Finite Group Databasebeta

← #6468064

Subgroups — #6468064

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic, complemented
2#34276C211yesmaximal, characteristic
3#29153C214yesmaximal, complemented
4#29151C214yesmaximal, complemented
5#29148C212yesmaximal, complemented
6#29147C212yesmaximal, complemented
7#29142C212yesmaximal, complemented
8#29141C212yesmaximal, complemented
9#23455C211yesmaximal, characteristic, complemented
10#23452C211yesmaximal, characteristic
11#23451C211yesmaximal, characteristic
12C2×#2024C211yesmaximal, characteristic, complemented
13#21205C212yesmaximal
14#20763C212yesmaximal, complemented
15#20652C212yesmaximal
16#20651C212yesmaximal, complemented
17#20430C211yesmaximal, characteristic, complemented
18C2×#536C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×#424C2212yes—
24C2×#417C2212yes—
25C2×#419C2212yes—
26C2×#416C2212yes—
27C22×#103C2212yes—
28C2×#405C2212yes—
29#2868C2212yes—
30#2862C2212yes—
31#2660C2214yes—
32#2567—21no—
33#2567—21no—
34#2546—21no—
35#2546—21no—
36#2502—22no—
37#2434—22no—
38#2362—21no—
39#2362—21no—
40#2360—21no—
41#2360—21no—
42#2359—22no—
43#2359—22no—
44#2327C2212yes—
45#2326C2212yes—
46#2317C2211yescharacteristic
47#2317C2212yes—
48#2317C2212yes—
49C4×C4⋊Q8C2212yes—
50C4×#124C2212yes—
51#2297C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2026C2212yes—
55#2026C2214yes—
56#2025C2212yes—
57#2024C2212yes—
58#2024C2212yes—
59#2024C2214yes—
60#2024C2211yescharacteristic
61#2024C2211yescharacteristic
62#2024C2212yes—
63#2024C2212yes—
64#2024C2212yes—
65#2024C2212yes—
66#2023C2211yescharacteristic
67#2023C2211yescharacteristic
68#2023C2212yes—
69#2023C2212yes—
70#1960C2214yes—
71#1959C2214yes—
72#1958C2214yes—
73#1956C2214yes—
74#1836C2214yes—
75#1836C2214yes—
76#1835C2214yes—
77#1834C2214yes—
78#1805C2214yes—
79#1805C2214yes—
80#1805C2214yes—
81#1804C2214yes—
82#1803C2214yes—
83#1803C2214yes—
84#1803C2214yes—
85#1802C2214yes—
86#1768C2214yes—
87#1767C2212yes—
88C4×C4⋊C8C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#536—24no—
101#536Q818yescomplemented
102#533—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—44no—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2312yes—
113C2×#125C2312yes—
114C2×#124C2312yes—
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120#520—28no—
121#519—28no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—22no—
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117—22no—
133C2×#117C2312yes—
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#424—28no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145#419—44no—
146#417—44no—
147#416—44no—
148C2×#103—44no—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159#405—28no—
160C2×#97C2314yes—
161C2×C4×C8C2314yes—
162#398—24no—
163#398—24no—
164#395—24no—
165#395—24no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—21no—
171#391—21no—
172#391C2311yescharacteristic
173#391C2312yes—
174#391C2312yes—
175#391C2311yescharacteristic
176#390C2311yescharacteristic
177#390C2311yescharacteristic
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#386—21no—
195#386—21no—
196#385—22no—
197#385—24no—
198#385—24no—
199#385—22no—
200#385—22no—
201#384—22no—
202#384—22no—
203#384C2312yes—
204#384C2314yes—
205#384C2314yes—
206#382C2314yes—
207#382C2312yes—
208#382C2314yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221#376C2312yes—
222C2×#95—21no—
223C2×#95—21no—
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2312yes—
239#336C2312yes—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41C2×Q814yes—
244C22×Q8—21no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8—44no—
253C22×Q8—82no—
254C22×Q8—82no—
255C22×Q8C2×Q814yes—
256C22×Q8C2412yes—
257C22×Q8C2411yescharacteristic
258C22×D4—21no—
259C22×D4C2411yescharacteristic
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2414yes—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—42no—
