Finite Group Databasebeta

← #6857029

Subgroups — #6857029

Order 512.

1643 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#32472C214yesmaximal, complemented
5#31802C214yesmaximal, complemented
6#29824C212yesmaximal
7#29346C212yesmaximal, complemented
8#29286C212yesmaximal, complemented
9#29219C211yesmaximal, characteristic, complemented
10#28279C211yesmaximal, characteristic, complemented
11#28185C212yesmaximal, complemented
12#26020C212yesmaximal, complemented
13#25469C212yesmaximal, complemented
14#24676C212yesmaximal, complemented
15Q8×C4⋊C8C211yesmaximal, characteristic, complemented
16#22110C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556—21no—
19C2×#533—21no—
20C2×#474C2212yes—
21C2×#474C2212yes—
22C2×#459C2211yescharacteristic
23C2×C8×Q8C2212yes—
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2858C2212yes—
27#2556—22no—
28#2551C2211yescharacteristic
29#2547—21no—
30#2546C2211yescharacteristic
31#2545—22no—
32#2542C2211yescharacteristic
33#2538—21no—
34#2538C2211yescharacteristic
35#2538C2212yes—
36#2512—22no—
37#2512C2214yes—
38#2506C2212yes—
39#2492—21no—
40#2452C2212yes—
41#2429—21no—
42#2396—22no—
43#2354—21no—
44#2354C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#127C2211yescharacteristic
49C4×C22⋊Q8—21no—
50#2293C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2099C2214yes—
54#2086C2214yes—
55#2086C2214yes—
56#2071C2214yes—
57#2067C2214yes—
58#2067C2214yes—
59#2067C2214yes—
60#2054C2214yes—
61#2054C2214yes—
62#2023C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1988C2214yes—
67#1988C2212yes—
68#1982C2212yes—
69#1982C2211yescharacteristic
70#1947C2212yes—
71#1947C2214yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1942C2214yes—
76#1939C2214yes—
77#1939C2212yes—
78#1934C2214yes—
79#1934C2214yes—
80#1934C2214yes—
81#1925C2214yes—
82#1925C2212yes—
83#1918C2214yes—
84#1879C2212yes—
85#1842C2212yes—
86#1842C2211yescharacteristic
87#1834C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C22×D8C2312yes—
96#556—48no—
97#533—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#126—24no—
107C2×#125—22no—
108C2×#124—22no—
109C2×#124—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117—22no—
129C22×C42—21no—
130#474—28no—
131#474—44no—
132#459—24no—
133C8×Q8—44no—
134C4×D8—44no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#399—22no—
156#398—22no—
157#398—22no—
158#397—24no—
159#396—22no—
160#396—22no—
161#396—24no—
162#396—24no—
163#395—22no—
164#393—22no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391—22no—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—22no—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#390C2314yes—
177#390C2314yes—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2314yes—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#388—24no—
191#388—24no—
192#387—22no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#384C2312yes—
201#380—22no—
202#380—22no—
203#380—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C43C2311yescharacteristic
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C2×D8—24no—
248C2×D8—42no—
249C2×D8—42no—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263#127—22no—
264#127—22no—
265#126—416no—
266#125—48no—
267#124—48no—
268#124—84no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—48no—
272C22⋊Q8—44no—
273C22⋊Q8—48no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—42no—
277C4×Q8—48no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8D818yescomplemented
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—24no—
287C4×D4—41no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4#4114yes—
291#117—48no—
292#117—48no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—28no—
359#107—42no—
360#107—44no—
361C4⋊C8—22no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—22no—
365C4⋊C8—42no—
366C4⋊C8—44no—
367C4⋊C8—44no—
368#102—28no—
369#102—24no—
370#102—24no—
371#102—44no—
372#102—42no—
373#102—42no—
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—22no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—82no—
404C2×Q8C2×D814yes—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4C2×#4112yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2511yescharacteristic
453D8—48no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4C2×D814yes—
499C4⋊C4C2×D818yes—
500C4⋊C4C2×#4112yes—
501C4⋊C42−1+412yes—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—82no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—21no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—22no—
525C42—41no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—48no—
533C42C2×D814yes—
534C42C2×D818yes—
535C42C2×#4112yes—
536C42H(5,𝔽2)12yes—
537C422−1+412yes—
538C422−1+412yes—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C23×D411yescharacteristic
554Q8—28no—
555Q8—88no—
556Q8—88no—
557Q8—84no—
558D4—22no—
559D4—22no—
560D4—42no—
561D4—42no—
562D4—41no—
563D4—42no—
564D4—41no—
565D4—42no—
566D4—84no—
567D4—84no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—88no—
612C2×C4#53712yes—
613C2×C4#53712yes—
614C2×C4#54812yes—
615C2×C4#54812yes—
616C2×C4C22×D814yes—
617C2×C4C22×D812yes—
618C2×C4C22×D814yes—
619C2×C4C22×D812yes—
620C2×C4C22×D812yes—
621C2×C4#57512yes—
622C2×C4#57712yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C2×C4#58511yescharacteristic
629C8—44no—
630C8—42no—
631C8—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22C2×#53711yescharacteristic
647C22C2×#54811yescharacteristic
648C22#348911yescharacteristic
649C22C23×D811yescharacteristic
650C22C2×#57511yescharacteristic
651C22C2×#57711yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—42no—
663C4—84no—
664C4—82no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4#324514yes—
669C4#335814yes—
670C4#350611yescharacteristic
671C4#350911yescharacteristic
672C4#355011yescharacteristic
673C4#355511yescharacteristic
674C2—82no—
675C2—162no—
676C2#3972711yescharacteristic
677C2C2×#324511yescharacteristic
678C2#6093111yescharacteristic
679C2C2×#335811yescharacteristic
680C2#6132311yescharacteristic
681C2#6209511yescharacteristic
682C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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. #36594C2semidirect
2. #34929C2semidirect
3. #33195C2semidirect
4. #32472C2semidirect
5. #31802C2semidirect
7. #29346C2semidirect
8. #29286C2semidirect
9. #29219C2semidirect
10. #28279C2semidirect
11. #28185C2semidirect
12. #26020C2semidirect
13. #25469C2semidirect
14. #24676C2semidirect
15. Q8×C4⋊C8C2semidirect
16. #22110C2semidirect
280. C4×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#365942
#349292
#331952
#324722
#318022
#298242
#293462
#292862
#292192
#282792
#281852
#260202
#254692
#246762
Q8×C4⋊C82
#221102
C4×#4592