Finite Group Databasebeta

← #6485470

Subgroups — #6485470

Order 512.

1561 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35322C211yesmaximal, characteristic, complemented
2#35288C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#33278C211yesmaximal, characteristic, complemented
5#33252C211yesmaximal, characteristic, complemented
6#33239C211yesmaximal, characteristic, complemented
7#30333C214yesmaximal, complemented
8#29508C214yesmaximal, complemented
9#28748C211yesmaximal, characteristic, complemented
10#28718C211yesmaximal, characteristic, complemented
11#28688C211yesmaximal, characteristic, complemented
12#28438C211yesmaximal, characteristic, complemented
13#25674C211yesmaximal, characteristic, complemented
14#25672C211yesmaximal, characteristic, complemented
15#25598C211yesmaximal, characteristic
16#25598C211yesmaximal, characteristic
17#24666C214yesmaximal, complemented
18#22162C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#492C2211yescharacteristic
25C2×#492C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#487C2211yescharacteristic
28C2×#486C2211yescharacteristic
29C2×#486C2211yescharacteristic
30C2×#461C2214yes—
31C2×#433C2212yes—
32#2868C2211yescharacteristic
33#2862C2211yescharacteristic
34#2862C2211yescharacteristic
35#2862C2211yescharacteristic
36#2859C2211yescharacteristic
37#2858C2211yescharacteristic
38#2857C2211yescharacteristic
39#2856C2211yescharacteristic
40#2644C2212yes—
41#2624C2212yes—
42#2621C2212yes—
43#2621C2212yes—
44#2565—21no—
45#2565—21no—
46#2565—21no—
47#2565—21no—
48#2513—21no—
49#2499—21no—
50#2499—21no—
51#2499—21no—
52#2499—21no—
53#2490—21no—
54#2454—21no—
55#2448—21no—
56#2447—21no—
57#2427—21no—
58#2426—21no—
59#2420—21no—
60#2327C2212yes—
61#2327C2212yes—
62#2324C2212yes—
63#2319C2212yes—
64C4×C4⋊Q8C2211yescharacteristic
65C4×#127C2211yescharacteristic
66C4×#117C2211yescharacteristic
67#2085C2214yes—
68#2073C2214yes—
69#2043C2214yes—
70#2036C2214yes—
71#2035C2214yes—
72#2013C2214yes—
73#2000C2214yes—
74#2000C2214yes—
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1981C2214yes—
80#1973C2214yes—
81#1973C2214yes—
82#1968C2214yes—
83#1968C2214yes—
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1956C2211yescharacteristic
88#1949C2214yes—
89#1935C2214yes—
90#1935C2214yes—
91#1933C2214yes—
92#1933C2214yes—
93#1880C2214yes—
94#1880C2214yes—
95#1865C2214yes—
96#1863C2214yes—
97#1804C2214yes—
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108C22×SD16C2314yes—
109#533—42no—
110#533—42no—
111#533—42no—
112#533—42no—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×#127C2311yescharacteristic
121C2×#125—21no—
122C2×#125—21no—
123C2×#125—21no—
124C2×#124—21no—
125C2×#124—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8C2312yes—
130C2×C22⋊Q8C2312yes—
131C2×#121—21no—
132C2×#121—21no—
133C2×#121C2312yes—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×D4—21no—
140C2×C4×D4—21no—
141C2×C4×D4C2312yes—
142C2×#117—21no—
143C2×#117—21no—
144C2×#117C2311yescharacteristic
145C2×#117C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#500—42no—
148#498—42no—
149#492—42no—
150#492—42no—
151#487—42no—
152#487—42no—
153#486—42no—
154#486—42no—
155#461—216no—
156#433—28no—
157C2×#107C2314yes—
158C2×#106C2314yes—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2312yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×#97C2312yes—
172C2×#97C2312yes—
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2312yes—
178C2×C4×C8C2312yes—
179#399—22no—
180#398—22no—
181#398—22no—
182#398—22no—
183#398—22no—
184#397—22no—
185#397—22no—
186#397—22no—
187#397—22no—
188#397—22no—
