Finite Group Databasebeta

← #6850654

Subgroups — #6850654

Order 512.

1587 conjugacy classes of subgroups.

The subgroup lattice has 767 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic
2#36181C211yesmaximal, characteristic, complemented
3#33291C211yesmaximal, characteristic, complemented
4#32059C214yesmaximal, complemented
5#32054C214yesmaximal, complemented
6#29807C211yesmaximal, characteristic, complemented
7#29807C211yesmaximal, characteristic, complemented
8#29144C211yesmaximal, characteristic, complemented
9#28514C211yesmaximal, characteristic, complemented
10#27974C211yesmaximal, characteristic, complemented
11#27973C211yesmaximal, characteristic, complemented
12#27892C211yesmaximal, characteristic, complemented
13#27889C211yesmaximal, characteristic, complemented
14#24686C214yesmaximal, complemented
15#24125C214yesmaximal, complemented
16#21984C211yesmaximal, characteristic, complemented
17#21981C211yesmaximal, characteristic, complemented
18C4×#492C211yesmaximal, characteristic, complemented
19#21264C211yesmaximal, characteristic, complemented
20C2×#552C2212yes—
21C2×#492C2211yescharacteristic
22C2×#478C2214yes—
23C2×C4×SD16C2214yes—
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2856C2211yescharacteristic
32#2565C2212yes—
33#2535C2212yes—
34#2513—21no—
35#2513—21no—
36#2506—21no—
37#2506—21no—
38#2503C2212yes—
39#2499C2212yes—
40#2496—21no—
41#2496—21no—
42#2490C2212yes—
43#2490C2212yes—
44#2486—21no—
45#2486—21no—
46#2420—21no—
47#2420—21no—
48#2414—21no—
49#2414—21no—
50#2363—21no—
51#2363—21no—
52#2357C2212yes—
53#2344—21no—
54#2344—21no—
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C42×D4C2211yescharacteristic
62#2092C2214yes—
63#2073C2214yes—
64#2071C2214yes—
65#2055C2214yes—
66#2055C2214yes—
67#2036C2214yes—
68#2035C2214yes—
69#2026C2214yes—
70#2000C2214yes—
71#2000C2214yes—
72#1991C2214yes—
73#1991C2214yes—
74#1968C2214yes—
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1954C2212yes—
82#1949C2214yes—
83#1948C2214yes—
84#1939C2212yes—
85#1928C2214yes—
86#1927C2214yes—
87#1923C2212yes—
88#1923C2212yes—
89#1875C2214yes—
90#1874C2214yes—
91#1854C2214yes—
92#1853C2214yes—
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1834C2211yescharacteristic
101#1834C2211yescharacteristic
102#1805C2214yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108#552—48no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#126—22no—
112C2×#125—22no—
113C2×#125C2312yes—
114C2×#125C2312yes—
115C2×#123—22no—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×#121—21no—
119C2×#121—21no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×#121C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—21no—
133C2×#117—22no—
134C2×#117C2312yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#492—24no—
141#478—48no—
142C4×SD16—48no—
143#431—24no—
144#431—24no—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2312yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2311yescharacteristic
167#400—22no—
168#400—22no—
169#398—22no—
170#398—22no—
171#397—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#397—22no—
176#394—22no—
177#394—22no—
178#392—22no—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#390—22no—
183#390—22no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
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—21no—
206#389—21no—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#386—22no—
216#386—22no—
217#384—21no—
218#384—21no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2312yes—
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#382—21no—
227#382—21no—
228#382C2312yes—
229#382C2312yes—
230#382C2312yes—
231#381—22no—
232#381—22no—
233#381—22no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—21no—
241#380—21no—
242#380C2312yes—
243#380C2312yes—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2311yescharacteristic
255C4×C22⋊C4—22no—
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4C2311yescharacteristic
261C4×C22⋊C4C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8C2412yes—
273C22×D4—22no—
274C22×D4—22no—
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C2×D411yescharacteristic
278C22×D4C2412yes—
279C23×C4—22no—
280C23×C4—22no—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2411yescharacteristic
286C23×C4C2411yescharacteristic
287C2×SD16—28no—
288C2×SD16—44no—
