Finite Group Databasebeta

← #6802526

Subgroups — #6802526

Order 512.

1877 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#32903C211yesmaximal, characteristic
4#32182C214yesmaximal, complemented
5#30757C214yesmaximal, complemented
6#29719C212yesmaximal, complemented
7#28473C212yesmaximal, complemented
8#27757C212yesmaximal, complemented
9#25726C212yesmaximal, complemented
10#25718C211yesmaximal, characteristic
11#25368C212yesmaximal, complemented
12#24632C212yesmaximal, complemented
13#24212C211yesmaximal, characteristic, complemented
14#24209C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22047C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C2×#497C2214yes—
22C2×#492C2212yes—
23C2×#462C2212yes—
24C2×#460C2212yes—
25C2×#461C2214yes—
26C2×#451C2211yescharacteristic
27C2×#451C2212yes—
28C2×#448C2212yes—
29C2×#448C2212yes—
30C2×#448C2212yes—
31C2×C4×Q16C2212yes—
32#2856C2211yescharacteristic
33#2850C2212yes—
34#2597—22no—
35#2571C2211yescharacteristic
36#2548C2211yescharacteristic
37#2546—22no—
38#2546—21no—
39#2546—21no—
40#2546C2212yes—
41#2546C2214yes—
42#2546C2212yes—
43#2513—21no—
44#2513C2212yes—
45#2504—21no—
46#2447C2212yes—
47#2444—22no—
48C2×#398—22no—
49#2366C2211yescharacteristic
50#2361—21no—
51#2357—21no—
52#2355C2211yescharacteristic
53C4×C22⋊Q8C2211yescharacteristic
54C4×#121C2212yes—
55C2×#391C2212yes—
56C2×#389—21no—
57C2×#389—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2074C2214yes—
60#2056C2214yes—
61#2055C2214yes—
62#2043C2214yes—
63#2043C2212yes—
64#2016C2212yes—
65#2016C2214yes—
66#2015C2214yes—
67#2013C2214yes—
68#2013C2214yes—
69#2013C2214yes—
70#1981C2212yes—
71#1981C2214yes—
72#1973C2214yes—
73#1960C2214yes—
74#1956C2212yes—
75#1939C2212yes—
76#1935C2214yes—
77#1928C2214yes—
78#1917C2214yes—
79#1880C2214yes—
80#1878C2212yes—
81#1877C2214yes—
82#1874C2212yes—
83#1872C2211yescharacteristic
84#1863C2212yes—
85#1852C2211yescharacteristic
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2212yes—
90#1787C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C22×SD16C2314yes—
100C22×SD16C2314yes—
101C22×SD16C2314yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—44no—
107C2×C4⋊Q8—48no—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×#127C2311yescharacteristic
113C2×#124C2312yes—
114C2×C22⋊Q8C2311yescharacteristic
115C2×#121—21no—
116C2×#121—21no—
117C2×#121C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—44no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#497—48no—
139#492—44no—
140#462—44no—
141#461—48no—
142#460—44no—
143#451—24no—
144#451—44no—
145#448—28no—
146#448—44no—
147#448—44no—
148C4×Q16—44no—
149C2×#107C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2311yescharacteristic
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2311yescharacteristic
166C2×#97C2314yes—
167C2×C4×C8C2312yes—
168#400—22no—
169#398—24no—
170#398—24no—
171#398—24no—
172#398—24no—
173#397—22no—
174#397—22no—
175#397—22no—
176#397—22no—
177#394—22no—
178#391—22no—
179#391—22no—
180#391C2314yes—
181#391C2312yes—
182#391C2314yes—
183#391C2311yescharacteristic
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#389—21no—
193#389—22no—
194#389—22no—
195#389—24no—
196#389—22no—
197#389—24no—
198#389—22no—
199#389C2312yes—
200#389C2311yescharacteristic
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—21no—
207#387—22no—
208#386—22no—
209#386—22no—
210#386—22no—
211#386—22no—
212#385—21no—
213#385—22no—
214#384C2311yescharacteristic
215#382—21no—
216#382—21no—
217#382C2312yes—
218#381—22no—
219#380C2311yescharacteristic
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—22no—
232C2×#95—22no—
233C2×#95—22no—
234C2×#95D412yes—
235#340C2312yes—
236#340C2314yes—
237#340C2312yes—
238#336C2314yes—
239#336C2312yes—
240#336C2312yes—
241#336C2314yes—
242C22×Q8—24no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—22no—
248C22×Q8—24no—
249C22×Q8—42no—
250C22×Q8—44no—
251C22×Q8—44no—
252C22×Q8C2411yescharacteristic
253C22×Q8C2414yes—
254C22×Q8C2412yes—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4C2412yes—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2411yescharacteristic
268C2×Q16—24no—
269C2×Q16—24no—
270C2×Q16—42no—
271C2×Q16—44no—
272C2×Q16—44no—
273C2×Q16—44no—
274C2×SD16—28no—
275C2×SD16—28no—
276C2×SD16—44no—
277C2×SD16—44no—
278C2×SD16—44no—
279C2×SD16—44no—
280C2×SD16—44no—
281C2×SD16—44no—
282C2×SD16—44no—
