Finite Group Databasebeta

← #211172

Subgroups — #211172

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#486C211yesmaximal, characteristic, complemented
2#21504C211yesmaximal, characteristic, complemented
3#21503C211yesmaximal, characteristic
4C4×C8×D4C211yesmaximal, characteristic, complemented
5C2×#1681C211yesmaximal, characteristic
6#9738C211yesmaximal, characteristic, complemented
7#9616C211yesmaximal, characteristic, complemented
8#9565C211yesmaximal, characteristic, complemented
9#9563C211yesmaximal, characteristic, complemented
10#9221C211yesmaximal, characteristic
11#9217C211yesmaximal, characteristic, complemented
12#8648C211yesmaximal, characteristic, complemented
13#8274C211yesmaximal, characteristic, complemented
14#8236C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#461C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×#442—21no—
20C2×#440—21no—
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C22×C4×C8—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C42×D4C2211yescharacteristic
27#1990C2211yescharacteristic
28#1982C2211yescharacteristic
29#1981C2211yescharacteristic
30#1957C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1934—21no—
34#1934—21no—
35#1928—21no—
36#1926—21no—
37#1879—21no—
38#1877—21no—
39#1834C2211yescharacteristic
40#1802C2211yescharacteristic
41#1802C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1767C2211yescharacteristic
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47#1763—21no—
48#1762—21no—
49C4×#102—21no—
50C4×#102—21no—
51C8×C22⋊C4—21no—
52C8×C22⋊C4—21no—
53C4×C22⋊C8—21no—
54C42×C8C2211yescharacteristic
55#1681C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×D8—21no—
70C23×C8—21no—
71C23×C8—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×#127C2311yescharacteristic
74C2×C4×Q8—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C22×C42—21no—
82#486—22no—
83#486C2×C414yescomplemented
84#461—22no—
85#461C2×C414yescomplemented
86#459—22no—
87#459C2×C414yes—
88#442—44no—
89#440—44no—
90C8×D4—22no—
91C8×D4—22no—
92C8×D4—22no—
93C8×D4C2×C414yescomplemented
94C2×#107—21no—
95C2×#106—21no—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—44no—
122C2×C4×C8D411yescharacteristic
123C2×C4×C8D411yescharacteristic
124C2×C4×C8D411yescharacteristic
125C2×C4×C8D411yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#391—21no—
129#390—21no—
130#390—21no—
131#390—21no—
132#389—21no—
133#389—21no—
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C43C2311yescharacteristic
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#335—22no—
159#335C2×C414yescomplemented
160#332—22no—
161#332C2×C414yescomplemented
162#331—22no—
163#331C2×C414yescomplemented
164C22×Q8—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C2×D411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—41no—
173C2×SD16—24no—
174C2×SD16—42no—
175C2×SD16—42no—
176C2×D8—24no—
177C2×D8—42no—
178C2×D8—42no—
179C22×C8—24no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—24no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8C22×C412yes—
199#127—21no—
200#127—21no—
201#127—21no—
202#127C22×C412yes—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—82no—
215C4×D4D814yes—
216C4×D4C22×C412yes—
217C4×D4#4112yes—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—41no—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42C2411yescharacteristic
249#107—42no—
250#106—42no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8C22×C412yes—
260C4⋊C8C22×C412yes—
261C4⋊C8C22×C412yes—
262C4⋊C8#4112yes—
263#103—24no—
264#103—42no—
265#103—42no—
266#102—24no—
267#102—24no—
268#102—24no—
269#102—42no—
270#102—42no—
271#102—42no—
272#102—42no—
273#102—42no—
274#102—42no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277#97—42no—
278#97—42no—
