Finite Group Databasebeta

← #6497745

Subgroups — #6497745

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#33277C211yesmaximal, characteristic, complemented
5#33207C211yesmaximal, characteristic, complemented
6#33042C211yesmaximal, characteristic
7#30305C211yesmaximal, characteristic, complemented
8#30277C211yesmaximal, characteristic, complemented
9#29877C211yesmaximal, characteristic, complemented
10#29843C211yesmaximal, characteristic
11#28949C214yesmaximal, complemented
12#28845C214yesmaximal, complemented
13#28062C214yesmaximal, complemented
14#28034C214yesmaximal, complemented
15#22246C211yesmaximal, characteristic, complemented
16#22219C211yesmaximal, characteristic, complemented
17C4×#498C211yesmaximal, characteristic, complemented
18C4×#486C211yesmaximal, characteristic, complemented
19C8×#117C211yesmaximal, characteristic, complemented
20C2×#542—21no—
21C2×#498C2211yescharacteristic
22C2×#498C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2647—21no—
36#2640—21no—
37#2629—21no—
38#2620—21no—
39#2512—21no—
40#2512C2212yes—
41#2507C2212yes—
42#2507C2212yes—
43#2497C2212yes—
44#2497C2212yes—
45#2491—21no—
46#2491C2212yes—
47#2452—21no—
48#2452C2212yes—
49#2446—21no—
50#2446—21no—
51#2429—21no—
52#2429—21no—
53#2422—21no—
54#2422C2212yes—
55#2330—21no—
56#2326—21no—
57#2314—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2094C2214yes—
64#2094C2214yes—
65#2088C2214yes—
66#2084C2214yes—
67#2077C2214yes—
68#2065C2214yes—
69#2058C2214yes—
70#2034C2214yes—
71#1990C2211yescharacteristic
72#1990C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1941C2214yes—
81#1941C2214yes—
82#1934C2214yes—
83#1934C2214yes—
84#1934C2214yes—
85#1934C2214yes—
86#1926C2214yes—
87#1920C2214yes—
88#1919C2214yes—
89#1893C2214yes—
90#1864C2214yes—
91#1844C2211yescharacteristic
92#1844C2211yescharacteristic
93#1834C2214yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1807C2214yes—
97C4×#106C2214yes—
98C8×C4⋊C4C2212yes—
99C8×C4⋊C4C2212yes—
100C4×#102C2214yes—
101C8×C22⋊C4C2212yes—
102C8×C22⋊C4C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C42×C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×C8C2312yes—
108#542—48no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×#127C2311yescharacteristic
118C2×#126—22no—
119C2×#125—21no—
120C2×#124—22no—
121C2×#124—21no—
122C2×#124—21no—
123C2×#123—22no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—22no—
128C2×#121—21no—
129C2×#121—21no—
130C2×#121C2312yes—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4C2312yes—
137C2×#117D411yescharacteristic
138C2×#117D411yescharacteristic
139C22×C42C2311yescharacteristic
140#498—24no—
141#498—24no—
142#486—24no—
143#486—24no—
144#431—24no—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C4×C8—24no—
156C2×C4×C8C2312yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165#400—22no—
166#400—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#397—22no—
172#397—22no—
173#396—22no—
174#396—22no—
175#396—22no—
176#396—22no—
177#394—22no—
178#394—22no—
179#393—22no—
180#393—22no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391—21no—
186#391—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390C2312yes—
194#390C2312yes—
195#390C2312yes—
196#390C2312yes—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—22no—
206#389—22no—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#385—22no—
224#385—22no—
225#384—21no—
226#384—21no—
227#384—21no—
228#384—21no—
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2312yes—
234#384C2312yes—
235#384C2312yes—
236#382—21no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380C2312yes—
241#380C2312yes—
242#380C2312yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257#376D411yescharacteristic
258#376D411yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×D4—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2412yes—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C22×C8—24no—
286C22×C8—24no—
287C22×C8—42no—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—82no—
301C4⋊Q8—82no—
302C4⋊Q8—82no—
303#127—22no—