189#397—22no—
190#397—22no—
191#397—22no—
192#396—22no—
193#394—22no—
194#394—22no—
195#391—21no—
196#391—21no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391C2312yes—
206#391C2312yes—
207#391C2312yes—
208#391C2312yes—
209#390—22no—
210#390—21no—
211#390—21no—
212#390—21no—
213#390—21no—
214#390—21no—
215#390—21no—
216#390C2312yes—
217#390C2312yes—
218#390C2312yes—
219#390C2312yes—
220#389—21no—
221#389—21no—
222#389—21no—
223#389—21no—
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#388—22no—
229#388—22no—
230#388—22no—
231#387—22no—
232#387—22no—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#386—22no—
240#385—22no—
241#384—21no—
242#384—21no—
243#384—21no—
244#384—21no—
245#384C2311yescharacteristic
246#384C2311yescharacteristic
247#382—21no—
248#382—21no—
249#382—21no—
250#382—21no—
251#382—21no—
252#382C2312yes—
253#382C2312yes—
254#381—22no—
255#380—21no—
256#380—22no—
257#380C2312yes—
258C4×C4⋊C4—21no—
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4D411yescharacteristic
261C4×C4⋊C4D411yescharacteristic
262C4×C4⋊C4D411yescharacteristic
263C4×C4⋊C4C2312yes—
264C4×C22⋊C4D411yescharacteristic
265C4×C22⋊C4D411yescharacteristic
266C4×C22⋊C4D411yescharacteristic
267C4×C22⋊C4D411yescharacteristic
268#376C2311yescharacteristic
269#376C2311yescharacteristic
270C43C2311yescharacteristic
271C43C2311yescharacteristic
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#340C2314yes—
276#336C2312yes—
277#336C2312yes—
278#336C2312yes—
279#336C2312yes—
280C22×Q8—21no—
281C22×Q8—21no—
282C22×Q8—21no—
283C22×Q8C2412yes—
284C22×Q8C2412yes—
285C22×D4—22no—
286C22×D4—21no—
287C22×D4—21no—
288C22×D4C2412yes—
289C23×C4—22no—
290C23×C4C2×D411yescharacteristic
291C23×C4C2×D411yescharacteristic
292C23×C4C2411yescharacteristic
293C2×SD16—28no—
294C2×SD16—28no—
295C2×SD16—44no—
296C2×SD16—44no—
297C2×SD16—44no—
298C2×SD16—44no—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C4⋊Q8—28no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—44no—
315#127—22no—
316#127—22no—
317#125—44no—
318#125—44no—
319#125—82no—
320#124—82no—
321#124—82no—
322C22⋊Q8—42no—
323C22⋊Q8—42no—
324C22⋊Q8—42no—
325C22⋊Q8—42no—
326C22⋊Q8—44no—
327C22⋊Q8—44no—
328C22⋊Q8—82no—
329#121—44no—
330#121—82no—
331#121—82no—
332C4×Q8—42no—
333C4×Q8—42no—
334C4×Q8—42no—
335C4×Q8—42no—
336C4×Q8—44no—
337C4×Q8—44no—
338C4×Q8—82no—
339C4×D4—28no—
340C4×D4—82no—
341C4×D4—82no—
342#117—24no—
343#117—42no—
344#117—44no—
345#117—44no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C4⋊C4C2×D411yescharacteristic
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2412yes—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4C2×D411yescharacteristic
392C2×C22⋊C4C2×D411yescharacteristic
393C2×C22⋊C4C2×D411yescharacteristic
394C2×C22⋊C4C2×D411yescharacteristic
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416#107—44no—
417#106—44no—
418C4⋊C8—24no—
419C4⋊C8#4118yes—
420#103—28no—
421#103—28no—
422#103—44no—
423#103—44no—
424#103—44no—
425#103—44no—
426#102—44no—
427#102—44no—
428C22⋊C8—24no—
429C22⋊C8—24no—
430#97—24no—
431#97—42no—
432C4×C8—24no—
433C4×C8—22no—
434C4×C8—22no—
435C4×C8—22no—
436C4×C8—22no—
437C4×C8—42no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95—22no—
450#95—22no—
451#95—22no—
452#95—22no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95C2×D411yescharacteristic
457#95C2×D411yescharacteristic
458#95C2×D411yescharacteristic
459#95C2×D411yescharacteristic
460#95C2×D411yescharacteristic
461#95C2×D411yescharacteristic
462#95C2×D411yescharacteristic
463#95C2×D411yescharacteristic
464C24—22no—
465C24C22×D411yescharacteristic
466C2×Q8—24no—
