Finite Group Databasebeta

← #6499208

Subgroups — #6499208

Order 512.

1740 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#33070C214yesmaximal, complemented
4#30000C214yesmaximal, complemented
5#29239C214yesmaximal, complemented
6#29236C214yesmaximal, complemented
7#29192C211yesmaximal, characteristic, complemented
8#29192C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic, complemented
10#29190C211yesmaximal, characteristic, complemented
11#29179C211yesmaximal, characteristic, complemented
12#29178C211yesmaximal, characteristic, complemented
13#29177C211yesmaximal, characteristic, complemented
14#29176C211yesmaximal, characteristic, complemented
15#21984C211yesmaximal, characteristic, complemented
16#21984C211yesmaximal, characteristic, complemented
17#21863C211yesmaximal, characteristic
18#21863C211yesmaximal, characteristic
19#20808C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×C4×#41—21no—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2864C2212yes—
25#2864C2212yes—
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2851—21no—
31#2851—21no—
32#2851—21no—
33#2851—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×#127C2211yescharacteristic
37C4×#127C2211yescharacteristic
38C4×#125—21no—
39C4×#125—21no—
40C4×#125—21no—
41C4×#125—21no—
42C4×#124—21no—
43C4×#124—21no—
44C4×#121—21no—
45C4×#121—21no—
46C4×#121C2212yes—
47C4×#121C2212yes—
48#2297—21no—
49#2296C2212yes—
50#2296C2212yes—
51#2293—21no—
52#2293C2212yes—
53#2285C2211yescharacteristic
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56C42×D4—21no—
57C42×D4C2212yes—
58C2×C43C2211yescharacteristic
59#2088C2214yes—
60#2088C2214yes—
61#2087C2214yes—
62#2086C2214yes—
63#2067C2214yes—
64#2067C2214yes—
65#2065C2214yes—
66#2065C2214yes—
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1957C2211yescharacteristic
72#1957C2211yescharacteristic
73#1957C2211yescharacteristic
74#1957C2211yescharacteristic
75#1949C2214yes—
76#1949C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1875C2214yes—
82#1875C2214yes—
83#1864C2214yes—
84#1864C2214yes—
85#1862C2214yes—
86#1862C2214yes—
87#1854C2214yes—
88#1854C2214yes—
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1844C2211yescharacteristic
92#1844C2211yescharacteristic
93#1842C2212yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1831C2212yes—
99#1804C2214yes—
100#1804C2214yes—
101#1797C2212yes—
102#1797C2212yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106C4×C4⋊C8C2211yescharacteristic
107C22×#41—21no—
108#532—24no—
109#532Q818yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#127C2311yescharacteristic
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—21no—
116C2×#124—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C4×#41—48no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×#117—21no—
133C2×#117—22no—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C22×C42—22no—
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#431—24no—
145#431—24no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166#391—21no—
167#391—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
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#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#386—22no—
207#386—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#382—21no—
219#382—21no—
220#382—21no—
221#382—21no—
222#382—21no—
223#382C2312yes—
224#382C2312yes—
225#380—21no—
226#380—21no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380C2312yes—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248C4×C22⋊C4—22no—
249C4×C22⋊C4—22no—
250C4×C22⋊C4—22no—
251C43D411yescharacteristic
252C43D411yescharacteristic
253C43D411yescharacteristic
254C43D411yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C2×#41—28no—
272C2×#41—44no—
273C2×#41—44no—
274C22×Q8—21no—
275C22×D4—22no—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4C2412yes—
279C23×C4—22no—
280C23×C4—22no—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2411yescharacteristic
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—22no—
294C4⋊Q8C2×Q814yes—
295#127—22no—
296#127C2×Q814yes—
297#125—44no—
298#125—44no—
299#125—44no—
300#125—44no—
301#124—28no—
302#121—28no—
303#121—24no—
