Finite Group Databasebeta

← #6557830

Subgroups — #6557830

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36454C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#30367C212yesmaximal, complemented
4#30205C211yesmaximal, characteristic, complemented
5#29213C211yesmaximal, characteristic, complemented
6#29160C212yesmaximal, complemented
7#28987C214yesmaximal, complemented
8#28382C211yesmaximal, characteristic, complemented
9#25495C212yesmaximal, complemented
10#25385C212yesmaximal, complemented
11#25338C212yesmaximal, complemented
12#25041C214yesmaximal, complemented
13#23819C211yesmaximal, characteristic, complemented
14#23576C212yesmaximal, complemented
15#23002C212yesmaximal, complemented
16#22968C212yesmaximal, complemented
17#20558C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C2×#474C2212yes—
21C2×#465C2214yes—
22C2×#461C2212yes—
23C2×#440C2214yes—
24C2×C8×D4C2212yes—
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2857C2212yes—
28#2855C2212yes—
29#2575—22no—
30#2547—21no—
31#2547C2211yescharacteristic
32#2542C2211yescharacteristic
33#2537—21no—
34#2537C2212yes—
35#2517—22no—
36#2511—21no—
37#2489C2212yes—
38#2447—21no—
39#2432—22no—
40#2422—22no—
41#2422C2214yes—
42#2421C2212yes—
43#2367—21no—
44#2365C2211yescharacteristic
45#2365C2211yescharacteristic
46#2355—21no—
47D4×C4⋊C4—21no—
48D4×C4⋊C4C2212yes—
49C4×#127C2211yescharacteristic
50C4×#125C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52C42×D4C2211yescharacteristic
53#2100C2211yescharacteristic
54#2100C2211yescharacteristic
55#2095C2212yes—
56#2075C2214yes—
57#2073C2212yes—
58#2072C2214yes—
59#2058C2212yes—
60#2057C2212yes—
61#2041C2211yescharacteristic
62#2041C2211yescharacteristic
63#2035C2214yes—
64#2033C2214yes—
65#2025C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#2014C2212yes—
69#2000C2214yes—
70#1992C2212yes—
71#1990C2212yes—
72#1973C2214yes—
73#1956C2212yes—
74#1948C2212yes—
75#1947C2212yes—
76#1943C2214yes—
77#1935C2214yes—
78#1921C2214yes—
79#1893C2214yes—
80#1892C2214yes—
81#1877C2214yes—
82#1864C2214yes—
83#1863C2214yes—
84#1847C2212yes—
85#1808C2214yes—
86#1770C2212yes—
87#1767C2211yescharacteristic
88#1762C2214yes—
89C4×#102C2212yes—
90C4×#102C2212yes—
91#1752C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41—22no—
96C22×SD16C2314yes—
97C23×C8C2314yes—
98#547—48no—
99#532—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125C2311yescharacteristic
107C2×#124—21no—
108C2×#124—22no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×#121—22no—
114C2×#121—22no—
115C2×#121C2314yes—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2311yescharacteristic
126C22×C4⋊C4—22no—
127C22×C42C2312yes—
128#474—28no—
129#465—48no—
130#461—28no—
131#440—48no—
132C8×D4—28no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2312yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#400—22no—
151#399—22no—
152#397—22no—
153#397—22no—
154#397—22no—
155#397—24no—
156#394—24no—
157#394—24no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—21no—
162#391—21no—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#389—21no—
171#389—22no—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2314yes—
176#388—22no—
177#388—22no—
178#387—21no—
179#387—21no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#385—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—21no—
197#382—21no—
198#382—22no—
199#382C2311yescharacteristic
200#381—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235C2×#41—48no—
236C2×#41—48no—
237C2×#41—84no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2412yes—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—44no—
254C22×C8—28no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265#127—24no—
266#125—22no—
267#125—41no—
268#124—48no—
269#124—44no—
270#123—416no—
271C22⋊Q8—24no—
272C22⋊Q8—42no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275#121—28no—
276#121—44no—
277#121—48no—
278#121—48no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—42no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—48no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—44no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—28no—
350#107—44no—
351#106—44no—
352C4⋊C8#4118yescomplemented
353#103—28no—
354#103—44no—
355#103—44no—
356#102—24no—
357#102—24no—
358#102—28no—
359#102—24no—
360#102—42no—
361C22⋊C8—28no—
362#97—42no—
363#97—44no—
364C4×C8—24no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—22no—
387C24C22×D412yes—
388#41—832no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—84no—
415C2×D4AGL(1,ℤ/8)14yes—
416C2×D4H(5,𝔽2)14yes—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—84no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2511yescharacteristic
451SD16—416no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—88no—
461C2×C8C2×#4118yes—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C42−1+412yes—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—82no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—21no—
515C42—21no—
516C42—42no—
517C42—42no—
518C42—44no—
519C42—42no—
520C42—42no—
521C42—82no—
522C42C2×#4114yes—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C2×AGL(1,ℤ/8)14yes—
537C23C23×D411yescharacteristic
538Q8—44no—
539Q8—44no—
540Q8—88no—
541Q8—84no—
542D4—28no—
543D4—24no—
544D4—24no—
545D4—48no—
546D4—48no—
547D4—42no—
548D4—42no—
549D4—816no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—816no—
600C2×C4—88no—
601C2×C4#53212yes—
602C2×C4#53212yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4#57512yes—
605C2×C4#57712yes—
606C2×C4#57812yes—
607C2×C4C22×#4112yes—
608C2×C4C2×H(5,𝔽2)12yes—
609C2×C4C2×2−1+411yescharacteristic
610C2×C4#58511yescharacteristic
611C8—28no—
612C8—42no—
613C8—42no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C2×#53211yescharacteristic
629C22#348411yescharacteristic
630C22#348911yescharacteristic
631C22C22×AGL(1,ℤ/8)11yescharacteristic
632C22C2×#57511yescharacteristic
633C22C2×#57711yescharacteristic
634C22C2×#57811yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—42no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—81no—
648C4—81no—
649C4—84no—
650C4—81no—
651C4#318014yes—
652C4#351312yes—
653C4#352212yes—
654C2—44no—
655C2—84no—
656C2#3591211yescharacteristic
657C2C2×#318011yescharacteristic
658C2#6072311yescharacteristic
659C2#6159511yescharacteristic
660C2#6160011yescharacteristic
661C2#6213211yescharacteristic
662C2#6213411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36454C2semidirect
3. #30367C2semidirect
4. #30205C2semidirect
5. #29213C2semidirect
6. #29160C2semidirect
7. #28987C2semidirect
8. #28382C2semidirect
9. #25495C2semidirect
10. #25385C2semidirect
11. #25338C2semidirect
12. #25041C2semidirect
13. #23819C2semidirect
14. #23576C2semidirect
15. #23002C2semidirect
16. #22968C2semidirect
17. #20558C2semidirect
352. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#364542
#361272
#303672
#302052
#292132
#291602
#289872
#283822
#254952
#253852
#253382
#250412
#238192
#235762
#230022
#229682
#205582