Finite Group Databasebeta

← #6848663

Subgroups — #6848663

Order 512.

1571 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32205C214yesmaximal, complemented
5#31498C214yesmaximal, complemented
6#29689C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28598C212yesmaximal, complemented
9#28283C211yesmaximal, characteristic, complemented
10#27630C212yesmaximal, complemented
11#26441C212yesmaximal, complemented
12#25339C212yesmaximal
13#25273C212yesmaximal, complemented
14#24838C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21887C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#487C2212yes—
20C2×#475C2212yes—
21C2×#476C2212yes—
22C2×#474C2211yescharacteristic
23C2×C8×D4C2212yes—
24C22×C4⋊C8C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2571—21no—
28#2563C2212yes—
29#2553—22no—
30#2552C2211yescharacteristic
31#2547C2211yescharacteristic
32#2542C2211yescharacteristic
33#2538—21no—
34#2537C2211yescharacteristic
35#2497C2214yes—
36#2496—21no—
37#2492—22no—
38#2489C2212yes—
39#2439—22no—
40#2427C2212yes—
41#2422—21no—
42#2395—22no—
43#2369C2212yes—
44#2355—21no—
45#2354—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49C4×#121—21no—
50#2285—21no—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2088C2214yes—
55#2077C2214yes—
56#2072C2214yes—
57#2066C2214yes—
58#2066C2214yes—
59#2059C2214yes—
60#2042C2214yes—
61#2036C2214yes—
62#2036C2214yes—
63#2034C2214yes—
64#1990C2212yes—
65#1989C2214yes—
66#1988C2211yescharacteristic
67#1987C2212yes—
68#1983C2214yes—
69#1981C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2211yescharacteristic
74#1939C2212yes—
75#1927C2214yes—
76#1918C2214yes—
77#1918C2212yes—
78#1893C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1862C2212yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1807C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C23×C8C2314yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101#529—28no—
102#529—44no—
103C2×#126—22no—
104C2×#125C2312yes—
105C2×#124C2312yes—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—22no—
123C22×C42—21no—
124C22×C42C2312yes—
125#487—44no—
126#476—44no—
127#475—44no—
128#474—24no—
129C8×D4—44no—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8—44no—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
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—22no—
157#396—24no—
158#394—22no—
159#394—22no—
160#393—24no—
161#391—21no—
162#391—22no—
163#391—24no—
164#391C2312yes—
165#391C2312yes—
166#390—22no—
167#390—21no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389C2312yes—
175#389C2314yes—
176#389C2312yes—
177#388—24no—
178#388—22no—
179#387—22no—
180#387—24no—
181#386—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—22no—
193#382—22no—
194#380—22no—
195#380—22no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380C2314yes—
201#380C2314yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C22×C8—44no—
249C22×C8—44no—
250C22×C8—44no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—82no—
261C4⋊Q8—82no—
262#126—48no—
263#125—24no—
264#125—42no—
265#124—24no—
266#124—42no—
267#123—48no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—84no—
273#121—24no—
274#121—42no—
275#121—44no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—41no—
288C4×D4—48no—
289C4×D4#4114yes—
290#117—48no—
291#117—84no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
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—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—24no—
356#107—24no—
357#106—44no—
358#106—44no—
359C4⋊C8—22no—
360C4⋊C8—22no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363C4⋊C8—44no—
364C4⋊C8—42no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—24no—
370#102—42no—
371#102—44no—
372C22⋊C8—44no—
373C4×C8—24no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—21no—
393C24C22×D412yes—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—82no—
416C2×D4#13714yes—
417C2×D4C2×#4112yes—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—81no—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2511yescharacteristic
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—88no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—84no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4C2×#4112yes—
503C4⋊C42−1+412yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—24no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—21no—
526C42—41no—
527C42—44no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—84no—
533C42#13514yes—
534C42C2×#4112yes—
535C42H(5,𝔽2)12yes—
536C42H(5,𝔽2)12yes—
537C422−1+412yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C2×#13714yes—
552C23C23×D411yescharacteristic
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—84no—
558Q8—164no—
559D4—28no—
560D4—22no—
561D4—22no—
562D4—41no—
563D4—41no—
564D4—88no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—48no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4#53712yes—
617C2×C4#53712yes—
618C2×C4#54612yes—
619C2×C4#54612yes—
620C2×C4C2×#13512yes—
621C2×C4C2×#13512yes—
622C2×C4C2×#13512yes—
623C2×C4C2×#13712yes—
624C2×C4#57712yes—
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C2×C4#58511yescharacteristic
631C8—42no—
632C8—44no—
633C8—42no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22C2×#53711yescharacteristic
649C22C2×#54611yescharacteristic
650C22#348911yescharacteristic
651C22C22×#13511yescharacteristic
652C22C22×#13711yescharacteristic
653C22C2×#57711yescharacteristic
654C22#358511yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—41no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—81no—
668C4—82no—
669C4—84no—
670C4#324814yes—
671C4#330314yes—
672C4#350711yescharacteristic
673C4#351111yescharacteristic
674C4#354611yescharacteristic
675C4#355011yescharacteristic
676C2—44no—
677C2—82no—
678C2#3946811yescharacteristic
679C2C2×#324811yescharacteristic
680C2#6092411yescharacteristic
681C2C2×#330311yescharacteristic
682C2#6116211yescharacteristic
683C2#6209811yescharacteristic
684C2#6213111yescharacteristic
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. #36194C2semidirect
3. #33197C2semidirect
4. #32205C2semidirect
5. #31498C2semidirect
6. #29689C2semidirect
7. #28717C2semidirect
8. #28598C2semidirect
9. #28283C2semidirect
10. #27630C2semidirect
11. #26441C2semidirect
13. #25273C2semidirect
14. #24838C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21887C2semidirect
17. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349322
#331972
#322052
#314982
#296892
#287172
#285982
#282832
#276302
#264412
#253392
#252732
#248382
D4×C4⋊C82
#218872
C4×#4742