Finite Group Databasebeta

← #6800018

Subgroups — #6800018

Order 512.

1855 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic
3#32953C211yesmaximal, characteristic
4#32089C214yesmaximal, complemented
5#30885C214yesmaximal, complemented
6#29957C212yesmaximal, complemented
7#28403C212yesmaximal, complemented
8#27768C212yesmaximal, complemented
9#25859C212yesmaximal
10#25856C211yesmaximal, characteristic, complemented
11#25275C212yesmaximal, complemented
12#24729C212yesmaximal, complemented
13#24217C212yesmaximal
14#24212C211yesmaximal, characteristic, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22044C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×D4C2211yescharacteristic
21C2×#488C2214yes—
22C2×#487C2212yescomplemented
23C2×#474C2212yes—
24C2×#464C2214yes—
25C2×#462C2212yescomplemented
26C2×#451C2212yes—
27C2×#450C2211yescharacteristic
28C2×C4×SD16C2212yes—
29#2857C2211yescharacteristic
30#2851C2212yes—
31#2590—22no—
32#2568C2211yescharacteristic, complemented
33#2552C2211yescharacteristic, complemented
34#2547—22no—
35#2547—21no—
36#2547—21no—
37#2547C2214yes—
38#2547C2212yes—
39#2547C2212yes—
40#2511—21no—
41#2508C2212yescomplemented
42#2505—22no—
43#2502—21no—
44#2453C2212yescomplemented
45C2×#397—22no—
46#2365C2211yescharacteristic, complemented
47#2360—21no—
48#2355—21no—
49Q8×C4⋊C4C2211yescharacteristic, complemented
50C4×C22⋊Q8C2212yes—
51C4×C22⋊Q8C2211yescharacteristic
52C2×#391C2212yes—
53C2×#380—21no—
54C2×#380—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2097C2214yescomplemented
57#2068C2214yescomplemented
58#2066C2214yescomplemented
59#2043C2214yescomplemented
60#2043C2212yescomplemented
61#2042C2214yes—
62#2042C2214yes—
63#2042C2214yes—
64#2026C2214yescomplemented
65#2016C2212yescomplemented
66#2015C2214yes—
67#1983C2214yes—
68#1981C2212yescomplemented
69#1969C2214yescomplemented
70#1958C2214yes—
71#1956C2212yescomplemented
72#1939C2212yescomplemented
73#1933C2214yescomplemented
74#1928C2214yescomplemented
75#1916C2214yescomplemented
76#1878C2214yescomplemented
77#1877C2212yes—
78#1872C2211yescharacteristic
79#1864C2212yes—
80#1862C2214yes—
81#1853C2211yescharacteristic
82#1852C2212yes—
83#1802C2212yes—
84#1797C2212yes—
85#1797C2211yescharacteristic
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89#1786C2211yescharacteristic
90#1786C2212yes—
91C8×C4⋊C4C2212yescomplemented
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8C2312yes—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×Q16C2314yes—
99C22×SD16C2312yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#125C2312yes—
107C2×#124—22no—
108C2×#124—22no—
109C2×#124—48no—
110C2×#124C2314yes—
111C2×#124C2314yes—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—44no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—22no—
129C22×C22⋊C4—22no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#488—48no—
133#487—44no—
134#474—44no—
135#464—48no—
136#462—44no—
137#451—28no—
138#450—24no—
139C4×SD16—44no—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161C2×C4×C8C2314yes—
162#400—22no—
163#400—22no—
164#400—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#397—24no—
169#397—24no—
170#397—24no—
171#397—24no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—22no—
177#391—22no—
178#391C2312yes—
179#391C2314yes—
180#391C2314yes—
181#391C2311yescharacteristic
182#389—22no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2311yescharacteristic
190#389C2312yes—
191#387—22no—
192#387—21no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#385—21no—
202#384—21no—
203#384—21no—
204#384C2311yescharacteristic
205#382—22no—
206#382C2312yes—
207#380—21no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—22no—
212#380—24no—
213#380—24no—
214#380C2311yescharacteristic
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D412yes—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—22no—
229C2×#95—22no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8C2414yes—
247C22×Q8C2411yescharacteristic
248C22×D4—24no—
249C22×D4—21no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—42no—
