Finite Group Databasebeta

← #6488839

Subgroups — #6488839

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33134C214yesmaximal, complemented
4#30382C212yesmaximal, complemented
5#30343C212yesmaximal, complemented
6#29451C214yesmaximal, complemented
7#29355C211yesmaximal, characteristic, complemented
8#29319C211yesmaximal, characteristic, complemented
9#29160C212yesmaximal, complemented
10#29130C211yesmaximal, characteristic, complemented
11#28574C212yesmaximal
12#28382C211yesmaximal, characteristic, complemented
13#24835C212yesmaximal, complemented
14#24751C212yesmaximal, complemented
15#22152C212yesmaximal, complemented
16#22116C212yesmaximal, complemented
17C2×#1767C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#474C2212yes—
20C2×#465C2214yes—
21C2×#461C2212yes—
22C22×C4⋊C8C2212yes—
23C22×#97C2212yes—
24#2864—21no—
25#2863—21no—
26#2863C2212yes—
27#2862C2211yescharacteristic
28#2861—21no—
29#2860—21no—
30#2860C2212yes—
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2857C2211yescharacteristic
37#2856C2211yescharacteristic
38#2855—21no—
39#2855C2212yes—
40#2854—21no—
41#2854C2212yes—
42#2851—21no—
43#2850—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#125—21no—
49C4×#124—21no—
50C4×C22⋊Q8—21no—
51C4×C22⋊Q8C2212yes—
52C4×#121—21no—
53C4×#121C2212yes—
54#2295—21no—
55#2295C2212yes—
56#2293—21no—
57#2293C2212yes—
58#2283—21no—
59C2×C43C2211yescharacteristic
60#2100C2212yes—
61#2095C2212yes—
62#2086C2214yes—
63#2084C2212yes—
64#2072C2212yes—
65#2067C2214yes—
66#2066C2214yes—
67#2057C2212yes—
68#2055C2214yes—
69#2041C2212yes—
70#2024C2212yes—
71#2014C2212yes—
72#1988C2212yes—
73#1981C2212yes—
74#1973C2214yes—
75#1973C2214yes—
76#1948C2212yes—
77#1947C2212yes—
78#1943C2214yes—
79#1943C2214yes—
80#1936C2214yes—
81#1936C2214yes—
82#1935C2214yes—
83#1935C2214yes—
84#1926C2214yes—
85#1921C2214yes—
86#1893C2214yes—
87#1892C2214yes—
88#1880C2214yes—
89#1863C2214yes—
90#1808C2214yes—
91#1802C2212yes—
92#1802C2212yes—
93#1767C2212yes—
94#1767C2212yes—
95#1767C2211yescharacteristic
96#1767C2212yes—
97#1767C2211yescharacteristic
98C2×#336C2212yes—
99C22×SD16C2314yes—
100C23×C8C2314yes—
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×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×#117—22no—
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#474—28no—
134#465—48no—
135#461—28no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8—28no—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#97—44no—
150C2×#97C2312yes—
151C2×#97C2314yes—
152C2×#97C2312yes—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
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#389—21no—
183#389—21no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#386—22no—
210#386—22no—
211#385—22no—
212#385—22no—
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#383—22no—
218#382—21no—
219#382—21no—
220#382—21no—
221#382—21no—
222#381—22no—
223#380—21no—
224#380—21no—
225#380—21no—
226#380—21no—
227#380C2312yes—
228#380C2312yes—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4C2312yes—
236C4×C4⋊C4C2312yes—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240#376—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—44no—
275C2×SD16—44no—
276C22×C8—28no—
277C22×C8—28no—
278C22×C8—44no—
279C22×C8C2414yes—
280C22×C8C2414yes—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—42no—
284#127—42no—
285#126—48no—
286#125—44no—
287#125—44no—
288#124—44no—
289#124—44no—
290#123—48no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—44no—
294C22⋊Q8—44no—
295#121—24no—
296#121—24no—
297#121—44no—
298#121—44no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—48no—
307#117—48no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
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—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⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
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×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—42no—
358C2×C42—42no—
359C2×C42—42no—
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×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×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388#107—28no—
389#107—44no—
390#106—44no—
391C4⋊C8—28no—
392C4⋊C8—24no—
393C4⋊C8#4118yes—
394#103—28no—
395#103—44no—
396#103—44no—
397#102—28no—
398#102—44no—
399#102—44no—
400#97—42no—
401#97—44no—
402#97—42no—
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—
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—
423C24—22no—
424C24C22×D411yescharacteristic
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—42no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
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—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4AGL(1,ℤ/8)14yes—
479C22×C4#13714yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4H(5,𝔽2)12yes—
494C22×C42−1+412yes—
495C22×C4C2511yescharacteristic
496SD16—88no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—416no—
503C2×C8C2×#4118yes—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
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—21no—
569C42—21no—
570C42—22no—
571C42—22no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—44no—
579C42—44no—
580C42—44no—
581C42C2×#4114yes—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23#53712yes—
594C23C2×AGL(1,ℤ/8)12yes—
595C23C2×#13712yes—
596C23C23×D411yescharacteristic
597Q8—44no—
598Q8—44no—
599Q8—84no—
600Q8—84no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—84no—
606D4—84no—
607D4—88no—
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—21no—
619C2×C4—21no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
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—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
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—88no—
668C2×C4#53212yes—
669C2×C4#53212yes—
670C2×C4#53712yes—
671C2×C4C2×AGL(1,ℤ/8)12yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C2×#13712yes—
674C2×C4C2×#13712yes—
675C2×C4#57512yes—
676C2×C4#57512yes—
677C2×C4C22×#4112yes—
678C2×C4C2×H(5,𝔽2)11yescharacteristic
679C2×C4C2×2−1+411yescharacteristic
680C2×C4#58512yes—
681C8—28no—
682C8—44no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22#325414yes—
697C22C2×#53211yescharacteristic
698C22C2×#53711yescharacteristic
699C22#344911yescharacteristic
700C22C22×AGL(1,ℤ/8)11yescharacteristic
701C22C22×#13711yescharacteristic
702C22C2×#57511yescharacteristic
703C22C2×#57511yescharacteristic
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4#318014yes—
724C4#351312yes—
725C4#351612yes—
726C2—24no—
727C2—162no—
728C2#3523411yescharacteristic
729C2C2×#318011yescharacteristic
730C2#6070111yescharacteristic
731C2C2×#325411yescharacteristic
732C2#6090711yescharacteristic
733C2#6096011yescharacteristic
734C2#6096211yescharacteristic
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. #33134C2semidirect
4. #30382C2semidirect
5. #30343C2semidirect
6. #29451C2semidirect
7. #29355C2semidirect
8. #29319C2semidirect
9. #29160C2semidirect
10. #29130C2semidirect
12. #28382C2semidirect
13. #24835C2semidirect
14. #24751C2semidirect
15. #22152C2semidirect
16. #22116C2semidirect
17. C2×#1767C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#331342
#303822
#303432
#294512
#293552
#293192
#291602
#291302
#285742
#283822
#248352
#247512
#221522
#221162
C2×#17672