279C4×C8—22no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—41no—
283C4×C8—41no—
284C4×C8—41no—
285C4×C8—42no—
286C4×C8—42no—
287C4×C8—42no—
288C4×C8C22×C412yes—
289C4×C8#4112yes—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—41no—
294#95—41no—
295C24—21no—
296C24—41no—
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—81no—
321C2×D4—81no—
322C2×D4C4×D412yes—
323C2×D4C2×D812yes—
324C2×D4#13512yes—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4111yescharacteristic
353SD16—48no—
354D8—48no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—48no—
369C2×C8—48no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8C4×D412yes—
385C2×C8C4×D412yes—
386C2×C8AGL(1,ℤ/8)12yes—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—81no—
403C4⋊C4—81no—
404C4⋊C4—82no—
405C4⋊C4C4×D412yes—
406C4⋊C4C2×D812yes—
407C4⋊C4#13512yes—
408C22⋊C4—24no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—24no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—22no—
422C42—41no—
423C42—42no—
424C42—42no—
425C42—41no—
426C42—41no—
427C42—82no—
428C42C2×D812yes—
429C42#13512yes—
430C42C23×C411yescharacteristic
431C42C2×#4111yescharacteristic
432C42C2×#4111yescharacteristic
433C42C2×#4111yescharacteristic
434C23—22no—
435C23—22no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23C2×#12111yescharacteristic
449Q8—42no—
450Q8—42no—
451Q8—81no—
452Q8—81no—
453D4—22no—
454D4—22no—
455D4—22no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—82no—
461D4—82no—
462D4—81no—
463D4—81no—
464D4—81no—
465D4—81no—
466D4C4×D814yes—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—164no—
503C2×C4#46612yes—
504C2×C4#46612yes—
505C2×C4#47112yes—
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C4×#4111yescharacteristic
509C2×C4#52511yescharacteristic
510C2×C4#52911yescharacteristic
511C2×C4#53211yescharacteristic
512C2×C4C22×D811yescharacteristic
513C2×C4C2×#13511yescharacteristic
514C2×C4C2×#13511yescharacteristic
515C2×C4C2×#13511yescharacteristic
516C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
517C2×C4#57511yescharacteristic
518C2×C4#57511yescharacteristic
519C2×C4#57511yescharacteristic
520C8—22no—
521C8—22no—
522C8—22no—
523C8—41no—
524C8—42no—
525C8—42no—
526C8—42no—
527C8—42no—
528C8—81no—
529C8—81no—
530C22—24no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—161no—
541C22—161no—
542C22—161no—
543C22—161no—
544C22C4×#12111yescharacteristic
545C22#303611yescharacteristic
546C22#304011yescharacteristic
547C22C2×#46611yescharacteristic
548C22#305011yescharacteristic
549C22C2×#47111yescharacteristic
550C22#306711yescharacteristic
551C4—21no—
552C4—21no—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—41no—
560C4—82no—
561C4—161no—
562C4—161no—
563C4C2×C4×D811yescharacteristic
564C4C4×#13511yescharacteristic
565C4#294311yescharacteristic
566C4C4×AGL(1,ℤ/8)11yescharacteristic
567C4#311111yescharacteristic
568C4#311411yescharacteristic
569C4#316811yescharacteristic
570C4#316911yescharacteristic
571C2—44no—
572C2—162no—
573C2#1925511yescharacteristic
574C2#1925711yescharacteristic
575C2#1927611yescharacteristic
576C2#1927811yescharacteristic
577C2#2150711yescharacteristic
578C2C4×#46611yescharacteristic
579C2C4×#47111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#486C2semidirect
2. #21504C2semidirect
4. C4×C8×D4C2semidirect
6. #9738C2semidirect
7. #9616C2semidirect
8. #9565C2semidirect
9. #9563C2semidirect
11. #9217C2semidirect
12. #8648C2semidirect
13. #8274C2semidirect
14. #8236C2semidirect
15. #8234C2semidirect
55. #1681C4semidirect
83. #486C2×C4semidirect
85. #461C2×C4semidirect
93. C8×D4C2×C4semidirect
159. #335C2×C4semidirect
161. #332C2×C4semidirect
163. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4×#4862
#215042
#215032
C4×C8×D42
C2×#16812
#97382
#96162
#95652
#95632
#92212
#92172
#86482
#82742
#82362
#82342