467C2×Q8—24no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—24no—
473C2×Q8—24no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—42no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—42no—
483C2×D4—24no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—24no—
487C2×D4—44no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—82no—
495C2×D4—82no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—44no—
514C22×C4—82no—
515C22×C4—82no—
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C2511yescharacteristic
530SD16—88no—
531SD16—88no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8C2×#4114yes—
543C2×C8C2×#4114yes—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—24no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—82no—
576C4⋊C4—82no—
577C4⋊C4—82no—
578C4⋊C4—82no—
579C22⋊C4—22no—
580C22⋊C4—22no—
581C22⋊C4—41no—
582C22⋊C4—41no—
583C22⋊C4—44no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C22⋊C4—44no—
587C22⋊C4—82no—
588C22⋊C4—82no—
589C22⋊C4—82no—
590C22⋊C4—82no—
591C22⋊C4—82no—
592C22⋊C4—82no—
593C22⋊C4—82no—
594C22⋊C4—82no—
595C22⋊C4—82no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42—21no—
608C42—21no—
609C42—22no—
610C42—22no—
611C42—44no—
612C42—44no—
613C42—82no—
614C42C2×#4114yes—
615C42H(5,𝔽2)12yes—
616C42H(5,𝔽2)12yes—
617C23—21no—
618C23—21no—
619C23—21no—
620C23—21no—
621C23—21no—
622C23—21no—
623C23—21no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23—42no—
628C23—42no—
629C23—42no—
630C23—42no—
631C23C23×D411yescharacteristic
632Q8—44no—
633Q8—44no—
634Q8—44no—
635Q8—44no—
636Q8—84no—
637Q8—84no—
638Q8—84no—
639D4—44no—
640D4—44no—
641D4—84no—
642D4—84no—
643D4—164no—
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—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—21no—
675C2×C4—44no—
676C2×C4—44no—
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—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—164no—
701C2×C4#53212yes—
702C2×C4#53212yes—
703C2×C4#53512yes—
704C2×C4#53512yes—
705C2×C4#57712yes—
706C2×C4#57712yes—
707C2×C4#57712yes—
708C2×C4#57812yes—
709C2×C4C22×#4112yes—
710C2×C4C2×H(5,𝔽2)11yescharacteristic
711C2×C4C2×H(5,𝔽2)11yescharacteristic
712C2×C4#58512yes—
713C8—24no—
714C8—24no—
715C8—42no—
716C8—42no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22C2×#53211yescharacteristic
732C22C2×#53511yescharacteristic
733C22#344911yescharacteristic
734C22C2×#57711yescharacteristic
735C22C2×#57711yescharacteristic
736C22C2×#57711yescharacteristic
737C22C2×#57811yescharacteristic
738C4—22no—
739C4—22no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—41no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4#322114yes—
757C4#322514yes—
758C4#351412yes—
759C4#351712yes—
760C2—81no—
761C2—162no—
762C2#3519111yescharacteristic
763C2#6071211yescharacteristic
764C2#6072311yescharacteristic
765C2C2×#322111yescharacteristic
766C2C2×#322511yescharacteristic
767C2#6100811yescharacteristic
768C2#6101211yescharacteristic
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. #35322C2semidirect
3. #33287C2semidirect
4. #33278C2semidirect
5. #33252C2semidirect
6. #33239C2semidirect
7. #30333C2semidirect
8. #29508C2semidirect
9. #28748C2semidirect
10. #28718C2semidirect
11. #28688C2semidirect
12. #28438C2semidirect
13. #25674C2semidirect
14. #25672C2semidirect
17. #24666C2semidirect
18. #22162C2semidirect
19. C4×#433C2semidirect

Maximal subgroups

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

subgroupindex
#353222
#352882
#332872
#332782
#332522
#332392
#303332
#295082
#287482
#287182
#286882
#284382
#256742
#256722
#255982
#246662
#221622
C4×#4332