281C4⋊Q8—42no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8—84no—
285C4⋊Q8—84no—
286C4⋊Q8C2×Q818yescomplemented
287#125—42no—
288#125—42no—
289#124—24no—
290#124C2×Q818yescomplemented
291#123—216no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296C4×Q8—28no—
297C4×Q8—28no—
298C4×Q8—28no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×D4—28no—
304C4×D4—48no—
305#117—24no—
306#117—24no—
307#117—24no—
308#117—44no—
309#117—48no—
310#117C2×Q814yes—
311#117#4114yes—
312#117#4118yes—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2412yes—
374C4⋊C8—28no—
375C4⋊C8—28no—
376#103—44no—
377#103—44no—
378#103—42no—
379#103—42no—
380#103—44no—
381#103—44no—
382#103—44no—
383#102—42no—
384#102—42no—
385#102—44no—
386#97—28no—
387C4×C8—28no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—21no—
407C24C22×D411yescharacteristic
408#41—216no—
409#41—48no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—216no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—48no—
428C2×Q8—42no—
429C2×Q8—44no—
430C2×Q8—42no—
431C2×Q8—82no—
432C2×Q8—82no—
433C2×Q8—816no—
434C2×Q8C22×Q814yes—
435C2×Q8C22×Q812yes—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4C22×Q812yes—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×Q812yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C4C2511yescharacteristic
486M16—48no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—48no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4C22⋊Q814yes—
526C4⋊C4C22⋊Q814yes—
527C4⋊C4#12414yes—
528C4⋊C4#12414yes—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—81no—
537C22⋊C4—81no—
538C22⋊C4C22⋊Q818yes—
539C22⋊C4#12418yes—
540C42—22no—
541C42—22no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—44no—
550C42—44no—
551C42—44no—
552C42—44no—
553C42C22×Q814yes—
554C42C2×#4114yes—
555C42C2×#4118yes—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23C2×C22⋊Q812yes—
568C23C2×#12412yes—
569C23#53512yes—
570C23C23×D411yescharacteristic
571Q8—28no—
572Q8—24no—
573Q8—48no—
574Q8—44no—
575Q8—42no—
576Q8—84no—
577Q8—84no—
578Q8—88no—
579Q8—88no—
580D4—24no—
581D4—48no—
582D4—42no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—88no—
630C2×C4—164no—
631C2×C4—162no—
632C2×C4C2×C22⋊Q814yes—
633C2×C4C2×C22⋊Q814yes—
634C2×C4C2×C22⋊Q812yes—
635C2×C4C2×#12414yes—
636C2×C4C2×#12414yes—
637C2×C4C2×#12412yes—
638C2×C4#53512yes—
639C2×C4#55418yes—
640C2×C4#55518yes—
641C2×C4#57712yes—
642C2×C4#57812yes—
643C2×C4C23×Q811yescharacteristic
644C2×C4C22×#4112yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×H(5,𝔽2)12yes—
647C8—42no—
648C8—42no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22#256314yes—
664C22C22×C22⋊Q811yescharacteristic
665C22C22×#12411yescharacteristic
666C22C2×#53511yescharacteristic
667C22C2×#57711yescharacteristic
668C22C2×#57811yescharacteristic
669C22#358511yescharacteristic
670C22#358611yescharacteristic
671C4—24no—
672C4—24no—
673C4—22no—
674C4—22no—
675C4—24no—
676C4—24no—
677C4—42no—
678C4—44no—
679C4—41no—
680C4—41no—
681C4—84no—
682C4—82no—
683C4—82no—
684C4—81no—
685C4—81no—
686C4—82no—
687C4#256314yes—
688C4C2×#55412yes—
689C4C2×#55512yes—
690C2—24no—
691C2—82no—
692C2C2×#256311yescharacteristic
693C2#6049711yescharacteristic
694C2#6049811yescharacteristic
695C2#6062311yescharacteristic
696C2#6062511yescharacteristic
697C2#6081211yescharacteristic
698C2#6081411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35281C2semidirect
3. #29153C2semidirect
4. #29151C2semidirect
5. #29148C2semidirect
6. #29147C2semidirect
7. #29142C2semidirect
8. #29141C2semidirect
9. #23455C2semidirect
12. C2×#2024C2semidirect
14. #20763C2semidirect
16. #20651C2semidirect
17. #20430C2semidirect
101. #536Q8semidirect
286. C4⋊Q8C2×Q8semidirect
290. #124C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352812
#342762
#291532
#291512
#291482
#291472
#291422
#291412
#234552
#234522
#234512
C2×#20242
#212052
#207632
#206522
#206512
#204302