253C22×D4C2412yes—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×Q16—28no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—42no—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281#125—44no—
282#124—28no—
283#124—28no—
284#124—44no—
285#124—44no—
286#124—48no—
287#124—48no—
288C22⋊Q8—22no—
289C22⋊Q8—44no—
290C22⋊Q8#4114yes—
291C4×Q8—24no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—44no—
299C4×D4—44no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—48no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361#107—44no—
362#106—42no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—28no—
366#103—42no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#102—28no—
374#102—24no—
375#102—24no—
376#102—42no—
377#102—44no—
378C22⋊C8—22no—
379C22⋊C8—24no—
380C22⋊C8—22no—
381C4×C8—24no—
382C4×C8—44no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—24no—
407C24—22no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24C22×D411yescharacteristic
415C2×Q8—21no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—88no—
431C2×Q8—88no—
432C2×Q8C2×#4112yes—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—48no—
440C2×D4—48no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4H(5,𝔽2)14yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—816no—
478C22×C4#13514yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491Q16—88no—
492SD16—48no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4C2×#4112yes—
534C22⋊C4—24no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—44no—
543C42—22no—
544C42—22no—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—42no—
549C42—42no—
550C42—44no—
551C42—42no—
552C42—42no—
553C42—82no—
554C42H(5,𝔽2)14yes—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—24no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23#54812yes—
574C23C2×#13512yes—
575C23#57812yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—42no—
579Q8—44no—
580Q8—44no—
581Q8—44no—
582Q8—44no—
583Q8—88no—
584Q8—84no—
585D4—24no—
586D4—24no—
587D4—48no—
588D4—42no—
589D4—42no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—88no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—88no—
644C2×C4—88no—
645C2×C4#53512yes—
646C2×C4#53512yes—
647C2×C4#54812yes—
648C2×C4C2×#13512yes—
649C2×C4C2×#13512yes—
650C2×C4#57512yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C2×H(5,𝔽2)12yes—
653C2×C4C2×H(5,𝔽2)12yes—
654C2×C4C2×2−1+411yescharacteristic
655C8—42no—
656C8—44no—
657C8—42no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—816no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22#336414yes—
673C22C2×#53511yescharacteristic
674C22C2×#54811yescharacteristic
675C22#348411yescharacteristic
676C22C22×#13511yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57811yescharacteristic
679C22#358611yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—41no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—81no—
694C4—81no—
695C4—84no—
696C4#321114yes—
697C4#351912yes—
698C4#353612yes—
699C2—24no—
700C2—84no—
701C2#3912011yescharacteristic
702C2C2×#321111yescharacteristic
703C2#6081411yescharacteristic
704C2C2×#336411yescharacteristic
705C2#6132311yescharacteristic
706C2#6156511yescharacteristic
707C2#6159511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36552C2semidirect
4. #32089C2semidirect
5. #30885C2semidirect
6. #29957C2semidirect
7. #28403C2semidirect
8. #27768C2semidirect
10. #25856C2semidirect
11. #25275C2semidirect
12. #24729C2semidirect
14. #24212C2semidirect
15. #22625C2semidirect
16. #22044C2semidirect
22. C2×#487C22semidirect
25. C2×#462C22semidirect
32. #2568C22semidirect
33. #2552C22semidirect
41. #2508C22semidirect
44. #2453C22semidirect
46. #2365C22semidirect
49. Q8×C4⋊C4C22semidirect
56. #2097C22semidirect
57. #2068C22semidirect
58. #2066C22semidirect
59. #2043C22semidirect
60. #2043C22semidirect
64. #2026C22semidirect
65. #2016C22semidirect
68. #1981C22semidirect
69. #1969C22semidirect
71. #1956C22semidirect
72. #1939C22semidirect
73. #1933C22semidirect
74. #1928C22semidirect
75. #1916C22semidirect
76. #1878C22semidirect
91. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×#25472
#329532
#320892
#308852
#299572
#284032
#277682
#258592
#258562
#252752
#247292
#242172
#242122
#226252
#220442
C4×#4502