283C22×C8C2412yes—
284C22×C8C2414yes—
285C22×C8C2412yes—
286C4⋊Q8—28no—
287C4⋊Q8—28no—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—48no—
296C4⋊Q8—48no—
297#127—24no—
298#124—44no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301#121—24no—
302#121—24no—
303#121—44no—
304#121—82no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—42no—
309C4×Q8—42no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4—84no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
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—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—48no—
337C2×C4⋊C4—48no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
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—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—44no—
370C2×C42—44no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379#107—42no—
380C4⋊C8—24no—
381C4⋊C8—44no—
382#103—24no—
383#103—28no—
384#103—24no—
385#103—42no—
386#103—42no—
387#103—44no—
388#103—44no—
389#103—44no—
390#103—44no—
391#103—44no—
392#102—44no—
393C22⋊C8—22no—
394C22⋊C8—24no—
395C22⋊C8#4114yes—
396#97—44no—
397C4×C8—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—21no—
403#95—22no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—22no—
408#95—22no—
409#95—21no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417C24—22no—
418C24C22×D411yescharacteristic
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—48no—
436C2×Q8—48no—
437C2×Q8—42no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—88no—
443C2×Q8—88no—
444C2×Q8H(5,𝔽2)14yes—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—81no—
457C2×D4—81no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—41no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4#13714yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4C2511yescharacteristic
504Q16—48no—
505Q16—88no—
506SD16—88no—
507SD16—88no—
508SD16—88no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—24no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—22no—
517C2×C8C2×#4114yes—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—81no—
546C4⋊C4—82no—
547C4⋊C4—81no—
548C4⋊C4#13714yes—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C42—22no—
562C42—22no—
563C42—24no—
564C42—24no—
565C42—22no—
566C42—42no—
567C42—42no—
568C42—44no—
569C42—44no—
570C42—42no—
571C42—42no—
572C42—82no—
573C42H(5,𝔽2)14yes—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23#55012yes—
586C23C2×#13712yes—
587C23#57712yes—
588C23C23×D411yescharacteristic
589Q8—24no—
590Q8—24no—
591Q8—48no—
592Q8—42no—
593Q8—42no—
594Q8—44no—
595Q8—42no—
596Q8—44no—
597Q8—44no—
598Q8—42no—
599Q8—44no—
600Q8—88no—
601D4—44no—
602D4—44no—
603D4—88no—
604D4—84no—
605D4—162no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—84no—
658C2×C4—88no—
659C2×C4—164no—
660C2×C4#53512yes—
661C2×C4#53512yes—
662C2×C4#55012yes—
663C2×C4C2×#13712yes—
664C2×C4C2×#13714yes—
665C2×C4C2×#13712yes—
666C2×C4C2×#13712yes—
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)12yes—
669C2×C4C2×H(5,𝔽2)12yes—
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C8—24no—
672C8—42no—
673C8—44no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22#340014yes—
688C22C2×#53511yescharacteristic
689C22C2×#55011yescharacteristic
690C22#348411yescharacteristic
691C22C22×#13711yescharacteristic
692C22C22×#13711yescharacteristic
693C22C2×#57711yescharacteristic
694C22#358611yescharacteristic
695C4—22no—
696C4—22no—
697C4—24no—
698C4—22no—
699C4—41no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—81no—
705C4—82no—
706C4—81no—
707C4—82no—
708C4—84no—
709C4—81no—
710C4—81no—
711C4#321814yes—
712C4#351912yes—
713C4#353712yes—
714C2—24no—
715C2—162no—
716C2#3913211yescharacteristic
717C2C2×#321811yescharacteristic
718C2#6081311yescharacteristic
719C2C2×#340011yescharacteristic
720C2#6140411yescharacteristic
721C2#6158411yescharacteristic
722C2#6159311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
4. #32182C2semidirect
5. #30757C2semidirect
6. #29719C2semidirect
7. #28473C2semidirect
8. #27757C2semidirect
9. #25726C2semidirect
11. #25368C2semidirect
12. #24632C2semidirect
13. #24212C2semidirect
14. #24209C2semidirect
15. #22629C2semidirect
16. #22047C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#25462
#329032
#321822
#307572
#297192
#284732
#277572
#257262
#257182
#253682
#246322
#242122
#242092
#226292
#220472
C4×#4512