Finite Group Databasebeta

← #6469759

Subgroups — #6469759

Order 512.

1791 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic, complemented
2#34226C211yesmaximal, characteristic
3#29568C214yesmaximal, complemented
4#29207C211yesmaximal, characteristic, complemented
5#29143C211yesmaximal, characteristic, complemented
6#28748C211yesmaximal, characteristic, complemented
7#28438C211yesmaximal, characteristic, complemented
8#27892C211yesmaximal, characteristic, complemented
9#27889C211yesmaximal, characteristic, complemented
10#25597C211yesmaximal, characteristic
11#25597C211yesmaximal, characteristic
12#24666C214yesmaximal, complemented
13#23584C214yesmaximal, complemented
14#21692C211yesmaximal, characteristic, complemented
15C4×#492C211yesmaximal, characteristic, complemented
16#21647C211yesmaximal, characteristic, complemented
17C4×#487C211yesmaximal, characteristic, complemented
18#21514C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic, complemented
20C2×#541C2212yes—
21C2×#532C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#492C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#461C2214yes—
27C2×#440C2214yes—
28C2×C4×SD16C2214yes—
29C2×#433C2212yes—
30#2868C2211yescharacteristic
31#2859C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2642C2212yes—
35#2621C2212yes—
36#2506—21no—
37#2506—21no—
38#2506—21no—
39#2506—21no—
40#2496—21no—
41#2496—21no—
42#2496—21no—
43#2496—21no—
44#2367—21no—
45#2367—21no—
46#2363—21no—
47#2363—21no—
48#2357—21no—
49#2349—21no—
50#2341—21no—
51#2339—21no—
52#2327C2212yes—
53#2324C2212yes—
54#2324C2212yes—
55#2319C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C4×#127C2211yescharacteristic
62#2285C2211yescharacteristic
63#2275C2212yes—
64C4×#117C2211yescharacteristic
65#2059C2214yes—
66#2043C2214yes—
67#2036C2214yes—
68#2035C2214yes—
69#2025C2214yes—
70#2013C2214yes—
71#2000C2214yes—
72#2000C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1981C2214yes—
76#1973C2214yes—
77#1968C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1949C2214yes—
85#1935C2214yes—
86#1933C2214yes—
87#1928C2214yes—
88#1927C2214yes—
89#1877C2214yes—
90#1877C2214yes—
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1804C2214yes—
96#1768C2212yes—
97C4×C4⋊C8C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99#1763C2214yes—
100C4×#103C2214yes—
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×#41—21no—
108C22×SD16C2314yes—
109C22×SD16C2314yes—
110#541—48no—
111#532—24no—
112#532—24no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×#127C2311yescharacteristic
119C2×#126—22no—
120C2×#126—22no—
121C2×#125—22no—
122C2×#125—22no—
123C2×#125C2312yes—
124C2×#124—21no—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×#121—21no—
128C2×#121C2312yes—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4C2312yes—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×#117—21no—
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#492—24no—
146#492—24no—
147#487—24no—
148#487—24no—
149#461—216no—
150#440—48no—
151C4×SD16—48no—
152#433—28no—
153C2×#107C2314yes—
154C2×#106C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2312yes—
167C2×#97C2312yes—
168C2×#97C2312yes—
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2312yes—
174C2×C4×C8C2312yes—
175#400—22no—
176#400—22no—
177#400—22no—
178#400—22no—
179#398—22no—
180#398—22no—
181#397—22no—
182#397—22no—
183#391—21no—
184#391—21no—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#390—22no—
190#390—22no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
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—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389C2312yes—
212#389C2312yes—
213#388—22no—
214#388—22no—
215#388—22no—
216#388—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#386—22no—
222#386—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#382—21no—
234#382C2312yes—
235#381—22no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—21no—
241#380—22no—
242#380C2312yes—
243C4×C4⋊C4—21no—
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⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C22⋊C4—22no—
258C4×C22⋊C4—22no—
259C4×C22⋊C4—22no—
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4D411yescharacteristic
264#376C2311yescharacteristic
265#376C2311yescharacteristic
266C2×#95—22no—
267C43C2311yescharacteristic
268C43C2311yescharacteristic
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#336C2312yes—
274#336C2312yes—
275#336C2312yes—
276#336C2312yes—
277C2×#41—28no—
278C2×#41—82no—
279C2×#41—82no—
280C22×Q8—21no—
281C22×Q8C2412yes—
282C22×Q8C2412yes—
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C2412yes—
287C23×C4—22no—
288C23×C4—22no—
289C23×C4—22no—
290C23×C4C2×D411yescharacteristic
291C23×C4C2×D411yescharacteristic
292C23×C4C2411yescharacteristic
293C2×SD16—28no—
294C2×SD16—28no—
295C2×SD16—28no—
296C2×SD16—28no—
297C2×SD16—44no—
298C2×SD16—44no—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C4⋊Q8—22no—
304C4⋊Q8—28no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—44no—
310C4⋊Q8#4114yes—
