Finite Group Databasebeta

← #6562260

Subgroups — #6562260

Order 512.

1919 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36287C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#30075C212yesmaximal, complemented
4#29939C211yesmaximal, characteristic, complemented
5#29367C212yesmaximal, complemented
6#29130C212yesmaximal, complemented
7#27541C214yesmaximal, complemented
8#26526C212yesmaximal, complemented
9#25855C211yesmaximal, characteristic
10#25322C212yesmaximal, complemented
11#24921C214yesmaximal, complemented
12#24211C211yesmaximal, characteristic
13#23757C211yesmaximal, characteristic, complemented
14#23684C212yesmaximal, complemented
15#23223C212yesmaximal, complemented
16#23003C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518—21no—
20C2×#486C2212yes—
21C2×#460C2212yes—
22C2×#450C2212yes—
23C2×#440C2214yescomplemented
24#2859C2212yescomplemented
25#2857C2211yescharacteristic
26#2612—22no—
27#2567C2211yescharacteristic, complemented
28#2556—22no—
29#2552C2211yescharacteristic, complemented
30#2506—21no—
31#2497C2212yes—
32#2441—21no—
33#2423C2212yes—
34#2370—21no—
35#2369C2211yescharacteristic, complemented
36#2366—21no—
37#2366—21no—
38#2366C2212yes—
39#2366C2212yes—
40Q8×C4⋊C4C2211yescharacteristic, complemented
41#2346—21no—
42D4×C22⋊C4—22no—
43D4×C22⋊C4C2214yes—
44#2328—22no—
45#2309—21no—
46C4×#127C2212yes—
47C4×#127C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49#2295C2212yescomplemented
50#2285C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2212yescomplemented
53#2100C2211yescharacteristic, complemented
54#2095C2211yescharacteristic, complemented
55#2093C2214yescomplemented
56#2083C2211yescharacteristic, complemented
57#2083C2212yescomplemented
58#2076C2214yes—
59#2076C2212yes—
60#2072C2211yescharacteristic, complemented
61#2071C2214yescomplemented
62#2067C2214yescomplemented
63#2055C2214yescomplemented
64#2041C2212yes—
65#2041C2212yes—
66#2041C2212yes—
67#2036C2212yes—
68#2014C2212yes—
69#1987C2212yescomplemented
70#1982C2212yes—
71#1957C2212yes—
72#1947C2212yescomplemented
73#1939C2212yescomplemented
74#1936C2214yescomplemented
75#1936C2214yescomplemented
76#1925C2214yescomplemented
77#1924C2214yes—
78#1880C2214yescomplemented
79#1877C2214yescomplemented
80#1862C2214yescomplemented
81#1862C2214yescomplemented
82#1862C2214yescomplemented
83#1839C2212yescomplemented
84#1802C2214yescomplemented
85#1802C2214yescomplemented
86#1797C2211yescharacteristic
87#1786C2214yes—
88#1786C2212yes—
89C4×#107C2212yescomplemented
90#1770C2212yescomplemented
91#1767C2212yescomplemented
92#1762C2214yescomplemented
93C4×#103C2212yescomplemented
94C4×C22⋊C8C2211yescharacteristic
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127—28no—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127C2312yes—
104C2×#127C2312yes—
105C2×#125—21no—
106C2×#125—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8C2311yescharacteristic
109#518—48no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4C2314yes—
119C2×C4×D4C2314yes—
120C2×#117—21no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C22×C22⋊C4—24no—
124C22×C42C2311yescharacteristic
125#486—28no—
126#460—28no—
127#450—44no—
128#440—48no—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147#400—22no—
148#399—22no—
149#398—22no—
150#398—22no—
151#396—22no—
152#395—22no—
153#391—21no—
154#391—22no—
155#391C2312yes—
156#391C2311yescharacteristic
157#390—22no—
158#390—22no—
159#390—22no—
160#390—22no—
161#390—22no—
162#390—22no—
163#390C2314yes—
164#390C2314yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2311yescharacteristic
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#386—24no—
186#386—24no—
187#386—24no—
188#386—24no—
189#385—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#384C2311yescharacteristic
196#382—21no—
197#382—22no—
198#380—21no—
199#380—21no—
200#380C2312yes—
201#380C2312yes—
202#380C2311yescharacteristic
203#379—24no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—22no—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234C25—22no—
235C22×Q8—21no—
236C22×Q8C2411yescharacteristic
237C22×D4—28no—
238C22×D4—28no—
239C22×D4—28no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—44no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4—44no—
249C22×D4—44no—
250C22×D4—82no—
251C22×D4—82no—
252C22×D4C2414yes—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×SD16—28no—
259C2×SD16—28no—
260C2×SD16—44no—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265#127—28no—
266#127—28no—
267#127—28no—
268#127—24no—
269#127#4118yes—
