863—21no—
23#2863C2212yes—
24#2862C2211yescharacteristic
25#2861—21no—
26#2860—21no—
27#2860C2212yes—
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2855—21no—
36#2855C2212yes—
37#2854—21no—
38#2854C2212yes—
39#2851—21no—
40#2850—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#125—21no—
46C4×#124—21no—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8C2212yes—
49C4×#121—21no—
50C4×#121C2212yes—
51#2295—21no—
52#2295C2212yes—
53#2293—21no—
54#2293C2212yes—
55#2283—21no—
56C2×C43C2211yescharacteristic
57#2100C2212yes—
58#2099C2212yes—
59#2095C2212yes—
60#2093C2212yes—
61#2087C2214yes—
62#2084C2212yes—
63#2083C2212yes—
64#2075C2214yes—
65#2072C2212yes—
66#2071C2212yes—
67#2068C2214yes—
68#2065C2214yes—
69#2058C2212yes—
70#2057C2212yes—
71#2055C2214yes—
72#2043C2212yes—
73#2041C2212yes—
74#2035C2214yes—
75#2033C2214yes—
76#2024C2212yes—
77#2023C2212yes—
78#2014C2212yes—
79#2013C2212yes—
80#2000C2214yes—
81#1973C2214yes—
82#1968C2214yes—
83#1943C2214yes—
84#1942C2214yes—
85#1936C2214yes—
86#1935C2214yes—
87#1934C2214yes—
88#1933C2214yes—
89#1776C2214yes—
90#1767C2212yes—
91C4×C4⋊C8C2212yes—
92#1763C2214yes—
93#1762C2214yes—
94#1749C2212yes—
95C4×C22⋊C8C2212yes—
96#1745C2211yescharacteristic
97#1745C2211yescharacteristic
98C4×#97C2212yes—
99C22×SD16C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104#529—48no—
105C2×#126—22no—
106C2×#125—21no—
107C2×#125—21no—
108C2×#124—21no—
109C2×#124—21no—
110C2×#123—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#440—216no—
133#432—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2314yes—
145C2×#97C2312yes—
146C2×#97C2314yes—
147C2×#97C2312yes—
148C2×C4×C8C2314yes—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—21no—
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#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#386—22no—
206#386—22no—
207#385—22no—
208#385—22no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#383—22no—
214#382—21no—
215#382—21no—
216#382—21no—
217#382—21no—
218#381—22no—
219#380—21no—
220#380—21no—
221#380—21no—
222#380—21no—
223#380C2312yes—
224#380C2312yes—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2312yes—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236#376—22no—
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2312yes—
254#336C2312yes—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8C2412yes—
258C22×D4—22no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4C2412yes—
262C23×C4—22no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×SD16—28no—
271C2×SD16—44no—
272C2×SD16—44no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—42no—
276C4⋊Q8—42no—
277C4⋊Q8—42no—
278#127—42no—
279#126—48no—
280#125—44no—
281#125—44no—
282#124—44no—
283#124—44no—
284#123—48no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288#121—44no—
289#121—44no—
290#121—44no—
291C4×Q8—28no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—28no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—48no—
298#117—48no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—28no—
379#107—44no—
380#106—44no—
381C4⋊C8—28no—
382#103—28no—
383#103—44no—
384#103—44no—
385#102—28no—
386#102—44no—
387#102—44no—
388C22⋊C8—28no—
389#97—24no—
390#97—44no—
391#97#4118yes—
392C4×C8—44no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413C24—22no—
414C24C22×D411yescharacteristic
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4AGL(1,ℤ/8)14yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2511yescharacteristic
483SD16—416no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8C2×#4118yes—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—44no—
560C42—44no—
561C42—44no—
562C42#13714yes—
563C42C2×#4114yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23C2×AGL(1,ℤ/8)12yes—
578C23C23×D411yescharacteristic
579Q8—44no—
580Q8—44no—
581Q8—84no—
582Q8—84no—
583D4—44no—
584D4—44no—
585D4—84no—
586D4—84no—
587D4—88no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—88no—
647C2×C4#53212yes—
648C2×C4#53212yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4C2×AGL(1,ℤ/8)12yes—
651C2×C4C2×#13712yes—
652C2×C4C2×#13712yes—
653C2×C4C2×#13712yes—
654C2×C4#57512yes—
655C2×C4#57512yes—
656C2×C4C22×#4112yes—
657C2×C4#58512yes—
658C2×C4#58512yes—
659C8—28no—
660C8—44no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22C2×#53211yescharacteristic
676C22#344911yescharacteristic
677C22#344911yescharacteristic
678C22C22×AGL(1,ℤ/8)11yescharacteristic
679C22C22×#13711yescharacteristic
680C22C2×#57511yescharacteristic
681C22C2×#57511yescharacteristic
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#345514yes—
703C2—42no—
704C2—162no—
705C2#3464311yescharacteristic
706C2#6070011yescharacteristic
707C2#6070111yescharacteristic
708C2#6095811yescharacteristic
709C2#6096311yescharacteristic
710C2#6096911yescharacteristic
711C2#6097011yescharacteristic
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
1. #36635C2semidirect
3. #30384C2semidirect
4. #30378C2semidirect
5. #30367C2semidirect
6. #30356C2semidirect
7. #30345C2semidirect
8. #30337C2semidirect
10. #30317C2semidirect
11. #23839C2semidirect
12. #23823C2semidirect
13. #23822C2semidirect
14. #23803C2semidirect
15. #23710C2semidirect
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
#366352
#366312
#303842
#303782
#303672
#303562
#303452
#303372
#303272
#303172
#238392
#238232
#238222
#238032
#237102
#212882
C4×#4322