Finite Group Databasebeta

← #6563117

Subgroups — #6563117

Order 512.

2029 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36481C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#30128C212yesmaximal, complemented
4#29923C211yesmaximal, characteristic, complemented
5#29498C212yesmaximal, complemented
6#28157C212yesmaximal, complemented
7#26526C212yesmaximal, complemented
8#25717C211yesmaximal, characteristic
9#25368C212yesmaximal, complemented
10#25176C214yesmaximal, complemented
11#24921C214yesmaximal, complemented
12#24211C211yesmaximal, characteristic
13#23803C212yesmaximal, complemented
14#23757C211yesmaximal, characteristic, complemented
15#23190C212yesmaximal, complemented
16#23003C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518—21no—
20C2×#492C2212yes—
21C2×#460C2212yes—
22C2×#461C2214yescomplemented
23C2×#450C2212yes—
24C2×#450C2214yes—
25C2×#440C2214yescomplemented
26C2×#440C2214yescomplemented
27#2863C2212yescomplemented
28#2857C2211yescharacteristic
29#2606—22no—
30#2552C2211yescharacteristic, complemented
31#2546C2211yescharacteristic, complemented
32#2522—22no—
33#2507—21no—
34#2441—21no—
35#2429C2212yes—
36#2423C2212yes—
37#2370—21no—
38#2366—21no—
39#2366—21no—
40#2366C2212yes—
41#2366C2212yes—
42#2365C2211yescharacteristic, complemented
43Q8×C4⋊C4C2211yescharacteristic, complemented
44#2347—21no—
45D4×C22⋊C4—22no—
46D4×C22⋊C4C2214yes—
47#2328—22no—
48#2309—21no—
49C4×C4⋊Q8C2212yescomplemented
50C4×#127C2212yes—
51C4×#127C2212yes—
52C4×C22⋊Q8C2211yescharacteristic
53#2285C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2100C2211yescharacteristic, complemented
56#2095C2212yescomplemented
57#2095C2211yescharacteristic, complemented
58#2093C2214yescomplemented
59#2083C2211yescharacteristic, complemented
60#2076C2212yes—
61#2072C2212yescomplemented
62#2072C2211yescharacteristic, complemented
63#2067C2214yescomplemented
64#2055C2214yescomplemented
65#2041C2212yes—
66#2036C2212yes—
67#2036C2214yes—
68#2014C2212yes—
69#2014C2212yes—
70#2014C2212yes—
71#2013C2214yescomplemented
72#1982C2212yes—
73#1981C2212yescomplemented
74#1973C2214yescomplemented
75#1957C2212yes—
76#1947C2212yescomplemented
77#1939C2212yescomplemented
78#1936C2214yescomplemented
79#1924C2214yes—
80#1880C2214yescomplemented
81#1880C2214yescomplemented
82#1877C2214yescomplemented
83#1877C2214yescomplemented
84#1862C2214yescomplemented
85#1862C2214yescomplemented
86#1831C2212yescomplemented
87#1802C2214yescomplemented
88#1797C2211yescharacteristic
89#1786C2212yes—
90C4×#107C2212yescomplemented
91#1770C2212yescomplemented
92C4×#103C2212yescomplemented
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yescomplemented
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C22×SD16C2314yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127—28no—
104C2×#127—21no—
105C2×#127—21no—
106C2×#127C2312yes—
107C2×#127C2312yes—
108C2×#125—21no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2312yes—
111#518—48no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4C2314yes—
122C2×C4×D4C2314yes—
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C22×C22⋊C4—24no—
127C22×C42C2311yescharacteristic
128#492—28no—
129#461—48no—
130#460—28no—
131#450—48no—
132#450—44no—
133#440—48no—
134#440—48no—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#399—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#396—22no—
159#391C2311yescharacteristic
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#390—22no—
164#390—22no—
165#390—22no—
166#390—22no—
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—22no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389C2312yes—
182#389C2311yescharacteristic
183#389C2312yes—
184#388—22no—
185#388—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#386—24no—
193#386—24no—
194#386—24no—
195#386—24no—
196#385—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2311yescharacteristic
202#383—22no—
203#382—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380C2311yescharacteristic
208#380C2312yes—
209#379—24no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225#376D412yes—
226C43C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C25—22no—
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4—28no—
241C22×D4—28no—
242C22×D4—28no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4—44no—
250C22×D4—42no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—82no—
254C22×D4—82no—
255C22×D4C2414yes—
256C23×C4—24no—
257C23×C4—24no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×SD16—28no—
262C2×SD16—28no—
263C2×SD16—28no—
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—44no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—42no—
276#127—28no—
277#127—28no—