270#125—44no—
271#125—84no—
272C22⋊Q8—22no—
273C22⋊Q8—41no—
274C22⋊Q8—82no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—42no—
278C4×D4—48no—
279C4×D4—48no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—48no—
286C4×D4—84no—
287C4×D4—84no—
288#117—24no—
289#117—24no—
290#117—44no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—416no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—42no—
351#107—42no—
352#107—44no—
353C4⋊C8—24no—
354C4⋊C8—44no—
355#103—24no—
356#103—42no—
357#102—28no—
358#102—28no—
359#102—28no—
360#102—44no—
361#102—44no—
362#102—44no—
363#102—44no—
364C22⋊C8—42no—
365C22⋊C8—42no—
366#97—44no—
367C4×C8—24no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—28no—
388C24—24no—
389C24—44no—
390C24—44no—
391C24—82no—
392C24—82no—
393C24—82no—
394C24—82no—
395C24C22×D411yescharacteristic
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8H(5,𝔽2)12yes—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—432no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—88no—
425C2×D4—88no—
426C2×D4—816no—
427C2×D4C2×#4118yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—41no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4#13714yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2511yescharacteristic
462SD16—416no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—82no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4AGL(1,ℤ/8)14yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—81no—
517C22⋊C4—82no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—42no—
528C42—44no—
529C42—42no—
530C42—44no—
531C42—42no—
532C42—82no—
533C42C2×#4114yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—28no—
541C23—416no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—88no—
549C23—88no—
550C23—164no—
551C23—164no—
552C23—164no—
553C23—164no—
554C23C2×#13712yes—
555C23C23×D411yescharacteristic
556Q8—24no—
557Q8—42no—
558Q8—162no—
559D4—28no—
560D4—28no—
561D4—416no—
562D4—416no—
563D4—416no—
564D4—44no—
565D4—44no—
566D4—88no—
567D4—88no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—168no—
617C2×C4—168no—
618C2×C4#52512yes—
619C2×C4#52512yes—
620C2×C4C2×AGL(1,ℤ/8)14yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4#57612yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4#58512yes—
628C2×C4#58511yescharacteristic
629C8—42no—
630C8—44no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—48no—
640C22—48no—
641C22—48no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—1616no—
647C22C2×#52511yescharacteristic
648C22#348811yescharacteristic
649C22#350011yescharacteristic
650C22C22×AGL(1,ℤ/8)11yescharacteristic
651C22C22×#13711yescharacteristic
652C22C2×#57611yescharacteristic
653C22#358511yescharacteristic
654C4—24no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—42no—
663C4—82no—
664C4—81no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4#311014yes—
671C4#351312yes—
672C4#352112yes—
673C2—42no—
674C2—88no—
675C2#3598211yescharacteristic
676C2C2×#311011yescharacteristic
677C2#6057811yescharacteristic
678C2#6196511yescharacteristic
679C2#6200111yescharacteristic
680C2#6253711yescharacteristic
681C2#6257611yescharacteristic
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. #36287C2semidirect
3. #30075C2semidirect
4. #29939C2semidirect
5. #29367C2semidirect
6. #29130C2semidirect
7. #27541C2semidirect
8. #26526C2semidirect
10. #25322C2semidirect
11. #24921C2semidirect
13. #23757C2semidirect
14. #23684C2semidirect
15. #23223C2semidirect
16. #23003C2semidirect
17. #22643C2semidirect
23. C2×#440C22semidirect
24. #2859C22semidirect
27. #2567C22semidirect
29. #2552C22semidirect
35. #2369C22semidirect
40. Q8×C4⋊C4C22semidirect
49. #2295C22semidirect
52. #2100C22semidirect
53. #2100C22semidirect
54. #2095C22semidirect
55. #2093C22semidirect
56. #2083C22semidirect
57. #2083C22semidirect
60. #2072C22semidirect
61. #2071C22semidirect
62. #2067C22semidirect
63. #2055C22semidirect
69. #1987C22semidirect
72. #1947C22semidirect
73. #1939C22semidirect
74. #1936C22semidirect
75. #1936C22semidirect
76. #1925C22semidirect
78. #1880C22semidirect
79. #1877C22semidirect
80. #1862C22semidirect
81. #1862C22semidirect
82. #1862C22semidirect
83. #1839C22semidirect
84. #1802C22semidirect
85. #1802C22semidirect
89. C4×#107C22semidirect
90. #1770C22semidirect
91. #1767C22semidirect
92. #1762C22semidirect
93. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#362872
#342102
#300752
#299392
#293672
#291302
#275412
#265262
#258552
#253222
#249212
#242112
#237572
#236842
#232232
#230032
#226432