Finite Group Databasebeta

← #6624515

Subgroups — #6624515

Order 512.

1671 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33110C212yesmaximal, complemented
4#32903C211yesmaximal, characteristic
5#28315C212yesmaximal, complemented
6#28115C212yesmaximal, complemented
7#27906C211yesmaximal, characteristic
8#27648C211yesmaximal, characteristic, complemented
9#27508C214yesmaximal, complemented
10#27210C214yesmaximal, complemented
11#26089C212yesmaximal, complemented
12#24633C212yesmaximal, complemented
13#23101C212yesmaximal, complemented
14#22814C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21580C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C22×C4×Q8C2211yescharacteristic
19C2×#484C2212yes—
20C2×#481C2212yes—
21C2×#481C2212yes—
22C2×#460C2212yes—
23C2×#451C2211yescharacteristic
24C2×#448C2212yes—
25C2×C4×Q16C2212yes—
26#2856C2211yescharacteristic
27#2850C2212yes—
28#2597—22no—
29#2571C2211yescharacteristic
30#2548C2211yescharacteristic
31#2513—21no—
32#2513C2212yes—
33#2504—21no—
34#2447C2212yes—
35#2444—22no—
36#2366C2211yescharacteristic
37#2361—21no—
38#2357—21no—
39#2355—22no—
40#2355—21no—
41#2355—21no—
42#2355C2211yescharacteristic
43#2355C2212yes—
44#2355C2212yes—
45#2355C2214yes—
46C4×C22⋊Q8C2211yescharacteristic
47C4×#121C2212yes—
48C2×#391C2212yes—
49C2×#387—22no—
50C2×#384C2211yescharacteristic
51C2×#382—21no—
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2097C2214yes—
56#2094C2214yes—
57#2083C2212yes—
58#2072C2212yes—
59#2066C2214yes—
60#2055C2214yes—
61#2015C2214yes—
62#1983C2214yes—
63#1973C2214yes—
64#1971C2214yes—
65#1957C2212yes—
66#1949C2214yes—
67#1947C2212yes—
68#1939C2212yes—
69#1927C2214yes—
70#1927C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1920C2214yes—
74#1917C2214yes—
75#1880C2214yes—
76#1878C2212yes—
77#1877C2214yes—
78#1874C2212yes—
79#1872C2211yescharacteristic
80#1863C2212yes—
81#1852C2211yescharacteristic
82#1844C2212yes—
83#1824C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1787C2211yescharacteristic
89#1776C2214yes—
90#1770C2212yes—
91#1762C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2312yes—
97C22×SD16C2314yes—
98C2×#127C2311yescharacteristic
99C2×#124C2312yes—
100C2×C22⋊Q8C2311yescharacteristic
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—28no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—22no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C42—21no—
121C22×C42—22no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#484—44no—
125#481—28no—
126#481—44no—
127#460—28no—
128#451—24no—
129#448—44no—
130C4×Q16—28no—
131C2×#107C2312yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#397—22no—
153#397—22no—
154#397—22no—
155#397—22no—
156#394—22no—
157#391—22no—
158#391—22no—
159#391C2314yes—
160#391C2311yescharacteristic
161#391C2312yes—
162#391C2314yes—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#389—21no—
172#389C2312yes—
173#389C2311yescharacteristic
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—21no—
180#387—22no—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—21no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—21no—
197#382—22no—
198#382—22no—
199#382—24no—
200#382—21no—
201#382C2312yes—
202#381—22no—
203#380C2311yescharacteristic
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C2×#95—22no—
224C2×#95—22no—
225C2×#95—22no—
226C2×#95D412yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2412yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C2×Q16—24no—
257C2×Q16—24no—
258C2×Q16—42no—
259C2×SD16—28no—
260C2×SD16—44no—
261C2×SD16—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265#127—24no—
266#124—44no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269#121—24no—
270#121—24no—
271#121—44no—
272#121—82no—
273C4×Q8—28no—
274C4×Q8—28no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8#4118yes—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—84no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42—28no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42—84no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348#107—24no—
349#107—44no—
350#106—28no—
351#106—44no—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#103—24no—
355#103—24no—
356#103—24no—
357#103—42no—
358#103—44no—
359#103—44no—
360#102—28no—
361#102—28no—
362#102—44no—
363C22⋊C8—24no—
364C22⋊C8—22no—
365C22⋊C8#4114yes—
366#97—44no—
367C4×C8—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—22no—
390C24C22×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8C2×#4114yes—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4#13714yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4C2511yescharacteristic
465Q16—48no—
466SD16—88no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8C2×#4114yes—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4C2×#4114yes—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C42—28no—
519C42—28no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—24no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—84no—
530C42—84no—
531C42—82no—
532C42C2×#4114yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23#55012yes—
545C23C2×#13712yes—
546C23#57712yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—24no—
550Q8—48no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555D4—44no—
556D4—44no—
557D4—88no—
558D4—84no—
559D4—162no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—84no—
613C2×C4—88no—
614C2×C4—164no—
615C2×C4#52512yes—
616C2×C4#52512yes—
617C2×C4#55012yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4#57512yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4112yes—
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4#58512yes—
625C8—24no—
626C8—42no—
627C8—44no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22#340014yes—
642C22C2×#52511yescharacteristic
643C22C2×#55011yescharacteristic
644C22#348511yescharacteristic
645C22C22×#13711yescharacteristic
646C22C2×#57511yescharacteristic
647C22C2×#57711yescharacteristic
648C22#358511yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—41no—
658C4—44no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—84no—
665C4#311714yes—
666C4#347512yes—
667C4#349412yes—
668C2—24no—
669C2—162no—
670C2#3651711yescharacteristic
671C2C2×#311711yescharacteristic
672C2#6058811yescharacteristic
673C2C2×#340011yescharacteristic
674C2#6140511yescharacteristic
675C2#6171911yescharacteristic
676C2#6173911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. C2×#2355C2semidirect
3. #33110C2semidirect
5. #28315C2semidirect
6. #28115C2semidirect
8. #27648C2semidirect
9. #27508C2semidirect
10. #27210C2semidirect
11. #26089C2semidirect
12. #24633C2semidirect
13. #23101C2semidirect
14. #22814C2semidirect
15. #22629C2semidirect
16. #21580C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#23552
#331102
#329032
#283152
#281152
#279062
#276482
#275082
#272102
#260892
#246332
#231012
#228142
#226292
#215802
C4×#4512