278#127—28no—
279#127—24no—
280#127#4118yes—
281#125—44no—
282C22⋊Q8—22no—
283C22⋊Q8—41no—
284C22⋊Q8—44no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—48no—
289C4×D4—48no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—48no—
295C4×D4—48no—
296C4×D4—84no—
297C4×D4—84no—
298#117—24no—
299#117—24no—
300#117—44no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—416no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—24no—
361#107—42no—
362#107—44no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—24no—
366#103—24no—
367#103—44no—
368#103—44no—
369#102—28no—
370#102—28no—
371#102—44no—
372#102—44no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—42no—
376#97—28no—
377C4×C8—24no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—28no—
398C24—24no—
399C24—44no—
400C24—44no—
401C24—82no—
402C24—82no—
403C24—82no—
404C24—82no—
405C24C22×D411yescharacteristic
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8H(5,𝔽2)12yes—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—432no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—88no—
436C2×D4—88no—
437C2×D4—816no—
438C2×D4C2×#4118yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—41no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4#13714yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2511yescharacteristic
473SD16—416no—
474SD16—416no—
475SD16—88no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8H(5,𝔽2)14yes—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4AGL(1,ℤ/8)14yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—81no—
529C22⋊C4—82no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—42no—
540C42—42no—
541C42—44no—
542C42—42no—
543C42—82no—
544C42—82no—
545C42C2×#4114yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—28no—
553C23—416no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—88no—
561C23—88no—
562C23—164no—
563C23—164no—
564C23—164no—
565C23—164no—
566C23C2×#13712yes—
567C23C23×D411yescharacteristic
568Q8—24no—
569Q8—42no—
570Q8—44no—
571Q8—44no—
572D4—28no—
573D4—28no—
574D4—416no—
575D4—416no—
576D4—416no—
577D4—44no—
578D4—44no—
579D4—88no—
580D4—88no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—168no—
630C2×C4—168no—
631C2×C4#52512yes—
632C2×C4#52512yes—
633C2×C4C2×AGL(1,ℤ/8)14yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×#13712yes—
636C2×C4C2×#13712yes—
637C2×C4#57712yes—
638C2×C4C22×#4112yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4#58511yescharacteristic
642C8—24no—
643C8—44no—
644C8—42no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—48no—
653C22—48no—
654C22—48no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—1616no—
660C22C2×#52511yescharacteristic
661C22#348811yescharacteristic
662C22#349811yescharacteristic
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C22×#13711yescharacteristic
665C22C2×#57711yescharacteristic
666C22#358511yescharacteristic
667C4—24no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—81no—
677C4—82no—
678C4—82no—
679C4—81no—
680C4—81no—
681C4—82no—
682C4—82no—
683C4#311014yes—
684C4#351312yes—
685C4#351912yes—
686C2—42no—
687C2—88no—
688C2#3599511yescharacteristic
689C2C2×#311011yescharacteristic
690C2#6057911yescharacteristic
691C2#6196511yescharacteristic
692C2#6198311yescharacteristic
693C2#6242211yescharacteristic
694C2#6245311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #36481C2semidirect
3. #30128C2semidirect
4. #29923C2semidirect
5. #29498C2semidirect
6. #28157C2semidirect
7. #26526C2semidirect
9. #25368C2semidirect
10. #25176C2semidirect
11. #24921C2semidirect
13. #23803C2semidirect
14. #23757C2semidirect
15. #23190C2semidirect
16. #23003C2semidirect
17. #22636C2semidirect
22. C2×#461C22semidirect
25. C2×#440C22semidirect
26. C2×#440C22semidirect
27. #2863C22semidirect
30. #2552C22semidirect
31. #2546C22semidirect
42. #2365C22semidirect
43. Q8×C4⋊C4C22semidirect
49. C4×C4⋊Q8C22semidirect
55. #2100C22semidirect
56. #2095C22semidirect
57. #2095C22semidirect
58. #2093C22semidirect
59. #2083C22semidirect
61. #2072C22semidirect
62. #2072C22semidirect
63. #2067C22semidirect
64. #2055C22semidirect
71. #2013C22semidirect
73. #1981C22semidirect
74. #1973C22semidirect
76. #1947C22semidirect
77. #1939C22semidirect
78. #1936C22semidirect
80. #1880C22semidirect
81. #1880C22semidirect
82. #1877C22semidirect
83. #1877C22semidirect
84. #1862C22semidirect
85. #1862C22semidirect
86. #1831C22semidirect
87. #1802C22semidirect
90. C4×#107C22semidirect
91. #1770C22semidirect
92. C4×#103C22semidirect
94. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#364812
#342102
#301282
#299232
#294982
#281572
#265262
#257172
#253682
#251762
#249212
#242112
#238032
#237572
#231902
#230032
#226362