311#127—22no—
312#127#4114yes—
313#126—48no—
314#126—48no—
315#125—44no—
316#125—48no—
317#125—48no—
318#124—28no—
319C22⋊Q8—44no—
320C22⋊Q8—44no—
321#121—28no—
322#121—24no—
323#121—24no—
324C4×Q8—28no—
325C4×Q8—28no—
326C4×Q8—82no—
327C4×Q8—82no—
328C4×D4—28no—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4—28no—
332C4×D4—28no—
333C4×D4—82no—
334C4×D4—82no—
335C4×D4—82no—
336C4×D4—82no—
337#117—24no—
338#117—24no—
339#117—82no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
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⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
386C2×C22⋊C4C2×D411yescharacteristic
387C2×C22⋊C4C2×D411yescharacteristic
388C2×C22⋊C4C2×D411yescharacteristic
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412#107—44no—
413#106—44no—
414C4⋊C8—24no—
415C4⋊C8#4118yescomplemented
416#103—28no—
417#103—28no—
418#103—28no—
419#103—28no—
420#103—44no—
421#103—44no—
422#102—44no—
423#102—44no—
424C22⋊C8—24no—
425C22⋊C8—24no—
426#97—24no—
427#97—24no—
428C4×C8—24no—
429C4×C8—22no—
430C4×C8—22no—
431C4×C8—22no—
432C4×C8—22no—
433C4×C8—24no—
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—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95—22no—
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455#95C2×D411yescharacteristic
456#95C2×D411yescharacteristic
457#95C2×D411yescharacteristic
458C24—22no—
459C24C22×D411yescharacteristic
460#41—816no—
461C2×Q8—24no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—24no—
465C2×Q8—24no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—24no—
469C2×Q8—24no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×D4—28no—
473C2×D4—24no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—22no—
477C2×D4—24no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—82no—
483C2×D4—82no—
484C2×D4C2×#4114yes—
485C22×C4—22no—
486C22×C4—28no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4—82no—
510C22×C4—82no—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
525SD16—416no—
526SD16—416no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8C2×#4114yes—
538C2×C8C2×#4114yes—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—82no—
572C4⋊C4—82no—
573C4⋊C4C2×#4114yes—
574C22⋊C4—28no—
575C22⋊C4—28no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—22no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C22⋊C4—82no—
587C22⋊C4—82no—
588C22⋊C4—82no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—21no—
601C42—21no—
602C42—22no—
603C42—44no—
604C42—44no—
605C42—82no—
606C42—82no—
607C42—82no—
608C42C2×#4112yes—
609C42C2×#4112yes—
610C42C2×#4114yes—
611C42H(5,𝔽2)14yes—
612C23—24no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23—21no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23C23×D411yescharacteristic
627Q8—48no—
628Q8—44no—
629Q8—44no—
630Q8—44no—
631Q8—44no—
632D4—28no—
633D4—48no—
634D4—48no—
635D4—44no—
636D4—88no—
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—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—24no—
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—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
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—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—88no—
692C2×C4—164no—
693C2×C4—164no—
694C2×C4#52512yes—
695C2×C4#52512yes—
696C2×C4#53212yes—
697C2×C4#53212yes—
698C2×C4#53512yes—
699C2×C4#53512yes—
700C2×C4#57712yes—
701C2×C4#57712yes—
702C2×C4#57712yes—
703C2×C4#57812yes—
704C2×C4C22×#4111yescharacteristic
705C2×C4C22×#4112yes—
706C2×C4C22×#4111yescharacteristic
707C2×C4C2×H(5,𝔽2)12yes—
708C8—24no—
709C8—24no—
710C8—42no—
711C8—42no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—44no—
716C22—41no—
717C22—44no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—44no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22C2×#52511yescharacteristic
727C22C2×#53211yescharacteristic
728C22C2×#53511yescharacteristic
729C22C2×#57711yescharacteristic
730C22C2×#57711yescharacteristic
731C22C2×#57711yescharacteristic
732C22C2×#57811yescharacteristic
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—44no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4#312014yes—
752C4#312314yes—
753C4#348112yes—
754C4#348312yes—
755C2—81no—
756C2—84no—
757C2#3497711yescharacteristic
758C2C2×#312011yescharacteristic
759C2C2×#312311yescharacteristic
760C2#6071211yescharacteristic
761C2#6072311yescharacteristic
762C2#6082011yescharacteristic
763C2#6082411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #35319C2semidirect
3. #29568C2semidirect
4. #29207C2semidirect
5. #29143C2semidirect
6. #28748C2semidirect
7. #28438C2semidirect
8. #27892C2semidirect
9. #27889C2semidirect
12. #24666C2semidirect
13. #23584C2semidirect
14. #21692C2semidirect
15. C4×#492C2semidirect
16. #21647C2semidirect
17. C4×#487C2semidirect
18. #21514C2semidirect
19. C4×#433C2semidirect
415. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#353192
#342262
#295682
#292072
#291432
#287482
#284382
#278922
#278892
#255972
#246662
#235842
#216922
C4×#4922
#216472
C4×#4872
#215142
C4×#4332