Finite Group Databasebeta

← #6796016

Subgroups — #6796016

Order 512.

1633 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32406C214yesmaximal, complemented
5#31602C214yesmaximal, complemented
6#29936C212yesmaximal, complemented
7#29286C211yesmaximal, characteristic, complemented
8#28704C212yesmaximal, complemented
9#28689C211yesmaximal, characteristic, complemented
10#27624C212yesmaximal, complemented
11#26506C212yesmaximal, complemented
12#25449C212yesmaximal, complemented
13#25338C212yesmaximal
14#24835C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21961C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#529—21no—
20C2×#500C2212yes—
21C2×#474C2211yescharacteristic
22C2×#474C2212yes—
23C2×C8×D4C2212yes—
24C22×C4⋊C8C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2568—21no—
28#2553—22no—
29#2552C2211yescharacteristic
30#2547—21no—
31#2547C2211yescharacteristic
32#2547C2212yes—
33#2542C2211yescharacteristic
34#2537C2211yescharacteristic
35#2499—22no—
36#2496—22no—
37#2496C2214yes—
38#2496C2212yes—
39#2489—21no—
40#2427—21no—
41#2422C2212yes—
42#2395—22no—
43#2365—21no—
44#2365C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8—21no—
48C4×#121C2212yes—
49#2285C2212yes—
50#2285C2212yes—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2099C2214yes—
54#2086C2214yes—
55#2086C2214yes—
56#2083C2214yes—
57#2083C2214yes—
58#2075C2214yes—
59#2075C2214yes—
60#2075C2214yes—
61#2067C2214yes—
62#2033C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1988C2212yes—
67#1988C2211yescharacteristic
68#1947C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1947C2211yescharacteristic
73#1947C2214yes—
74#1947C2214yes—
75#1947C2212yes—
76#1947C2212yes—
77#1939C2212yes—
78#1925C2214yes—
79#1918C2212yes—
80#1918C2214yes—
81#1893C2214yes—
82#1862C2212yes—
83#1850C2214yes—
84#1842C2211yescharacteristic
85#1808C2214yes—
86#1808C2214yes—
87#1808C2214yes—
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C23×C8C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102#529—48no—
103C2×#126—22no—
104C2×#126—24no—
105C2×#125—21no—
106C2×#124—21no—
107C2×#124—22no—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117—22no—
126C22×C42—21no—
127C22×C42C2312yes—
128#500—44no—
129#474—24no—
130#474—44no—
131C8×D4—44no—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×C4⋊C8—28no—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2312yes—
150#400—24no—
151#400—22no—
152#399—22no—
153#398—22no—
154#397—22no—
155#397—22no—
156#397—24no—
157#397—24no—
158#397—22no—
159#394—22no—
160#394—22no—
161#391—22no—
162#391—22no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—22no—
167#390C2312yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#389C2312yes—
176#388—22no—
177#387—24no—
178#387—24no—
179#387—22no—
180#386—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191#382—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—21no—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199#380C2314yes—
200#380C2314yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×D4—22no—
239C22×D4C2×D411yescharacteristic
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C22×C8—28no—
248C22×C8—44no—
249C22×C8—44no—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261#126—48no—
262#126—416no—
263#125—44no—
264#124—44no—
265#124—48no—
266#123—48no—
267C22⋊Q8—44no—
268C22⋊Q8—48no—
269C22⋊Q8—44no—
270#121—24no—
271#121—42no—
272#121—44no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—42no—
284C4×D4—41no—
285C4×D4—48no—
286C4×D4D818yes—
287#117—48no—
288#117—48no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—24no—
353#107—42no—
354#107—44no—
355#107—44no—
356#107—44no—
357#107—44no—
358C4⋊C8—28no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361C4⋊C8—22no—
362C4⋊C8—42no—
363C4⋊C8#4114yes—
364#102—24no—
365#102—24no—
366#102—44no—
367C22⋊C8—44no—
368C4×C8—24no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24C22×D412yes—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4C2×D814yes—
409C2×D42−1+412yes—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—81no—
438C22×C4C2×D818yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—88no—
458C2×C8C2×#4114yes—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4C2×D814yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4C2×D818yes—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42—41no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—44no—
524C42—48no—
525C42—44no—
526C42C2×D814yes—
527C42C2×#4112yes—
528C42H(5,𝔽2)12yes—
529C422−1+412yes—
530C422−1+412yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C22×D814yes—
545C23C23×D411yescharacteristic
546Q8—84no—
547Q8—84no—
548Q8—88no—
549D4—28no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554D4—88no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—216no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4#53712yes—
606C2×C4#53712yes—
607C2×C4#54712yes—
608C2×C4#54712yes—
609C2×C4C22×D814yes—
610C2×C4C22×D812yes—
611C2×C4C22×D812yes—
612C2×C4C22×D812yes—
613C2×C4#57512yes—
614C2×C4#57812yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58511yescharacteristic
621C8—24no—
622C8—44no—
623C8—42no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22C2×#53711yescharacteristic
639C22C2×#54711yescharacteristic
640C22#348911yescharacteristic
641C22C23×D811yescharacteristic
642C22C2×#57511yescharacteristic
643C22C2×#57811yescharacteristic
644C22#358611yescharacteristic
645C4—24no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—41no—
652C4—81no—
653C4—82no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—82no—
659C4—84no—
660C4#324514yes—
661C4#333414yes—
662C4#351411yescharacteristic
663C4#352311yescharacteristic
664C4#355211yescharacteristic
665C4#355511yescharacteristic
666C2—44no—
667C2—82no—
668C2#3892311yescharacteristic
669C2C2×#324511yescharacteristic
670C2#6092511yescharacteristic
671C2C2×#333411yescharacteristic
672C2#6125711yescharacteristic
673C2#6209511yescharacteristic
674C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36188C2semidirect
3. #33195C2semidirect
4. #32406C2semidirect
5. #31602C2semidirect
6. #29936C2semidirect
7. #29286C2semidirect
8. #28704C2semidirect
9. #28689C2semidirect
10. #27624C2semidirect
11. #26506C2semidirect
12. #25449C2semidirect
14. #24835C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21961C2semidirect
17. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349322
#331952
#324062
#316022
#299362
#292862
#287042
#286892
#276242
#265062
#254492
#253382
#248352
D4×C4⋊C82
#219612
C4×#4742