Finite Group Databasebeta

← #6800848

Subgroups — #6800848

Order 512.

1669 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
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34934C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32226C214yesmaximal, complemented
5#30805C214yesmaximal, complemented
6#29980C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic, complemented
8#28789C212yesmaximal, complemented
9#28716C212yesmaximal, complemented
10#28279C211yesmaximal, characteristic, complemented
11#28201C212yesmaximal, complemented
12#27767C212yesmaximal
13#26316C212yesmaximal, complemented
14#24990C212yesmaximal, complemented
15#22821C211yesmaximal, characteristic, complemented
16#22110C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×Q82—21no—
19C2×#529—21no—
20C2×#486C2214yes—
21C2×#474C2211yescharacteristic
22C2×#474C2212yes—
23C2×#474C2212yes—
24#2868C2212yes—
25#2868C2211yescharacteristic
26#2567—22no—
27#2567—21no—
28#2554—22no—
29#2552C2211yescharacteristic
30#2551—21no—
31#2551—22no—
32#2551C2214yes—
33#2551C2212yes—
34#2548C2211yescharacteristic
35#2541C2211yescharacteristic
36#2538C2211yescharacteristic
37#2512C2212yes—
38#2490—21no—
39#2445C2212yes—
40#2426—21no—
41#2392—22no—
42Q8×C4⋊C4—21no—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2212yes—
47C4×#127C2211yescharacteristic
48C4×#121—21no—
49#2293C2212yes—
50C42×Q8C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2099C2214yes—
53#2093C2214yes—
54#2083C2214yes—
55#2071C2214yes—
56#2071C2214yes—
57#2067C2214yes—
58#2057C2214yes—
59#2041C2214yes—
60#2023C2214yes—
61#2023C2214yes—
62#1990C2214yes—
63#1988C2211yescharacteristic
64#1988C2212yes—
65#1984C2212yes—
66#1982C2212yes—
67#1957C2212yes—
68#1947C2211yescharacteristic
69#1947C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1942C2214yes—
73#1934C2214yes—
74#1925C2214yes—
75#1925C2214yes—
76#1925C2214yes—
77#1918C2214yes—
78#1918C2212yes—
79#1916C2214yes—
80#1862C2212yes—
81#1842C2211yescharacteristic
82#1834C2212yes—
83#1831C2212yes—
84#1802C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95Q82—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2314yes—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#127C2312yes—
107#529—48no—
108C2×#126—22no—
109C2×#125—21no—
110C2×#124—21no—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×#121—21no—
114C2×#121—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C22×C42—21no—
131#486—48no—
132#474—24no—
133#474—28no—
134#474—44no—
135C2×#107C2314yes—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#398—22no—
154#397—22no—
155#396—22no—
156#396—22no—
157#395—24no—
158#395—24no—
159#395—24no—
160#394—22no—
161#394—22no—
162#394—22no—
163#392—22no—
164#391—22no—
165#391—21no—
166#390—21no—
167#390—21no—
168#390—22no—
169#390—22no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#389—24no—
177#389—22no—
178#389—22no—
179#389—24no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389C2312yes—
184#389C2314yes—
185#389C2312yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#386—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384—22no—
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2314yes—
201#380—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C43C2312yes—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—24no—
237C22×Q8C2×D411yescharacteristic
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4C2412yes—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—22no—
252C4⋊Q8—42no—
253C4⋊Q8—44no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—48no—
257C4⋊Q8—416no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260C4⋊Q8—82no—
261C4⋊Q8—84no—
262C4⋊Q8#4114yes—
263#127—24no—
264#127—22no—
265#127—22no—
266#127—42no—
267#126—48no—
268#125—44no—
269#124—44no—
270#123—48no—
271C22⋊Q8—44no—
272#121—44no—
273#121—44no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—48no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8D818yes—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—41no—
289C4×D4—44no—
290C4×D4—48no—
291C4×D4—82no—
292C4×D4—84no—
293#117—48no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
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—22no—
340C2×C42—21no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—28no—
360#107—24no—
361#107—42no—
362#107—44no—
363C4⋊C8—22no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—22no—
367C4⋊C8—42no—
368#102—28no—
369#102—24no—
370#102—24no—
371#102—28no—
372#102—44no—
373#102—44no—
374#102—44no—
375C4×C8—24no—
376C4×C8—44no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—82no—
399C2×Q8—84no—
400C2×Q8—82no—
401C2×Q8—84no—
402C2×Q8C2×D814yes—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—82no—
423C2×D4—81no—
424C2×D4—81no—
425C2×D4—82no—
426C2×D4H(5,𝔽2)12yes—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—81no—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2511yescharacteristic
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4C2×D814yes—
502C4⋊C4C2×D818yes—
503C4⋊C4#13714yes—
504C4⋊C4C2×#4114yes—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—24no—
520C42—22no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—22no—
529C42—41no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—48no—
535C42—44no—
536C42C2×D814yes—
537C42C2×D818yes—
538C42C2×#4112yes—
539C422−1+412yes—
540C422−1+412yes—
541C422−1+412yes—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23C23×D411yescharacteristic
557Q8—28no—
558Q8—48no—
559Q8—164no—
560Q8—168no—
561D4—22no—
562D4—22no—
563D4—41no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—41no—
568D4—42no—
569D4—88no—
570D4—88no—
571D4—84no—
572D4—84no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4#53612yes—
617C2×C4#53612yes—
618C2×C4#54812yes—
619C2×C4#54812yes—
620C2×C4C22×D814yes—
621C2×C4C22×D812yes—
622C2×C4C22×D814yes—
623C2×C4C22×D812yes—
624C2×C4C22×D812yes—
625C2×C4C2×#13714yes—
626C2×C4C2×#13712yes—
627C2×C4#57712yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4#58511yescharacteristic
634C8—42no—
635C8—44no—
636C8—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22C2×#53611yescharacteristic
652C22C2×#54811yescharacteristic
653C22#348811yescharacteristic
654C22C23×D811yescharacteristic
655C22C22×#13711yescharacteristic
656C22C2×#57711yescharacteristic
657C22#358511yescharacteristic
658C4—24no—
659C4—22no—
660C4—24no—
661C4—22no—
662C4—22no—
663C4—24no—
664C4—41no—
665C4—44no—
666C4—42no—
667C4—81no—
668C4—82no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4—84no—
673C4#322614yes—
674C4#335814yes—
675C4#351811yescharacteristic
676C4#352211yescharacteristic
677C4#354411yescharacteristic
678C4#355511yescharacteristic
679C2—82no—
680C2—162no—
681C2#3912511yescharacteristic
682C2C2×#322611yescharacteristic
683C2#6089211yescharacteristic
684C2C2×#335811yescharacteristic
685C2#6131911yescharacteristic
686C2#6194711yescharacteristic
687C2#6198011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. Q8×C4⋊Q8C2semidirect
3. #33195C2semidirect
4. #32226C2semidirect
5. #30805C2semidirect
6. #29980C2semidirect
7. #29139C2semidirect
8. #28789C2semidirect
9. #28716C2semidirect
10. #28279C2semidirect
11. #28201C2semidirect
13. #26316C2semidirect
14. #24990C2semidirect
15. #22821C2semidirect
16. #22110C2semidirect
17. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349342
#331952
#322262
#308052
#299802
#291392
#287892
#287162
#282792
#282012
#277672
#263162
#249902
#228212
#221102
C4×#4742