304#127—22no—
305#126—48no—
306#125—44no—
307#124—48no—
308#124—44no—
309#124—44no—
310#123—48no—
311C22⋊Q8—44no—
312C22⋊Q8—48no—
313C22⋊Q8—82no—
314C22⋊Q8—82no—
315#121—44no—
316#121—44no—
317#121—82no—
318C4×Q8—44no—
319C4×Q8—82no—
320C4×Q8—82no—
321C4×D4—44no—
322C4×D4—44no—
323C4×D4—82no—
324#117—24no—
325#117SD1618yes—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
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—28no—
403#106—44no—
404#106—44no—
405C4⋊C8—24no—
406#102—28no—
407#102—28no—
408#102—28no—
409#102—44no—
410C22⋊C8—24no—
411C4×C8—24no—
412C4×C8—24no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—42no—
417C4×C8—42no—
418C4×C8#4114yes—
419C4×C8#4114yes—
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—
431#95—22no—
432#95—22no—
433#95—22no—
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442C24—22no—
443C24C22×D411yescharacteristic
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—82no—
455C2×Q8—82no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—82no—
468C2×D4—82no—
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—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—44no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4C2×SD1614yes—
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×C4C22×D411yescharacteristic
502C22×C4C2511yescharacteristic
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—48no—
513C2×C8C2×#4114yes—
514C2×C8C2×#4114yes—
515C2×C82−1+414yes—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4C2×SD1614yes—
555C4⋊C4C2×SD1614yes—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4C2×SD1614yes—
572C22⋊C4C2×SD1614yes—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—21no—
581C42—21no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—44no—
587C42—82no—
588C42—82no—
589C42C2×SD1614yes—
590C42C2×SD1614yes—
591C42#13514yes—
592C42C2×#4112yes—
593C42C2×#4112yes—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23C22×SD1612yes—
608C23C23×D411yescharacteristic
609Q8—84no—
610Q8—84no—
611Q8—84no—
612Q8—164no—
613D4—44no—
614D4—44no—
615D4—84no—
616D4—84no—
617D4—164no—
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—28no—
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—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
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—42no—
668C2×C4—42no—
669C2×C4—164no—
670C2×C4#53212yes—
671C2×C4#53212yes—
672C2×C4C22×SD1612yes—
673C2×C4C22×SD1612yes—
674C2×C4C22×SD1612yes—
675C2×C4C22×SD1612yes—
676C2×C4C22×SD1612yes—
677C2×C4C22×SD1612yes—
678C2×C4C2×#13512yes—
679C2×C4C2×#13512yes—
680C2×C4C2×#13512yes—
681C2×C4#57612yes—
682C2×C4#57712yes—
683C2×C4C22×#4111yescharacteristic
684C2×C4C22×#4111yescharacteristic
685C2×C4C2×2−1+412yes—
686C2×C4#58512yes—
687C8—24no—
688C8—24no—
689C8—24no—
690C8—42no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22C2×#53211yescharacteristic
706C22#349911yescharacteristic
707C22#350011yescharacteristic
708C22C23×SD1611yescharacteristic
709C22C22×#13511yescharacteristic
710C22C2×#57611yescharacteristic
711C22C2×#57711yescharacteristic
712C4—22no—
713C4—22no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4#316714yes—
731C4#317914yes—
732C4#346112yes—
733C4#349512yes—
734C2—42no—
735C2—162no—
736C2#3531811yescharacteristic
737C2C2×#316711yescharacteristic
738C2C2×#317911yescharacteristic
739C2#6246911yescharacteristic
740C2#6247411yescharacteristic
741C2#6252511yescharacteristic
742C2#6253211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35284C2semidirect
3. #33289C2semidirect
4. #33277C2semidirect
5. #33207C2semidirect
7. #30305C2semidirect
8. #30277C2semidirect
9. #29877C2semidirect
11. #28949C2semidirect
12. #28845C2semidirect
13. #28062C2semidirect
14. #28034C2semidirect
15. #22246C2semidirect
16. #22219C2semidirect
17. C4×#498C2semidirect
18. C4×#486C2semidirect
19. C8×#117C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#352772
#332892
#332772
#332072
#330422
#303052
#302772
#298772
#298432
#289492
#288452
#280622
#280342
#222462
#222192
C4×#4982
C4×#4862
C8×#1172