Finite Group Databasebeta

← #6792915

Subgroups — #6792915

Order 512.

1713 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36126C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#33107C211yesmaximal, characteristic, complemented
4#31601C214yesmaximal, complemented
5#31483C214yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#28799C212yesmaximal, complemented
8#27973C211yesmaximal, characteristic, complemented
9#27755C212yesmaximal, complemented
10#27677C211yesmaximal, characteristic, complemented
11#26529C212yesmaximal, complemented
12#26107C212yesmaximal, complemented
13#24607C212yesmaximal, complemented
14#24061C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16#21865C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C2×#481C2211yescharacteristic
21C2×#481C2212yes—
22C2×#432C2212yes—
23#2856C2211yescharacteristic
24#2854C2212yes—
25#2571—21no—
26#2571—22no—
27#2571—21no—
28#2571C2212yes—
29#2571C2214yes—
30#2571C2212yes—
31#2551C2211yescharacteristic
32#2541C2211yescharacteristic
33#2491—21no—
34#2461—22no—
35#2452C2212yes—
36#2445C2212yes—
37#2430—22no—
38#2416—21no—
39C2×#400—22no—
40#2366C2211yescharacteristic
41#2361—21no—
42D4×C4⋊C4C2211yescharacteristic
43C4×#121C2211yescharacteristic
44C2×#389—21no—
45C2×#389C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382—21no—
48C2×#382C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yes—
52#2099C2212yes—
53#2093C2212yes—
54#2092C2214yes—
55#2088C2214yes—
56#2085C2214yes—
57#2073C2214yes—
58#2071C2214yes—
59#2067C2214yes—
60#2058C2214yes—
61#2041C2214yes—
62#1993C2214yes—
63#1982C2212yes—
64#1961C2214yes—
65#1957C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1943C2214yes—
69#1939C2212yes—
70#1936C2214yes—
71#1926C2212yes—
72#1926C2214yes—
73#1919C2212yes—
74#1917C2214yes—
75#1874C2212yes—
76#1874C2212yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1864C2214yes—
81#1864C2214yes—
82#1853C2211yescharacteristic
83#1852C2212yes—
84#1834C2212yes—
85#1824C2211yescharacteristic
86#1817C2212yes—
87#1804C2214yes—
88#1804C2214yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92#1770C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—22no—
96#547—48no—
97#532—28no—
98#532—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#127C2312yes—
103C2×#124—22no—
104C2×#123—22no—
105C2×#123—22no—
106C2×C22⋊Q8—22no—
107C2×#121—21no—
108C2×#121—22no—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×#121C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#481—24no—
133#481—44no—
134#432—44no—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2311yescharacteristic
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#400—24no—
154#400—24no—
155#400—24no—
156#400—24no—
157#396—22no—
158#395—22no—
159#394—22no—
160#394—22no—
161#393—22no—
162#392—22no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2311yescharacteristic
170#389—22no—
171#389—22no—
172#389—24no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#389C2314yes—
178#389C2312yes—
179#389C2312yes—
180#389C2311yescharacteristic
181#388—22no—
182#388—22no—
183#388—21no—
184#386—22no—
185#386—21no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2312yes—
194#384C2311yescharacteristic
195#382—22no—
196#382—22no—
197#382—24no—
198#382—22no—
199#382—22no—
200#382C2314yes—
201#382C2314yes—
202#380—21no—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C2×#95—21no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2314yes—
234C2×#41—48no—
235C2×#41—48no—
236C2×#41—84no—
237C22×Q8—22no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4C2412yes—
245C22×D4C2411yescharacteristic
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263#127—24no—
264#127—24no—
265#127—42no—
266#124—48no—
267#123—48no—
268#123—48no—
269C22⋊Q8—48no—
270#121—22no—
271#121—24no—
272#121—24no—
273#121—44no—
274#121—44no—
275#121—48no—
276#121—44no—
277#121—82no—
278#121#4114yes—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—48no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—44no—
286C4×D4—44no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
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—21no—
338C2×C22⋊C4—22no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—44no—
355#106—24no—
356#106—42no—
357#106—44no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360#102—24no—
361#102—44no—
362#102—44no—
363#102—44no—
364#102—42no—
365#102—44no—
366#102—44no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—24no—
370#97—44no—
371C4×C8—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24—22no—
391C24C22×D411yescharacteristic
392#41—832no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—82no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4C2×#4112yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4AGL(1,ℤ/8)14yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4C2511yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—81no—
514C4⋊C4C2×#4112yes—
515C22⋊C4—24no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—81no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C42—22no—
530C42—22no—
531C42—24no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—44no—
539C42—42no—
540C42—44no—
541C422−1+414yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23#54612yes—
561C23C2×AGL(1,ℤ/8)12yes—
562C23#57712yes—
563C23C23×D411yescharacteristic
564Q8—88no—
565D4—24no—
566D4—48no—
567D4—44no—
568D4—42no—
569D4—44no—
570D4—88no—
571D4—88no—
572D4—88no—
573D4—84no—
574D4—162no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4—88no—
625C2×C4—88no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4#53612yes—
630C2×C4#53612yes—
631C2×C4#54612yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4#57512yes—
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+412yes—
638C2×C4#58512yes—
639C8—42no—
640C8—42no—
641C8—44no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#332414yes—
663C22C2×#53611yescharacteristic
664C22C2×#54611yescharacteristic
665C22#348711yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57711yescharacteristic
669C22#358511yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—81no—
680C4—81no—
681C4—84no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#323414yes—
686C4#351512yes—
687C4#355912yes—
688C2—24no—
689C2—82no—
690C2—162no—
691C2#3890711yescharacteristic
692C2C2×#323411yescharacteristic
693C2#6087311yescharacteristic
694C2C2×#332411yescharacteristic
695C2#6116611yescharacteristic
696C2#6189111yescharacteristic
697C2#6189911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36126C2
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. #33107C2semidirect
4. #31601C2semidirect
5. #31483C2semidirect
6. #29912C2semidirect
7. #28799C2semidirect
8. #27973C2semidirect
9. #27755C2semidirect
10. #27677C2semidirect
11. #26529C2semidirect
12. #26107C2semidirect
13. #24607C2semidirect
14. #24061C2semidirect
15. #22603C2semidirect
16. #21865C2semidirect
17. C4×#481C2semidirect

Maximal subgroups

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

subgroupindex
#361262
C2×#25712
#331072
#316012
#314832
#299122
#287992
#279732
#277552
#276772
#265292
#261072
#246072
#240612
#226032
#218652
C4×#4812