Finite Group Databasebeta

← #6625121

Subgroups — #6625121

Order 512.

1885 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33106C211yesmaximal, characteristic
4#32909C212yesmaximal, complemented
5#28306C212yesmaximal, complemented
6#28118C212yesmaximal, complemented
7#27069C214yesmaximal, complemented
8#26589C214yesmaximal, complemented
9#26531C212yesmaximal, complemented
10#25718C211yesmaximal, characteristic
11#25369C212yesmaximal, complemented
12#24212C211yesmaximal, characteristic, complemented
13#23182C212yesmaximal, complemented
14#23096C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16C4×#483C211yesmaximal, characteristic, complemented
17C4×#448C212yesmaximal, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#492C2212yes—
21C2×#483C2211yescharacteristic
22C2×#480C2212yes—
23C2×#464C2214yes—
24C2×#462C2212yes—
25C2×#451C2212yes—
26C2×#448C2212yes—
27C2×#448C2212yes—
28C2×C4×SD16C2214yes—
29#2856C2211yescharacteristic
30#2850C2212yes—
31#2597—22no—
32#2571C2211yescharacteristic
33#2548C2211yescharacteristic
34#2513—21no—
35#2513C2212yes—
36#2504—21no—
37#2447C2212yes—
38#2444—22no—
39#2366C2211yescharacteristic
40#2365—22no—
41#2365—21no—
42#2365—21no—
43#2365C2212yes—
44#2365C2212yes—
45#2365C2214yes—
46#2361—21no—
47#2357—21no—
48#2355C2211yescharacteristic
49C4×C22⋊Q8C2211yescharacteristic
50C4×#121C2212yes—
51C2×#391C2212yes—
52C2×#389—21no—
53C2×#387—22no—
54C2×C4×C4⋊C4—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#2097C2214yes—
58#2068C2214yes—
59#2055C2214yes—
60#2043C2212yes—
61#2016C2212yes—
62#2015C2214yes—
63#2013C2214yes—
64#1982C2212yes—
65#1981C2212yes—
66#1969C2214yes—
67#1968C2214yes—
68#1956C2212yes—
69#1947C2212yes—
70#1939C2212yes—
71#1928C2214yes—
72#1927C2212yes—
73#1925C2214yes—
74#1920C2212yes—
75#1916C2214yes—
76#1880C2214yes—
77#1880C2214yes—
78#1878C2214yes—
79#1877C2214yes—
80#1874C2212yes—
81#1872C2211yescharacteristic
82#1865C2214yes—
83#1853C2211yescharacteristic
84#1823C2211yescharacteristic
85#1805C2214yes—
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1787C2212yes—
89C4×#106C2212yes—
90C8×C4⋊C4C2212yes—
91C4×#103C2212yes—
92C4×#103C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×SD16C2314yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#124C2312yes—
106C2×C22⋊Q8C2311yescharacteristic
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—48no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2314yes—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—22no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#492—28no—
131#483—24no—
132#480—44no—
133#464—48no—
134#462—28no—
135#451—44no—
136#448—28no—
137#448—44no—
138C4×SD16—48no—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2311yescharacteristic
156C2×C4×C8C2312yes—
157C2×C4×C8C2314yes—
158#400—22no—
159#397—22no—
160#397—22no—
161#397—22no—
162#397—22no—
163#394—22no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#391C2312yes—
168#391C2314yes—
169#391C2311yescharacteristic
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—24no—
182#389C2312yes—
183#389C2311yescharacteristic
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—21no—
190#387—22no—
191#387—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#386—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#385—21no—
200#385—22no—
201#384C2311yescharacteristic
202#382—21no—
203#382—21no—
204#382C2312yes—
205#381—22no—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—24no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C2×#95—22no—
227C2×#95—21no—
228C2×#95—22no—
229C2×#95D412yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—28no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8—44no—
244C22×Q8C2411yescharacteristic
245C22×Q8C2414yes—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×Q16—28no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×SD16—28no—
263C2×SD16—28no—
264C2×SD16—28no—
265C2×SD16—44no—
266C2×SD16—44no—
267C2×SD16—44no—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273C4⋊Q8—28no—
274C4⋊Q8—24no—
275C4⋊Q8#4118yes—
276#127—24no—
277#124—44no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280#121—24no—
281#121—24no—
282#121—44no—
283#121—82no—
284C4×Q8—28no—
285C4×Q8—28no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—48no—
290C4×Q8—48no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—84no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C42—24no—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42—48no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—44no—
360#106—24no—
361#106—24no—
362C4⋊C8—24no—
363C4⋊C8—44no—
364#103—28no—
365#103—28no—
366#103—24no—
367#103—24no—
368#103—44no—
369#103—44no—
370#103—44no—
371#102—28no—
372#102—44no—
373C22⋊C8—22no—
374C22⋊C8—24no—
375C22⋊C8#4114yes—
376C4×C8—24no—
377C4×C8—44no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—22no—
400C24C22×D411yescharacteristic
401C2×Q8—28no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—416no—
412C2×Q8—416no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8C2×#4118yes—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—81no—
428C2×D4—81no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4#13514yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4C2511yescharacteristic
475Q16—88no—
476SD16—416no—
477SD16—88no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8C2×#4114yes—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4—81no—
515C4⋊C4#13714yes—
516C4⋊C4H(5,𝔽2)14yes—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—42no—
535C42—42no—
536C42—48no—
537C42—48no—
538C42—42no—
539C42—44no—
540C42—44no—
541C42—42no—
542C42—82no—
543C42C2×#4114yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23#55012yes—
556C23C2×#13512yes—
557C23#57712yes—
558C23C23×D411yescharacteristic
559Q8—28no—
560Q8—28no—
561Q8—416no—
562Q8—44no—
563Q8—44no—
564Q8—42no—
565Q8—42no—
566D4—44no—
567D4—44no—
568D4—88no—
569D4—84no—
570D4—162no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—416no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—164no—
625C2×C4#52512yes—
626C2×C4#52512yes—
627C2×C4#55012yes—
628C2×C4C2×#13512yes—
629C2×C4C2×#13512yes—
630C2×C4C2×#13714yes—
631C2×C4C2×#13712yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4112yes—
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C8—24no—
637C8—42no—
638C8—44no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22#339414yes—
653C22C2×#52511yescharacteristic
654C22C2×#55011yescharacteristic
655C22#348411yescharacteristic
656C22C22×#13511yescharacteristic
657C22C22×#13711yescharacteristic
658C22C2×#57711yescharacteristic
659C22#358611yescharacteristic
660C4—22no—
661C4—24no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—42no—
668C4—41no—
669C4—48no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—82no—
674C4—82no—
675C4—81no—
676C4#310814yes—
677C4#348112yes—
678C4#351912yes—
679C2—24no—
680C2—162no—
681C2#3652911yescharacteristic
682C2C2×#310811yescharacteristic
683C2#6058211yescharacteristic
684C2C2×#339411yescharacteristic
685C2#6140411yescharacteristic
686C2#6156511yescharacteristic
687C2#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×#2365C2semidirect
4. #32909C2semidirect
5. #28306C2semidirect
6. #28118C2semidirect
7. #27069C2semidirect
8. #26589C2semidirect
9. #26531C2semidirect
11. #25369C2semidirect
12. #24212C2semidirect
13. #23182C2semidirect
14. #23096C2semidirect
15. #22625C2semidirect
16. C4×#483C2semidirect
17. C4×#448C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#23652
#331062
#329092
#283062
#281182
#270692
#265892
#265312
#257182
#253692
#242122
#231822
#230962
#226252
C4×#4832
C4×#4482