Finite Group Databasebeta

← #6855930

Subgroups — #6855930

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36596C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33106C211yesmaximal, characteristic, complemented
4#32257C214yesmaximal, complemented
5#30835C214yesmaximal, complemented
6#29754C212yesmaximal, complemented
7#28798C212yesmaximal, complemented
8#27973C211yesmaximal, characteristic, complemented
9#27775C212yesmaximal, complemented
10#27678C211yesmaximal, characteristic, complemented
11#26467C212yesmaximal, complemented
12#26025C212yesmaximal, complemented
13#25913C212yesmaximal, complemented
14#24305C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal, complemented
17C4×#480C211yesmaximal, characteristic, complemented
18C2×#553—21no—
19C22×#125C2212yes—
20C2×#490C2214yes—
21C2×#480C2211yescharacteristic
22C2×#480C2212yes—
23C2×#478C2214yes—
24#2856C2212yes—
25#2856C2211yescharacteristic
26#2554—22no—
27#2552—21no—
28#2552—21no—
29#2552—22no—
30#2552C2211yescharacteristic
31#2552C2214yes—
32#2552C2212yes—
33#2552C2212yes—
34#2543—22no—
35#2541C2211yescharacteristic
36#2512C2212yes—
37#2490—21no—
38#2445C2212yes—
39#2416—21no—
40C2×#398—22no—
41#2366C2211yescharacteristic
42#2363—21no—
43#2354C2211yescharacteristic
44C4×#121C2211yescharacteristic
45C4×#121C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384—21no—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2212yes—
53#2094C2212yes—
54#2083C2214yes—
55#2071C2214yes—
56#2067C2214yes—
57#2067C2214yes—
58#2059C2214yes—
59#2041C2214yes—
60#2025C2214yes—
61#2025C2214yes—
62#2025C2214yes—
63#1992C2214yes—
64#1989C2214yes—
65#1982C2212yes—
66#1957C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1943C2214yes—
70#1939C2212yes—
71#1936C2214yes—
72#1925C2214yes—
73#1925C2212yes—
74#1921C2214yes—
75#1918C2212yes—
76#1917C2214yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1874C2212yes—
81#1874C2212yes—
82#1864C2214yes—
83#1853C2212yes—
84#1852C2211yescharacteristic
85#1834C2212yes—
86#1823C2211yescharacteristic
87#1802C2212yes—
88#1797C2211yescharacteristic
89#1797C2212yes—
90C4×#107C2212yes—
91#1770C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#553—48no—
96C2×#127C2311yescharacteristic
97C2×#127C2312yes—
98C2×#126—22no—
99C2×#125—21no—
100C2×#125—22no—
101C2×#125—22no—
102C2×#125—48no—
103C2×#125C2314yes—
104C2×#125C2314yes—
105C2×#123—22no—
106C2×#121—21no—
107C2×#121—22no—
108C2×#121—22no—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121C2311yescharacteristic
113C2×#121C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×#117—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#490—48no—
134#480—24no—
135#480—28no—
136#478—48no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#398—24no—
156#398—24no—
157#398—24no—
158#398—24no—
159#398—22no—
160#397—22no—
161#396—22no—
162#394—22no—
163#394—22no—
164#392—22no—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390—21no—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#390C2311yescharacteristic
174#390C2312yes—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—24no—
179#389—22no—
180#389—22no—
181#389C2312yes—
182#389C2314yes—
183#389C2314yes—
184#389C2311yescharacteristic
185#388—22no—
186#388—21no—
187#388—22no—
188#388—22no—
189#386—22no—
190#386—21no—
191#386—22no—
192#386—22no—
193#384—22no—
194#384—22no—
195#384—24no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2312yes—
200#384C2312yes—
201#384C2311yescharacteristic
202#380—21no—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C2×#95—21no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×D4—21no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255#127—24no—
256#127—44no—
257#126—48no—
258#125—28no—
259#125—28no—
260#125—44no—
261#125—44no—
262#125—48no—
263#125—48no—
264#125—44no—
265#123—48no—
266#121—22no—
267#121—24no—
268#121—22no—
269#121—24no—
270#121—44no—
271#121—44no—
272#121—48no—
273#121—44no—
274#121—82no—
275#121—84no—
276C4×D4—24no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—48no—
280C4×D4—44no—
281#117—44no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4#4114yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—24no—
356#107—28no—
357#107—24no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361#102—24no—
362#102—28no—
363#102—28no—
364#102—24no—
365#102—44no—
366#102—44no—
367#102—44no—
368C22⋊C8—22no—
369C22⋊C8—24no—
370C22⋊C8—22no—
371#97—44no—
372C4×C8—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24—22no—
388C24C22×D411yescharacteristic
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—24no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4—81no—
414C2×D4—82no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4AGL(1,ℤ/8)14yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4112yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4C2×#4114yes—
508C4⋊C4H(5,𝔽2)14yes—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—81no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—44no—
533C42—44no—
534C42—42no—
535C422−1+414yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23#54812yes—
555C23C2×AGL(1,ℤ/8)12yes—
556C23#57612yes—
557C23C23×D411yescharacteristic
558D4—44no—
559D4—42no—
560D4—44no—
561D4—42no—
562D4—88no—
563D4—84no—
564D4—84no—
565D4—162no—
566D4—164no—
567D4—164no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4—84no—
620C2×C4—164no—
621C2×C4#53612yes—
622C2×C4#53612yes—
623C2×C4#54812yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4#57512yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×2−1+412yes—
630C2×C4C2×2−1+411yescharacteristic
631C8—44no—
632C8—42no—
633C8—42no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—81no—
641C22—82no—
642C22—82no—
643C22—81no—
644C22—81no—
645C22—82no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22—82no—
650C22—81no—
651C22—82no—
652C22—82no—
653C22—81no—
654C22#336614yes—
655C22C2×#53611yescharacteristic
656C22C2×#54811yescharacteristic
657C22#348411yescharacteristic
658C22C22×AGL(1,ℤ/8)11yescharacteristic
659C22C2×#57511yescharacteristic
660C22C2×#57611yescharacteristic
661C22#358611yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—44no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—41no—
672C4—82no—
673C4—81no—
674C4—81no—
675C4—84no—
676C4—81no—
677C4#323414yes—
678C4#351112yes—
679C4#355212yes—
680C2—24no—
681C2—162no—
682C2—161no—
683C2#3972011yescharacteristic
684C2C2×#323411yescharacteristic
685C2#6087111yescharacteristic
686C2C2×#336611yescharacteristic
687C2#6132011yescharacteristic
688C2#6158511yescharacteristic
689C2#6160511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36596C2
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×#2552C2semidirect
3. #33106C2semidirect
4. #32257C2semidirect
5. #30835C2semidirect
6. #29754C2semidirect
7. #28798C2semidirect
8. #27973C2semidirect
9. #27775C2semidirect
10. #27678C2semidirect
11. #26467C2semidirect
12. #26025C2semidirect
13. #25913C2semidirect
14. #24305C2semidirect
15. #22629C2semidirect
16. #22081C2semidirect
17. C4×#480C2semidirect

Maximal subgroups

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

subgroupindex
#365962
C2×#25522
#331062
#322572
#308352
#297542
#287982
#279732
#277752
#276782
#264672
#260252
#259132
#243052
#226292
#220812
C4×#4802