289C2×SD16—44no—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296#127—22no—
297#127—22no—
298#126—48no—
299#125—28no—
300#125—44no—
301#125—48no—
302#123—48no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305#121—44no—
306#121—44no—
307#121—44no—
308#121—44no—
309#121—82no—
310#121—82no—
311C4×Q8—28no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—28no—
315C4×D4—42no—
316C4×D4—82no—
317C4×D4—82no—
318#117—48no—
319#117—44no—
320#117—82no—
321#117—82no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
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×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#107—44no—
400#106—44no—
401#106—44no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405#103—44no—
406#103—44no—
407#103—44no—
408#102—44no—
409#102—44no—
410#102—44no—
411C22⋊C8—24no—
412C22⋊C8—24no—
413C22⋊C8—42no—
414C22⋊C8—42no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8—24no—
418C4×C8—22no—
419C4×C8—42no—
420C4×C8#4114yes—
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#95—22no—
435#95—22no—
436#95—22no—
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443C24—22no—
444C24C22×D411yescharacteristic
445C24C22×D411yescharacteristic
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—41no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—82no—
466C2×D4—82no—
467C2×D4—82no—
468C2×D4—82no—
469C22×C4—22no—
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—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4#13514yes—
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×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C2511yescharacteristic
515SD16—416no—
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—
526C2×C8—22no—
527C2×C8C2×#4114yes—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—41no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—41no—
566C22⋊C4—41no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—21no—
590C42—21no—
591C42—21no—
592C42—21no—
593C42—22no—
594C42—44no—
595C42—44no—
596C42—82no—
597C42—82no—
598C42C2×#4112yes—
599C42H(5,𝔽2)14yes—
600C422−1+412yes—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23C2×#13512yes—
622C23C23×D411yescharacteristic
623Q8—44no—
624Q8—44no—
625D4—44no—
626D4—44no—
627D4—44no—
628D4—84no—
629D4—84no—
630D4—164no—
631D4—164no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—28no—
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—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
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—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—88no—
689C2×C4—164no—
690C2×C4#53512yes—
691C2×C4#53512yes—
692C2×C4#54712yes—
693C2×C4#54712yes—
694C2×C4C2×#13512yes—
695C2×C4C2×#13512yes—
696C2×C4#57512yes—
697C2×C4#57712yes—
698C2×C4#57712yes—
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×H(5,𝔽2)12yes—
701C2×C4C2×2−1+411yescharacteristic
702C2×C4#58512yes—
703C8—24no—
704C8—42no—
705C8—42no—
706C8—42no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22C2×#53511yescharacteristic
729C22C2×#54711yescharacteristic
730C22#350011yescharacteristic
731C22C22×#13511yescharacteristic
732C22C2×#57511yescharacteristic
733C22C2×#57711yescharacteristic
734C22C2×#57711yescharacteristic
735C4—22no—
736C4—22no—
737C4—22no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4#333714yes—
753C4#335114yes—
754C4#350612yes—
755C4#354712yes—
756C2—42no—
757C2—81no—
758C2—162no—
759C2#3949111yescharacteristic
760C2#6078211yescharacteristic
761C2#6082011yescharacteristic
762C2C2×#333711yescharacteristic
763C2C2×#335111yescharacteristic
764C2#6253111yescharacteristic
765C2#6254411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36211C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36181C2semidirect
3. #33291C2semidirect
4. #32059C2semidirect
5. #32054C2semidirect
6. #29807C2semidirect
7. #29807C2semidirect
8. #29144C2semidirect
9. #28514C2semidirect
10. #27974C2semidirect
11. #27973C2semidirect
12. #27892C2semidirect
13. #27889C2semidirect
14. #24686C2semidirect
15. #24125C2semidirect
16. #21984C2semidirect
17. #21981C2semidirect
18. C4×#492C2semidirect
19. #21264C2semidirect

Maximal subgroups

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

subgroupindex
#362112
#361812
#332912
#320592
#320542
#298072
#291442
#285142
#279742
#279732
#278922
#278892
#246862
#241252
#219842
#219812
C4×#4922
#212642