Finite Group Databasebeta

← #6571298

Subgroups — #6571298

Order 512.

1671 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36493C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29928C212yesmaximal, complemented
4#29921C211yesmaximal, characteristic, complemented
5#27976C212yesmaximal, complemented
6#27973C211yesmaximal, characteristic, complemented
7#27650C212yesmaximal, complemented
8#27647C211yesmaximal, characteristic, complemented
9#26239C214yesmaximal, complemented
10#26237C212yesmaximal, complemented
11#24974C214yesmaximal, complemented
12#24972C212yesmaximal, complemented
13#23757C211yesmaximal, characteristic, complemented
14#23755C212yesmaximal, complemented
15#22741C212yesmaximal, complemented
16#22543C212yesmaximal, complemented
17#22530C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#482C2214yes—
21C2×#483C2212yes—
22C2×C8×Q8C2212yes—
23#2856C2211yescharacteristic
24#2850C2212yes—
25#2541C2211yescharacteristic
26#2540C2212yes—
27#2538C2211yescharacteristic
28#2535C2212yes—
29#2471—22no—
30#2456—22no—
31#2448—21no—
32#2431—21no—
33#2366—21no—
34#2363—21no—
35#2355—21no—
36#2355C2212yes—
37#2355C2211yescharacteristic
38#2354—21no—
39#2354C2212yes—
40#2352—22no—
41#2352C2214yes—
42#2314—22no—
43#2311—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C4×#121C2212yes—
46#2285C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2100C2211yescharacteristic
52#2095C2211yescharacteristic
53#2094C2212yes—
54#2084C2212yes—
55#2083C2211yescharacteristic
56#2077C2214yes—
57#2076C2212yes—
58#2072C2211yescharacteristic
59#2072C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2071C2212yes—
63#2036C2212yes—
64#2035C2214yes—
65#2034C2214yes—
66#2034C2214yes—
67#2033C2214yes—
68#1983C2214yes—
69#1982C2212yes—
70#1958C2214yes—
71#1957C2212yes—
72#1954C2212yes—
73#1954C2214yes—
74#1949C2214yes—
75#1949C2214yes—
76#1948C2212yes—
77#1948C2212yes—
78#1875C2214yes—
79#1874C2214yes—
80#1873C2214yes—
81#1835C2214yes—
82#1834C2212yes—
83#1825C2214yes—
84#1823C2212yes—
85#1804C2214yes—
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90C4×#106C2214yes—
91C4×#103C2212yes—
92C4×#102C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96#559—28no—
97C2×#127—21no—
98C2×#127C2311yescharacteristic
99C2×#124—21no—
100C2×#124C2312yes—
101C2×#123—22no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#121C2312yes—
109C4×#41—28no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4D412yes—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117—21no—
121C2×#117—22no—
122C2×#117D412yes—
123C2×#117D411yescharacteristic
124C2×#117C2314yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#483—44no—
131#482—48no—
132C8×Q8—28no—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2314yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#399—22no—
153#397—22no—
154#396—22no—
155#392—22no—
156#392—22no—
157#392—22no—
158#391—22no—
159#391—21no—
160#391C2311yescharacteristic
161#390—22no—
162#390—21no—
163#390—21no—
164#390—22no—
165#390—22no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#389—21no—
170#389C2311yescharacteristic
171#388—24no—
172#388—24no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—24no—
177#387—24no—
178#387—22no—
179#387—21no—
180#386—24no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#385—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—21no—
195#382—22no—
196#382C2314yes—
197#382C2312yes—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—22no—
202#380C2311yescharacteristic
203#380C2312yes—
204#380C2312yes—
205#380C2314yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—24no—
220C4×C22⋊C4—22no—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—42no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2×D412yes—
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257#127—24no—
258#127—44no—
259#124—28no—
260#124—44no—
261#123—48no—
262C22⋊Q8—22no—
263C22⋊Q8—41no—
264C22⋊Q8—44no—
265#121—24no—
266#121—42no—
267#121—44no—
268#121—44no—
269#121—44no—
270#121—82no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8#4118yes—
274C4×D4—24no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—28no—
279C4×D4—44no—
280C4×D4—84no—
281#117—24no—
282#117—28no—
283#117—44no—
284#117—44no—
285#117—44no—
286#117—84no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—22no—
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—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—28no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350#107—24no—
351#107—42no—
352#106—28no—
353#106—44no—
354C4⋊C8—28no—
355C4⋊C8—24no—
356#103—42no—
357#103—42no—
358#103—44no—
359#102—44no—
360#102—44no—
361#102—44no—
362#102—44no—
363C22⋊C8—44no—
364C22⋊C8—42no—
365C22⋊C8—44no—
366C22⋊C8—42no—
367C4×C8—28no—
368C4×C8—24no—
369#95—21no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—24no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—24no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384C24—22no—
385C24C22×D412yes—
386C24C22×D411yescharacteristic
387#41—416no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8#12314yes—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4#13514yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4#12314yes—
500C4⋊C4#12318yes—
501C4⋊C4H(5,𝔽2)12yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—81no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C42—24no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—42no—
526C42—44no—
527C42—42no—
528C42—84no—
529C42—84no—
530C42—82no—
531C42C2×#4118yes—
532C42C2×#4114yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C2×#13512yes—
554C23C23×D411yescharacteristic
555Q8—28no—
556Q8—42no—
557Q8—42no—
558D4—48no—
559D4—44no—
560D4—44no—
561D4—44no—
562D4—88no—
563D4—84no—
564D4—162no—
565D4—164no—
566D4—164no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
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—164no—
616C2×C4—164no—
617C2×C4C2×#12314yes—
618C2×C4C2×#12312yes—
619C2×C4C2×#12314yes—
620C2×C4C2×#12312yes—
621C2×C4C2×#13512yes—
622C2×C4C2×#13512yes—
623C2×C4#57512yes—
624C2×C4#57712yes—
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—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22C22×#12311yescharacteristic
654C22#348811yescharacteristic
655C22#350011yescharacteristic
656C22C22×#13511yescharacteristic
657C22C2×#57511yescharacteristic
658C22C2×#57711yescharacteristic
659C22#358511yescharacteristic
660C4—22no—
661C4—24no—
662C4—22no—
663C4—24no—
664C4—22no—
665C4—42no—
666C4—42no—
667C4—41no—
668C4—84no—
669C4—82no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4—81no—
675C4#309114yes—
676C4#351712yes—
677C4#352412yes—
678C2—42no—
679C2—82no—
680C2—162no—
681C2#3607611yescharacteristic
682C2C2×#309111yescharacteristic
683C2#6052511yescharacteristic
684C2#6195311yescharacteristic
685C2#6199611yescharacteristic
686C2#6253111yescharacteristic
687C2#6259211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #34188C2semidirect
3. #29928C2semidirect
4. #29921C2semidirect
5. #27976C2semidirect
6. #27973C2semidirect
7. #27650C2semidirect
8. #27647C2semidirect
9. #26239C2semidirect
10. #26237C2semidirect
11. #24974C2semidirect
12. #24972C2semidirect
13. #23757C2semidirect
14. #23755C2semidirect
15. #22741C2semidirect
16. #22543C2semidirect
17. #22530C2semidirect

Maximal subgroups

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

subgroupindex
#364932
#341882
#299282
#299212
#279762
#279732
#276502
#276472
#262392
#262372
#249742
#249722
#237572
#237552
#227412
#225432
#225302