Finite Group Databasebeta

← #6528582

Subgroups — #6528582

Order 512.

1593 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35238C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#29336C212yesmaximal, complemented
4#29328C212yesmaximal, complemented
5#27982C212yesmaximal, complemented
6#27977C212yesmaximal, complemented
7#27759C214yesmaximal, complemented
8#27755C212yesmaximal, complemented
9#27750C212yesmaximal, complemented
10#27743C214yesmaximal, complemented
11#27738C212yesmaximal, complemented
12#27728C212yesmaximal, complemented
13#21867C211yesmaximal, characteristic, complemented
14#21859C211yesmaximal, characteristic, complemented
15#21395C211yesmaximal, characteristic, complemented
16#21388C211yesmaximal, characteristic, complemented
17#20535C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2856C2212yes—
23#2850C2212yes—
24#2662—22no—
25#2571—21no—
26#2571—21no—
27#2571—22no—
28#2571C2212yes—
29#2571C2212yes—
30#2571C2214yes—
31#2567C2211yescharacteristic
32#2566C2212yes—
33#2563C2211yescharacteristic
34#2548C2211yescharacteristic
35#2544C2212yes—
36#2541C2211yescharacteristic
37#2430—22no—
38#2416—21no—
39#2416—21no—
40C2×#400—22no—
41#2324—21no—
42#2324—21no—
43C2×#389—21no—
44C2×#389C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382—21no—
47C2×#382C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2099C2214yes—
52#2074C2214yes—
53#2073C2214yes—
54#2073C2214yes—
55#2073C2214yes—
56#2042C2214yes—
57#2041C2214yes—
58#1992C2212yes—
59#1990C2212yes—
60#1989C2212yes—
61#1987C2212yes—
62#1983C2212yes—
63#1982C2212yes—
64#1961C2214yes—
65#1959C2212yes—
66#1958C2214yes—
67#1957C2212yes—
68#1936C2214yes—
69#1935C2214yes—
70#1934C2214yes—
71#1934C2214yes—
72#1874C2212yes—
73#1874C2212yes—
74#1874C2211yescharacteristic
75#1874C2212yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1874C2212yes—
79#1872C2211yescharacteristic
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1836C2214yes—
83#1835C2214yes—
84#1803C2212yes—
85#1802C2212yes—
86#1802C2214yes—
87#1771C2214yes—
88#1770C2212yes—
89C8×C4⋊C4C2212yes—
90#1762C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2212yes—
95C22×#41—21no—
96#532—24no—
97#532—24no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#127C2312yes—
101#529—24no—
102#529—24no—
103C2×#126—21no—
104C2×#125C2311yescharacteristic
105C2×#124—21no—
106C2×#124C2311yescharacteristic
107C2×#124C2312yes—
108C2×#123—21no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×#121—21no—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×#121C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C42—21no—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#432—24no—
135#431—24no—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2312yes—
151C2×#97C2312yes—
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2314yes—
155#400—24no—
156#400—24no—
157#400—24no—
158#400—24no—
159#396—22no—
160#396—22no—
161#395—22no—
162#394—22no—
163#394—22no—
164#394—22no—
165#393—22no—
166#392—22no—
167#391C2312yes—
168#390—21no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—24no—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#388—22no—
185#388—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#382—22no—
198#382—22no—
199#382—24no—
200#382—22no—
201#382—22no—
202#382C2312yes—
203#382C2314yes—
204#382C2314yes—
205#380—21no—
206#380—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234C2×#41—44no—
235C2×#41—82no—
236C2×#41—82no—
237C22×Q8—21no—
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262#127—22no—
263#127—22no—
264#127—44no—
265#126—44no—
266#125—22no—
267#125—22no—
268#124—22no—
269#124—22no—
270#124—44no—
271#124—44no—
272#123—44no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275#121—22no—
276#121—22no—
277#121—24no—
278#121—24no—
279#121—44no—
280#121—44no—
281#121—44no—
282#121—84no—
283#121—84no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—44no—
291C4×D4—44no—
292#117—44no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42—44no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42#4114yes—
354C2×C42#4114yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C4⋊C8—24no—
360C4⋊C8—24no—
361#103—42no—
362#103—44no—
363#103—42no—
364#102—44no—
365#102—44no—
366#102—44no—
367#102—44no—
368#102—42no—
369#102—44no—
370#102—42no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—24no—
374#97—24no—
375#97—44no—
376C4×C8—24no—
377C4×C8—44no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—21no—
395C24—22no—
396C24C22×D411yescharacteristic
397#41—816no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4#12414yes—
466C22×C4#12414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2511yescharacteristic
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C42—24no—
538C42—24no—
539C42—42no—
540C42—42no—
541C42—44no—
542C42—42no—
543C42—44no—
544C42#12414yes—
545C42#12414yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23C2×#12412yes—
569C23#57612yes—
570C23#57712yes—
571C23C23×D411yescharacteristic
572Q8—42no—
573Q8—42no—
574Q8—84no—
575D4—42no—
576D4—42no—
577D4—44no—
578D4—44no—
579D4—84no—
580D4—84no—
581D4—88no—
582D4—84no—
583D4—164no—
584D4—164no—
585C2×C4—28no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—88no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4C2×#12412yes—
638C2×C4C2×#12412yes—
639C2×C4C2×#12412yes—
640C2×C4C2×#12412yes—
641C2×C4C2×#12412yes—
642C2×C4#57512yes—
643C2×C4#57512yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4#58512yes—
647C2×C4#58512yes—
648C8—42no—
649C8—42no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—81no—
658C22—82no—
659C22—81no—
660C22—82no—
661C22—81no—
662C22—82no—
663C22—82no—
664C22—81no—
665C22—81no—
666C22—82no—
667C22—82no—
668C22—81no—
669C22—82no—
670C22—81no—
671C22#313614yes—
672C22C22×#12411yescharacteristic
673C22#348511yescharacteristic
674C22#348711yescharacteristic
675C22C2×#57511yescharacteristic
676C22C2×#57511yescharacteristic
677C22C2×#57611yescharacteristic
678C22C2×#57711yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—82no—
688C4—81no—
689C4—82no—
690C4—81no—
691C4—81no—
692C4—84no—
693C4—82no—
694C4#314214yes—
695C4#351212yes—
696C4#351212yes—
697C2—24no—
698C2—161no—
699C2—162no—
700C2#3564911yescharacteristic
701C2C2×#313611yescharacteristic
702C2C2×#314211yescharacteristic
703C2#6173911yescharacteristic
704C2#6174011yescharacteristic
705C2#6189811yescharacteristic
706C2#6189911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35238C2
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×#2571C2semidirect
3. #29336C2semidirect
4. #29328C2semidirect
5. #27982C2semidirect
6. #27977C2semidirect
7. #27759C2semidirect
8. #27755C2semidirect
9. #27750C2semidirect
10. #27743C2semidirect
11. #27738C2semidirect
12. #27728C2semidirect
13. #21867C2semidirect
14. #21859C2semidirect
15. #21395C2semidirect
16. #21388C2semidirect
17. #20535C2semidirect

Maximal subgroups

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

subgroupindex
#352382
C2×#25712
#293362
#293282
#279822
#279772
#277592
#277552
#277502
#277432
#277382
#277282
#218672
#218592
#213952
#213882
#205352