Finite Group Databasebeta

← #6800083

Subgroups — #6800083

Order 512.

1863 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic
3#32956C211yesmaximal, characteristic
4#32105C214yesmaximal, complemented
5#30901C214yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#28419C212yesmaximal, complemented
8#27776C212yesmaximal, complemented
9#25864C212yesmaximal
10#25856C211yesmaximal, characteristic, complemented
11#25273C212yesmaximal, complemented
12#24729C212yesmaximal, complemented
13#24217C212yesmaximal
14#24212C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22050C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×D4C2211yescharacteristic
21C2×#489C2214yes—
22C2×#487C2212yescomplemented
23C2×#476C2212yes—
24C2×#464C2214yes—
25C2×#462C2212yescomplemented
26C2×#451C2212yes—
27C2×#450C2211yescharacteristic
28C2×C4×SD16C2212yes—
29#2857C2211yescharacteristic
30#2857C2212yes—
31#2586—22no—
32#2567C2211yescharacteristic, complemented
33#2551C2211yescharacteristic, complemented
34#2549—22no—
35#2547—22no—
36#2547—21no—
37#2547—21no—
38#2547C2212yes—
39#2547C2214yes—
40#2547C2212yes—
41#2507—21no—
42#2507C2212yescomplemented
43#2451C2212yescomplemented
44#2440—21no—
45C2×#397—22no—
46#2369—21no—
47#2369C2211yescharacteristic, complemented
48Q8×C4⋊C4C2211yescharacteristic, complemented
49#2345—21no—
50C4×C22⋊Q8C2212yes—
51C4×C22⋊Q8C2211yescharacteristic
52C2×#391C2212yes—
53C2×#380—21no—
54C2×#380—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2096C2214yescomplemented
57#2066C2214yescomplemented
58#2066C2214yescomplemented
59#2043C2214yescomplemented
60#2043C2212yescomplemented
61#2042C2214yes—
62#2042C2214yes—
63#2042C2214yes—
64#2024C2214yescomplemented
65#2016C2212yescomplemented
66#2015C2214yes—
67#1983C2214yes—
68#1981C2212yescomplemented
69#1971C2214yescomplemented
70#1959C2214yes—
71#1956C2212yescomplemented
72#1939C2212yescomplemented
73#1935C2214yescomplemented
74#1927C2214yescomplemented
75#1917C2214yescomplemented
76#1878C2214yescomplemented
77#1877C2212yes—
78#1872C2211yescharacteristic
79#1864C2212yes—
80#1864C2214yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1802C2212yes—
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89#1786C2211yescharacteristic
90#1786C2212yes—
91#1770C2212yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×Q16C2314yes—
99C22×SD16C2312yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#124—22no—
109C2×#124—22no—
110C2×#124—48no—
111C2×#124C2314yes—
112C2×#124C2314yes—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—44no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#489—48no—
134#487—44no—
135#476—44no—
136#464—48no—
137#462—44no—
138#451—28no—
139#450—24no—
140C4×SD16—44no—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2311yescharacteristic
161C2×#97C2314yes—
162C2×C4×C8C2312yes—
163#398—22no—
164#398—22no—
165#398—22no—
166#397—24no—
167#397—24no—
168#397—24no—
169#397—24no—
170#395—22no—
171#395—22no—
172#395—22no—
173#391—21no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391C2312yes—
180#391C2314yes—
181#391C2314yes—
182#391C2311yescharacteristic
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2311yescharacteristic
195#389C2312yes—
196#387—22no—
197#387—22no—
198#387—21no—
199#387—22no—
200#385—22no—
201#385—21no—
202#384—21no—
203#384C2312yes—
204#384C2311yescharacteristic
205#380—21no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—24no—
211#380—22no—
212#380—24no—
213#380C2312yes—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
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—22no—
241C22×Q8—22no—
242C22×Q8—21no—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2414yes—
248C22×Q8C2411yescharacteristic
249C22×D4—24no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—42no—
254C22×D4C2412yes—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2411yescharacteristic
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—42no—
273C22×C8C2412yes—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—24no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284#124—28no—
285#124—28no—
286#124—44no—
287#124—44no—
288#124—48no—
289#124—48no—
290C22⋊Q8—22no—
291C22⋊Q8—82no—
292C22⋊Q8—84no—
293C22⋊Q8#4114yes—
294C4×Q8—24no—
295C4×Q8—44no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—42no—
