Finite Group Databasebeta

← #6852334

Subgroups — #6852334

Order 512.

1705 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3#32956C211yesmaximal, characteristic
4#32373C214yesmaximal, complemented
5#31549C214yesmaximal, complemented
6#29950C212yesmaximal
7#28539C212yesmaximal, complemented
8#27956C212yesmaximal, complemented
9#27821C212yesmaximal, complemented
10#27686C212yesmaximal, complemented
11#25813C211yesmaximal, characteristic, complemented
12#25450C212yesmaximal, complemented
13#24814C212yesmaximal, complemented
14#24302C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22049C212yesmaximal
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#500C2212yescomplemented
21C2×#477C2212yescomplemented
22C2×#476C2212yes—
23C2×#450C2211yescharacteristic
24C2×C4×SD16C2212yes—
25#2857C2212yes—
26#2857C2211yescharacteristic
27#2586—22no—
28#2567C2211yescharacteristic, complemented
29#2563—22no—
30#2563—21no—
31#2563—21no—
32#2563C2214yes—
33#2563C2212yes—
34#2563C2212yes—
35#2551C2211yescharacteristic, complemented
36#2549—22no—
37#2507—21no—
38#2507C2212yescomplemented
39#2451C2212yescomplemented
40#2440—21no—
41C2×#393—22no—
42#2369—21no—
43#2369C2211yescharacteristic, complemented
44Q8×C4⋊C4C2211yescharacteristic, complemented
45#2345—21no—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#390C2212yes—
49C2×#382C2212yes—
50C2×#380—21no—
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yescomplemented
54#2097C2214yescomplemented
55#2093C2214yes—
56#2085C2214yescomplemented
57#2085C2214yescomplemented
58#2067C2214yes—
59#2067C2214yes—
60#2057C2214yescomplemented
61#2057C2212yescomplemented
62#2024C2212yescomplemented
63#2023C2214yes—
64#1993C2214yescomplemented
65#1990C2212yescomplemented
66#1987C2212yescomplemented
67#1973C2214yes—
68#1949C2214yescomplemented
69#1939C2212yescomplemented
70#1936C2214yes—
71#1925C2214yes—
72#1917C2214yescomplemented
73#1880C2214yes—
74#1877C2212yes—
75#1872C2212yes—
76#1872C2211yescharacteristic
77#1864C2214yes—
78#1864C2212yes—
79#1853C2212yes—
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1847C2214yescomplemented
85#1805C2214yescomplemented
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1786C2212yes—
90#1786C2211yescharacteristic
91#1770C2212yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—44no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4—22no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#500—44no—
128#477—44no—
129#476—44no—
130#450Q818yescomplemented
131C4×SD16—44no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#398—22no—
152#398—22no—
153#398—22no—
154#395—22no—
155#395—22no—
156#395—22no—
157#393—24no—
158#393—24no—
159#393—24no—
160#393—24no—
161#391—21no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391C2312yes—
166#391C2311yescharacteristic
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2311yescharacteristic
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#387—22no—
185#387—22no—
186#387—21no—
187#387—22no—
188#385—22no—
189#385—21no—
190#384—21no—
191#384C2312yes—
192#384C2311yescharacteristic
193#382—22no—
194#382—22no—
195#382C2314yes—
196#382C2314yes—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—24no—
203#380—22no—
204#380—24no—
205#380C2312yes—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C25—21no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×Q8C2411yescharacteristic
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4C2×Q814yes—
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×SD16—42no—
263C2×SD16—42no—
264C2×SD16C2×Q818yescomplemented
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C22⋊Q8—22no—
277C22⋊Q8—82no—
278C22⋊Q8—84no—
279C22⋊Q8C2×Q814yes—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—44no—
287C4×D4—44no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
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—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—84no—
335C2×C22⋊C4—84no—
336C2×C42—24no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350#107—44no—
351#107—44no—
352#107—44no—
353#106—42no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#103—42no—
358#103—44no—
359#102—44no—
360#102—42no—
361#102—42no—
362#102—44no—
363#102C2×Q818yescomplemented
364C22⋊C8—22no—
365C22⋊C8—24no—
366C22⋊C8C2×Q814yes—
367#97—44no—
368C4×C8—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—24no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24C22×D411yescharacteristic
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—81no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—81no—
414C2×Q8—816no—
415C2×Q8C22×Q812yes—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—48no—
426C2×D4C22×Q814yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—816no—
461C22×C4#13714yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×Q812yes—
472C22×C42−1+412yes—
473C22×C4C2511yescharacteristic
474SD16—48no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8C22×Q814yes—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—82no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4C22×Q812yes—
518C22⋊C4—24no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—24no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—82no—
537C42—82no—
538C42H(5,𝔽2)14yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—24no—
544C23—48no—
545C23—48no—
546C23—42no—
547C23—48no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23D4×Q812yes—
558C23C2×#13712yes—
559C23#57612yes—
560C23C23×D411yescharacteristic
561Q8—24no—
562Q8—42no—
563Q8—88no—
564Q8—88no—
565Q8—162no—
566Q8—164no—
567D4—24no—
568D4—24no—
569D4—48no—
570D4—42no—
571D4—42no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—88no—
626C2×C4—88no—
627C2×C4#53512yes—
628C2×C4#53512yes—
629C2×C4D4×Q812yes—
630C2×C4C2×#13712yes—
631C2×C4C2×#13712yes—
632C2×C4#57512yes—
633C2×C4C23×Q811yescharacteristic
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4C2×2−1+411yescharacteristic
636C2×C4#58512yes—
637C8—24no—
638C8—42no—
639C8—44no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—816no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22#338514yes—
655C22C2×#53511yescharacteristic
656C22C2×D4×Q811yescharacteristic
657C22#348511yescharacteristic
658C22C22×#13711yescharacteristic
659C22C2×#57511yescharacteristic
660C22C2×#57611yescharacteristic
661C22#358511yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—84no—
677C4—82no—
678C4#321514yes—
679C4#350112yes—
680C4#354112yes—
681C2—24no—
682C2—84no—
683C2#3959411yescharacteristic
684C2C2×#321511yescharacteristic
685C2#6081111yescharacteristic
686C2C2×#338511yescharacteristic
687C2#6136711yescharacteristic
688C2#6172111yescharacteristic
689C2#6174011yescharacteristic
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. #32373C2semidirect
5. #31549C2semidirect
7. #28539C2semidirect
8. #27956C2semidirect
9. #27821C2semidirect
10. #27686C2semidirect
11. #25813C2semidirect
12. #25450C2semidirect
13. #24814C2semidirect
14. #24302C2semidirect
15. #22629C2semidirect
20. C2×#500C22semidirect
21. C2×#477C22semidirect
28. #2567C22semidirect
35. #2551C22semidirect
38. #2507C22semidirect
39. #2451C22semidirect
43. #2369C22semidirect
44. Q8×C4⋊C4C22semidirect
53. #2101C22semidirect
54. #2097C22semidirect
56. #2085C22semidirect
57. #2085C22semidirect
60. #2057C22semidirect
61. #2057C22semidirect
62. #2024C22semidirect
64. #1993C22semidirect
65. #1990C22semidirect
66. #1987C22semidirect
68. #1949C22semidirect
69. #1939C22semidirect
72. #1917C22semidirect
84. #1847C22semidirect
85. #1805C22semidirect
91. #1770C22semidirect
130. #450Q8semidirect
264. C2×SD16C2×Q8semidirect
363. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#25632
#329562
#323732
#315492
#299502
#285392
#279562
#278212
#276862
#258132
#254502
#248142
#243022
#226292
#220492
C4×#4502