Finite Group Databasebeta

← #6643299

Subgroups — #6643299

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×#2513C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#33107C211yesmaximal, characteristic, complemented
5#32533C214yesmaximal, complemented
6#32487C214yesmaximal, complemented
7#31913C214yesmaximal, complemented
8#31815C214yesmaximal, complemented
9#29952C211yesmaximal, characteristic, complemented
10#29712C211yesmaximal, characteristic
11#28117C212yesmaximal, complemented
12#27684C212yesmaximal, complemented
13#24603C212yesmaximal, complemented
14#24500C212yesmaximal, complemented
15#21951C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C2×#481C2212yes—
18C2×#447C2212yes—
19C2×#431C2212yes—
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2854C2212yes—
23#2639—21no—
24#2618—21no—
25#2560—22no—
26#2558—22no—
27#2513—22no—
28#2513—22no—
29#2513C2214yes—
30#2513C2214yes—
31#2512C2212yescomplemented
32#2506C2212yescomplemented
33#2452C2212yescomplemented
34#2445C2212yescomplemented
35C2×#397—21no—
36C2×#397—21no—
37#2325—21no—
38C4×#121C2212yes—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#388—21no—
45C2×#382C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#2101C2214yescomplemented
49#2094C2212yes—
50#2092C2214yescomplemented
51#2088C2214yescomplemented
52#2071C2212yes—
53#2067C2214yes—
54#2065C2214yes—
55#2065C2214yes—
56#2065C2214yescomplemented
57#2054C2214yes—
58#2054C2214yes—
59#2054C2214yes—
60#2023C2212yes—
61#2011C2212yes—
62#1987C2212yescomplemented
63#1982C2212yescomplemented
64#1970C2214yes—
65#1949C2214yescomplemented
66#1948C2212yescomplemented
67#1947C2212yescomplemented
68#1941C2214yescomplemented
69#1939C2214yescomplemented
70#1939C2214yescomplemented
71#1934C2214yes—
72#1934C2214yescomplemented
73#1926C2214yes—
74#1926C2214yescomplemented
75#1921C2214yescomplemented
76#1879C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1824C2212yes—
82#1804C2214yescomplemented
83#1797C2211yescharacteristic
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1786C2212yes—
87C23×D4C2311yescharacteristic
88C22×D8C2314yes—
89#539—48no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#125—22no—
94C2×#123—22no—
95C2×#123—22no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121C2312yes—
101C2×C4×D4—22no—
102C2×C4×D4C2312yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#481—44no—
120#447—44no—
121#431—44no—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C4×C8C2314yes—
138#400—22no—
139#398—22no—
140#397—24no—
141#397—24no—
142#397—22no—
143#397—24no—
144#397—24no—
145#396—22no—
146#396—22no—
147#394—22no—
148#394—22no—
149#393—22no—
150#391—21no—
151#390—21no—
152#390—21no—
153#390—22no—
154#390—22no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#390C2312yes—
161#390C2314yes—
162#390C2314yes—
163#390C2312yes—
164#390C2312yes—
165#389—21no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2312yes—
171#389C2312yes—
172#388—24no—
173#388—24no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—24no—
179#388—24no—
180#387—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#385—22no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382—22no—
188#382C2314yes—
189#380—21no—
190#380C2312yes—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4D412yes—
194C4×C4⋊C4D412yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2314yes—
215#336C2312yes—
216C25—21no—
217C22×Q8—21no—
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—42no—
225C22×D4—42no—
226C22×D4C2412yes—
227C22×D4C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×D8—44no—
240C2×D8—44no—
241C2×D8—44no—
242C22×C8C2414yes—
243C22×C8C2414yes—
244C4⋊Q8—42no—
245#127—42no—
246#126—48no—
247#125—48no—
248#123—48no—
249#123—48no—
250C22⋊Q8—48no—
251C22⋊Q8—44no—
252#121—24no—
253#121—24no—
254#121—44no—
255#121—44no—
256C4×D4—44no—
257C4×D4—48no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—82no—
310C2×C22⋊C4—82no—
311C2×C22⋊C4—82no—
312C2×C22⋊C4—82no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—82no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2412yes—
329#107—44no—
330#106—44no—
331#106—44no—
332C4⋊C8—44no—
333C4⋊C8—42no—
334C4⋊C8—42no—
335C4⋊C8—44no—
336#102—44no—
337#102—44no—
338#102—44no—
339#102—44no—
340#102—44no—
341#102—44no—
342C22⋊C8—24no—
343C22⋊C8—24no—
344C4×C8—44no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24—22no—
361C24—22no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24C22×D411yescharacteristic
370C2×Q8—42no—
371C2×Q8—81no—
372C2×Q8—81no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—81no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4—81no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4#13514yes—
432C22×C4AGL(1,ℤ/8)14yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444C22×C4C2511yescharacteristic
445D8—88no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—81no—
476C4⋊C4—84no—
477C4⋊C4—81no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—41no—
504C42—41no—
505C42—42no—
506C42—44no—
507C42—82no—
508C42—82no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23#53712yes—
535C23C2×#13512yes—
536C23C2×AGL(1,ℤ/8)12yes—
537C23C23×D411yescharacteristic
538Q8—162no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—44no—
543D4—84no—
544D4—88no—
545D4—162no—
546D4—162no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—164no—
600C2×C4#53712yes—
601C2×C4C2×#13512yes—
602C2×C4C2×#13512yes—
603C2×C4C2×AGL(1,ℤ/8)14yes—
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4#58512yes—
607C2×C4#58512yes—
608C8—44no—
609C8—44no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—168no—
631C22#324714yes—
632C22C2×#53711yescharacteristic
633C22#348511yescharacteristic
634C22#348711yescharacteristic
635C22C22×#13511yescharacteristic
636C22C22×AGL(1,ℤ/8)11yescharacteristic
637C22#358511yescharacteristic
638C22#358511yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—24no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—41no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4#356012yes—
655C4#356412yes—
656C2—24no—
657C2—162no—
658C2—162no—
659C2#3697111yescharacteristic
660C2C2×#324711yescharacteristic
661C2#6092011yescharacteristic
662C2#6169911yescharacteristic
663C2#6172011yescharacteristic
664C2#6188511yescharacteristic
665C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#29712C2
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
1. #35006C2semidirect
2. C2×#2513C2semidirect
3. #33149C2semidirect
4. #33107C2semidirect
5. #32533C2semidirect
6. #32487C2semidirect
7. #31913C2semidirect
8. #31815C2semidirect
9. #29952C2semidirect
11. #28117C2semidirect
12. #27684C2semidirect
13. #24603C2semidirect
14. #24500C2semidirect
15. #21951C2semidirect
31. #2512C22semidirect
32. #2506C22semidirect
33. #2452C22semidirect
34. #2445C22semidirect
48. #2101C22semidirect
50. #2092C22semidirect
51. #2088C22semidirect
56. #2065C22semidirect
62. #1987C22semidirect
63. #1982C22semidirect
65. #1949C22semidirect
66. #1948C22semidirect
67. #1947C22semidirect
68. #1941C22semidirect
69. #1939C22semidirect
70. #1939C22semidirect
72. #1934C22semidirect
74. #1926C22semidirect
75. #1921C22semidirect
82. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#25132
#331492
#331072
#325332
#324872
#319132
#318152
#299522
#297122
#281172
#276842
#246032
#245002
#219512