304#121C2×Q818yes—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×D4—28no—
308C4×D4—24no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—48no—
314C4×D4—48no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4C2×Q818yes—
318#117—48no—
319#117—44no—
320#117—44no—
321#117—44no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—42no—
371C2×C42—42no—
372C2×C42—42no—
373C2×C42—42no—
374C2×C42—48no—
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×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×Q814yes—
398C2×C42#4114yes—
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—44no—
406#107—44no—
407#106—44no—
408#106—44no—
409C4⋊C8—24no—
410C4⋊C8—24no—
411C4⋊C8—42no—
412C4⋊C8—42no—
413C4⋊C8—42no—
414C4⋊C8—42no—
415#102—44no—
416#102—44no—
417#102—44no—
418#102—44no—
419C22⋊C8—24no—
420C22⋊C8—24no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8—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#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443C24—22no—
444C24C22×D411yescharacteristic
445#41—432no—
446C2×Q8—24no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×D4—28no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4C22×Q814yes—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4C22⋊Q814yes—
497C22×C4C22⋊Q814yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×Q812yes—
512C22×C4C2×#4112yes—
513C22×C4C2511yescharacteristic
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4C22×Q814yes—
557C22⋊C4—28no—
558C22⋊C4—28no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—44no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42C22⋊Q814yes—
596C42C22⋊Q814yes—
597C42#13514yes—
598C42AGL(1,ℤ/8)14yes—
599C42C22×Q812yes—
600C42C22×Q812yes—
601C42C2×#4112yes—
602C42C2×#4112yes—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—24no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23C2×C22⋊Q812yes—
617C23C23×D411yescharacteristic
618Q8—48no—
619D4—28no—
620D4—416no—
621D4—48no—
622D4—48no—
623D4—44no—
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—28no—
639C2×C4—28no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
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—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—416no—
659C2×C4—44no—
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—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—88no—
681C2×C4—88no—
682C2×C4C2×C22⋊Q812yes—
683C2×C4C2×C22⋊Q812yes—
684C2×C4C2×C22⋊Q812yes—
685C2×C4C2×C22⋊Q812yes—
686C2×C4C2×C22⋊Q812yes—
687C2×C4C2×#13512yes—
688C2×C4C2×#13512yes—
689C2×C4C2×#13512yes—
690C2×C4C2×AGL(1,ℤ/8)12yes—
691C2×C4C2×AGL(1,ℤ/8)12yes—
692C2×C4C2×AGL(1,ℤ/8)12yes—
693C2×C4#57512yes—
694C2×C4#57512yes—
695C2×C4C23×Q811yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4#58512yes—
698C2×C4#58512yes—
699C8—42no—
700C8—42no—
701C8—42no—
702C8—42no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22C22×C22⋊Q811yescharacteristic
718C22#344911yescharacteristic
719C22#344911yescharacteristic
720C22C22×#13511yescharacteristic
721C22C22×AGL(1,ℤ/8)11yescharacteristic
722C22C2×#57511yescharacteristic
723C22C2×#57511yescharacteristic
724C4—22no—
725C4—22no—
726C4—22no—
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—44no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4#307614yes—
743C4#307714yes—
744C4#350412yes—
745C4#351212yes—
746C2—42no—
747C2—84no—
748C2#3533311yescharacteristic
749C2C2×#307611yescharacteristic
750C2C2×#307711yescharacteristic
751C2#6094411yescharacteristic
752C2#6094611yescharacteristic
753C2#6096511yescharacteristic
754C2#6097211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #36640C2semidirect
3. #33070C2semidirect
4. #30000C2semidirect
5. #29239C2semidirect
6. #29236C2semidirect
7. #29192C2semidirect
8. #29192C2semidirect
9. #29190C2semidirect
10. #29190C2semidirect
11. #29179C2semidirect
12. #29178C2semidirect
13. #29177C2semidirect
14. #29176C2semidirect
15. #21984C2semidirect
16. #21984C2semidirect
19. #20808C2semidirect

Maximal subgroups

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

subgroupindex
#366402
C4×#5322
#330702
#300002
#292392
#292362
#291922
#291902
#291792
#291782
#291772
#291762
#219842
#218632
#208082