Finite Group Databasebeta

← #6802840

Subgroups — #6802840

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 692 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#32374C214yesmaximal, complemented
5#31570C214yesmaximal, complemented
6#29754C212yesmaximal, complemented
7#28827C212yesmaximal, complemented
8#27909C211yesmaximal, characteristic, complemented
9#27909C212yesmaximal, complemented
10#27747C212yesmaximal, complemented
11#27644C212yesmaximal, complemented
12#27644C211yesmaximal, characteristic, complemented
13#26526C212yesmaximal, complemented
14#26086C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22075C212yesmaximal, complemented
17C4×#480C211yesmaximal, characteristic, complemented
18C2×#553—21no—
19C22×#125—21no—
20C2×#480C2212yes—
21C2×#480C2211yescharacteristic
22C2×#480C2212yes—
23C2×#480C2212yes—
24C2×#480C2212yes—
25#2856C2212yes—
26#2856C2211yescharacteristic
27#2554—22no—
28#2552—21no—
29#2552—21no—
30#2552—22no—
31#2552C2214yes—
32#2552C2212yes—
33#2552C2212yes—
34#2552C2211yescharacteristic
35#2543—22no—
36#2541C2211yescharacteristic
37#2512C2212yes—
38#2490—21no—
39#2445C2212yes—
40#2416—21no—
41C2×#398—22no—
42#2366C2211yescharacteristic
43#2363—21no—
44#2354C2211yescharacteristic
45C4×#121C2212yes—
46C4×#121C2211yescharacteristic
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#384C2212yes—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2093C2214yes—
54#2093C2214yes—
55#2093C2214yes—
56#2083C2214yes—
57#2083C2214yes—
58#2083C2212yes—
59#2071C2212yes—
60#2071C2214yes—
61#2067C2214yes—
62#2067C2214yes—
63#2041C2214yes—
64#1982C2212yes—
65#1957C2214yes—
66#1957C2212yes—
67#1947C2212yes—
68#1947C2214yes—
69#1947C2212yes—
70#1942C2214yes—
71#1939C2212yes—
72#1934C2214yes—
73#1925C2212yes—
74#1925C2214yes—
75#1918C2214yes—
76#1918C2212yes—
77#1916C2214yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1862C2214yes—
81#1852C2211yescharacteristic
82#1845C2214yes—
83#1845C2212yes—
84#1823C2212yes—
85#1823C2211yescharacteristic
86#1802C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#553—48no—
96C2×#127C2311yescharacteristic
97C2×#127C2312yes—
98C2×#126—22no—
99C2×#125—22no—
100C2×#125—22no—
101C2×#125—24no—
102C2×#125—21no—
103C2×#125—48no—
104C2×#123—22no—
105C2×#121—21no—
106C2×#121—22no—
107C2×#121—22no—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×#121C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#480—28no—
133#480—24no—
134#480—44no—
135#480—44no—
136#480—44no—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#398—24no—
158#398—24no—
159#398—24no—
160#398—24no—
161#398—22no—
162#397—22no—
163#396—22no—
164#394—22no—
165#394—22no—
166#392—22no—
167#390—21no—
168#390—21no—
169#390—22no—
170#390—22no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#390C2311yescharacteristic
175#390C2312yes—
176#390C2312yes—
177#389—21no—
178#389—22no—
179#389—24no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2314yes—
187#388—22no—
188#388—21no—
189#388—22no—
190#388—22no—
191#386—22no—
192#386—21no—
193#386—22no—
194#386—22no—
195#384—21no—
196#384—21no—
197#384—22no—
198#384—22no—
199#384C2314yes—
200#384C2312yes—
201#384C2312yes—
202#384C2314yes—
203#384C2312yes—
204#384C2311yescharacteristic
205#380—21no—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C2×#95—21no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256#127—24no—
257#127—44no—
258#126—48no—
259#125—48no—
260#125—48no—
261#125—416no—
262#125—44no—
263#123—48no—
264#121—24no—
265#121—22no—
266#121—24no—
267#121—22no—
268#121—48no—
269#121—44no—
270#121—44no—
271#121—44no—
272#121—84no—
273#121—82no—
274C4×D4—24no—
275C4×D4—44no—
276C4×D4—48no—
277C4×D4—44no—
278C4×D4—44no—
279#117—44no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
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—48no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—84no—
310C2×C4⋊C4D818yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—28no—
355#107—24no—
356#107—42no—
357#107—44no—
358#107—44no—
359#107—44no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#102—28no—
363#102—24no—
364#102—44no—
365#102—42no—
366#102—44no—
367#102—44no—
368#102—44no—
369C22⋊C8—22no—
370C22⋊C8—24no—
371C22⋊C8#4114yes—
372C4×C8—24no—
373C4×C8—44no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24—22no—
389C24C22×D411yescharacteristic
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—82no—
411C2×D4—81no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4—81no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4C2×D814yes—
451C22×C4C2×D818yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4C2511yescharacteristic
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8C2×#4114yes—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—84no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4C2×D814yes—
507C4⋊C4C2×D814yes—
508C4⋊C4C2×D818yes—
509C4⋊C4#13714yes—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—81no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—44no—
534C42—44no—
535C42—42no—
536C422−1+414yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23#55012yes—
556C23C22×D812yes—
557C23#57612yes—
558C23C23×D411yescharacteristic
559D4—42no—
560D4—44no—
561D4—42no—
562D4—44no—
563D4—88no—
564D4—84no—
565D4—84no—
566D4—164no—
567D4—162no—
568D4—164no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4—164no—
619C2×C4#53612yes—
620C2×C4#53612yes—
621C2×C4#55012yes—
622C2×C4C22×D812yes—
623C2×C4C22×D814yes—
624C2×C4C22×D814yes—
625C2×C4C22×D812yes—
626C2×C4C2×#13714yes—
627C2×C4C2×#13712yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+412yes—
631C2×C4#58512yes—
632C8—24no—
633C8—42no—
634C8—44no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—81no—
642C22—82no—
643C22—81no—
644C22—82no—
645C22—82no—
646C22—81no—
647C22—81no—
648C22—82no—
649C22—81no—
650C22—82no—
651C22—81no—
652C22—82no—
653C22—82no—
654C22—81no—
655C22#339114yes—
656C22C2×#53611yescharacteristic
657C22C2×#55011yescharacteristic
658C22#348511yescharacteristic
659C22C23×D811yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57611yescharacteristic
662C22#358611yescharacteristic
663C4—24no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—24no—
668C4—24no—
669C4—42no—
670C4—41no—
671C4—81no—
672C4—82no—
673C4—84no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—81no—
678C4#322614yes—
679C4#351512yes—
680C4#355912yes—
681C2—24no—
682C2—161no—
683C2—162no—
684C2#3913411yescharacteristic
685C2C2×#322611yescharacteristic
686C2#6087111yescharacteristic
687C2C2×#339111yescharacteristic
688C2#6140411yescharacteristic
689C2#6169411yescharacteristic
690C2#6172611yescharacteristic
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. #32374C2semidirect
5. #31570C2semidirect
6. #29754C2semidirect
7. #28827C2semidirect
8. #27909C2semidirect
9. #27909C2semidirect
10. #27747C2semidirect
11. #27644C2semidirect
12. #27644C2semidirect
13. #26526C2semidirect
14. #26086C2semidirect
15. #22625C2semidirect
16. #22075C2semidirect
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
#323742
#315702
#297542
#288272
#279092
#277472
#276442
#265262
#260862
#226252
#220752
C4×#4802