299C4×D4—42no—
300C4×D4—44no—
301C4×D4—44no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—48no—
349C2×C22⋊C4—48no—
350C2×C42—24no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364#106—42no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—44no—
368#103—28no—
369#103—42no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376#102—28no—
377#102—24no—
378#102—24no—
379#102—42no—
380#102—44no—
381C22⋊C8—22no—
382C22⋊C8—24no—
383C22⋊C8—22no—
384#97—44no—
385C4×C8—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—22no—
399#95—21no—
400#95—22no—
401#95—21no—
402#95—22no—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24—24no—
408C24—22no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24C22×D411yescharacteristic
416C2×Q8—21no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8—44no—
430C2×Q8—81no—
431C2×Q8—82no—
432C2×Q8—82no—
433C2×Q8—81no—
434C2×Q8—88no—
435C2×Q8—88no—
436C2×Q8C2×#4112yes—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—48no—
444C2×D4—48no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4H(5,𝔽2)14yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—816no—
482C22×C4#13714yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C2×#4112yes—
493C22×C42−1+412yes—
494C22×C4C2511yescharacteristic
495Q16—88no—
496SD16—48no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—81no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4C2×#4112yes—
539C22⋊C4—24no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C42—22no—
549C42—22no—
550C42—24no—
551C42—24no—
552C42—22no—
553C42—42no—
554C42—42no—
555C42—42no—
556C42—42no—
557C42—82no—
558C42—82no—
559C42H(5,𝔽2)14yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—24no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23#54812yes—
579C23C2×#13712yes—
580C23#57712yes—
581C23C23×D411yescharacteristic
582Q8—24no—
583Q8—42no—
584Q8—44no—
585Q8—44no—
586Q8—44no—
587Q8—44no—
588Q8—88no—
589Q8—162no—
590Q8—164no—
591D4—24no—
592D4—24no—
593D4—48no—
594D4—42no—
595D4—42no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—88no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—88no—
650C2×C4—88no—
651C2×C4#53512yes—
652C2×C4#53512yes—
653C2×C4#54812yes—
654C2×C4C2×#13712yes—
655C2×C4C2×#13712yes—
656C2×C4#57512yes—
657C2×C4C22×#4111yescharacteristic
658C2×C4C2×H(5,𝔽2)12yes—
659C2×C4C2×H(5,𝔽2)12yes—
660C2×C4C2×2−1+411yescharacteristic
661C8—42no—
662C8—44no—
663C8—42no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—816no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22#336914yes—
679C22C2×#53511yescharacteristic
680C22C2×#54811yescharacteristic
681C22#348411yescharacteristic
682C22C22×#13711yescharacteristic
683C22C2×#57511yescharacteristic
684C22C2×#57711yescharacteristic
685C22#358611yescharacteristic
686C4—22no—
687C4—22no—
688C4—22no—
689C4—42no—
690C4—41no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—81no—
698C4—82no—
699C4—82no—
700C4—81no—
701C4—84no—
702C4#321514yes—
703C4#351912yes—
704C4#353612yes—
705C2—24no—
706C2—84no—
707C2#3912011yescharacteristic
708C2C2×#321511yescharacteristic
709C2#6081311yescharacteristic
710C2C2×#336911yescharacteristic
711C2#6132311yescharacteristic
712C2#6158611yescharacteristic
713C2#6160411yescharacteristic
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. #36564C2semidirect
4. #32105C2semidirect
5. #30901C2semidirect
6. #29951C2semidirect
7. #28419C2semidirect
8. #27776C2semidirect
10. #25856C2semidirect
11. #25273C2semidirect
12. #24729C2semidirect
14. #24212C2semidirect
15. #22629C2semidirect
16. #22050C2semidirect
22. C2×#487C22semidirect
25. C2×#462C22semidirect
32. #2567C22semidirect
33. #2551C22semidirect
42. #2507C22semidirect
43. #2451C22semidirect
47. #2369C22semidirect
48. Q8×C4⋊C4C22semidirect
56. #2096C22semidirect
57. #2066C22semidirect
58. #2066C22semidirect
59. #2043C22semidirect
60. #2043C22semidirect
64. #2024C22semidirect
65. #2016C22semidirect
68. #1981C22semidirect
69. #1971C22semidirect
71. #1956C22semidirect
72. #1939C22semidirect
73. #1935C22semidirect
74. #1927C22semidirect
75. #1917C22semidirect
76. #1878C22semidirect
91. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#25472
#329562
#321052
#309012
#299512
#284192
#277762
#258642
#258562
#252732
#247292
#242172
#242122
#226292
#220502
C4×#4502