Finite Group Databasebeta

← #211456

Subgroups — #211456

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#486C211yesmaximal, characteristic, complemented
2#21614C211yesmaximal, characteristic, complemented
3#21503C211yesmaximal, characteristic
4C4×#436C211yesmaximal, characteristic, complemented
5C2×#1697C211yesmaximal, characteristic
6#9747C211yesmaximal, characteristic, complemented
7#9619C211yesmaximal, characteristic, complemented
8#9575C211yesmaximal, characteristic, complemented
9#9571C211yesmaximal, characteristic, complemented
10#9229C211yesmaximal, characteristic
11#9223C211yesmaximal, characteristic, complemented
12#8654C211yesmaximal, characteristic, complemented
13#8274C211yesmaximal, characteristic, complemented
14#8235C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×#442—21no—
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×C4×M16—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#127C2211yescharacteristic
25C42×D4C2211yescharacteristic
26#1990C2211yescharacteristic
27#1987C2211yescharacteristic
28#1982C2211yescharacteristic
29#1957C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1934—21no—
33#1934—21no—
34#1926—21no—
35#1919—21no—
36#1879—21no—
37#1862—21no—
38#1834C2211yescharacteristic
39#1802C2211yescharacteristic
40#1802C2211yescharacteristic
41#1776—21no—
42#1770C2211yescharacteristic
43#1770C2211yescharacteristic
44#1767C2211yescharacteristic
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47#1762—21no—
48#1762—21no—
49C4×#102—21no—
50C4×#102—21no—
51#1753—21no—
52#1753—21no—
53C4×C22⋊C8—21no—
54C42×C8C2211yescharacteristic
55#1697C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×D8—21no—
69C22×M16—21no—
70C22×M16—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C22×C42—21no—
81#486—22no—
82#486C2×C414yescomplemented
83#476—22no—
84#476C2×C414yescomplemented
85#459—22no—
86#459C2×C414yes—
87#442—44no—
88#436—22no—
89#436—22no—
90#436—22no—
91#436C2×C414yescomplemented
92C2×#107—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C4×M16—44no—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8D411yescharacteristic
120C2×C4×C8D411yescharacteristic
121C2×C4×C8D411yescharacteristic
122C2×C4×C8D411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391—21no—
126#390—21no—
127#390—21no—
128#390—21no—
129#389—21no—
130#389—21no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C43C2311yescharacteristic
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#335—22no—
158#335C2×C414yescomplemented
159#333—22no—
160#333C2×C414yescomplemented
161#331—22no—
162#331C2×C414yescomplemented
163C22×Q8—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—41no—
172C2×D8—24no—
173C2×D8—42no—
174C2×D8—42no—
175C2×M16—24no—
176C2×M16—24no—
177C2×M16—42no—
178C2×M16—42no—
179C2×M16—42no—
180C2×M16—42no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8C22×C412yes—
195#127—21no—
196#127—21no—
197#127—21no—
198#127C22×C412yes—
199C4×Q8—82no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—82no—
211C4×D4C22×C412yes—
212C4×D4#4112yes—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—41no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42C2411yescharacteristic
244#107—42no—
245#106—24no—
246#106—42no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8C22×C412yes—
256C4⋊C8C22×C412yes—
257C4⋊C8C22×C412yes—
258C4⋊C8#4112yes—
259#102—24no—
260#102—24no—
261#102—24no—
262#102—24no—
263#102—42no—
264#102—42no—
265#102—42no—
266#102—42no—
267#102—42no—
268#102—42no—
269C22⋊C8—24no—
270C22⋊C8—24no—
271#97—42no—
272#97—42no—
273#97—42no—
274#97—42no—
275C4×C8—22no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—41no—
279C4×C8—41no—
280C4×C8—41no—
281C4×C8—42no—
282C4×C8C22×C412yes—
283C4×C8#4112yes—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—41no—
288#95—41no—
289C24—21no—
290C24—41no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—81no—
294C2×Q8—81no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—81no—
315C2×D4—81no—
316C2×D4C4×D412yes—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12111yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4111yescharacteristic
345D8—48no—
346M16—48no—
347M16—48no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8C4×D412yes—
376C2×C8C4×D412yes—
377C2×C8AGL(1,ℤ/8)12yes—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—81no—
396C4⋊C4—81no—
397C4⋊C4—82no—
398C4⋊C4C4×D412yes—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—22no—
413C42—41no—
414C42—42no—
415C42—42no—
416C42—41no—
417C42—41no—
418C42—82no—
419C42AGL(1,ℤ/8)12yes—
420C42#13712yes—
421C42C23×C411yescharacteristic
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C42C2×#4111yescharacteristic
425C23—22no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23C2×#12111yescharacteristic
440Q8—82no—
441Q8—82no—
442D4—22no—
443D4—22no—
444D4—41no—
445D4—41no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—82no—
451D4—82no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—81no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—164no—
492C2×C4#46812yes—
493C2×C4#46812yes—
494C2×C4#47212yes—
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C4×#4111yescharacteristic
498C2×C4#52511yescharacteristic
499C2×C4#52911yescharacteristic
500C2×C4#53211yescharacteristic
501C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4C2×#13711yescharacteristic
504C2×C4#57511yescharacteristic
505C2×C4#57511yescharacteristic
506C2×C4#57511yescharacteristic
507C2×C4#57511yescharacteristic
508C2×C4#57511yescharacteristic
509C8—22no—
510C8—22no—
511C8—22no—
512C8—41no—
513C8—42no—
514C8—42no—
515C8—42no—
516C8—42no—
517C8—81no—
518C8—81no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22—161no—
533C22C4×#12111yescharacteristic
534C22#303611yescharacteristic
535C22#304011yescharacteristic
536C22C2×#46811yescharacteristic
537C22#305411yescharacteristic
538C22C2×#47211yescharacteristic
539C22#306711yescharacteristic
540C4—21no—
541C4—21no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—42no—
546C4—41no—
547C4—82no—
548C4—81no—
549C4—81no—
550C4—161no—
551C4—161no—
552C4#294311yescharacteristic
553C4C4×AGL(1,ℤ/8)11yescharacteristic
554C4C4×AGL(1,ℤ/8)11yescharacteristic
555C4C4×#13711yescharacteristic
556C4#311111yescharacteristic
557C4#311411yescharacteristic
558C4#317111yescharacteristic
559C4#317211yescharacteristic
560C2—82no—
561C2—162no—
562C2#1927911yescharacteristic
563C2#1928311yescharacteristic
564C2#1929911yescharacteristic
565C2#1930311yescharacteristic
566C2#2150711yescharacteristic
567C2C4×#46811yescharacteristic
568C2C4×#47211yescharacteristic
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. #21614C2semidirect
4. C4×#436C2semidirect
6. #9747C2semidirect
7. #9619C2semidirect
8. #9575C2semidirect
9. #9571C2semidirect
11. #9223C2semidirect
12. #8654C2semidirect
13. #8274C2semidirect
14. #8235C2semidirect
15. #8234C2semidirect
55. #1697C4semidirect
82. #486C2×C4semidirect
84. #476C2×C4semidirect
91. #436C2×C4semidirect
158. #335C2×C4semidirect
160. #333C2×C4semidirect
162. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4×#4862
#216142
#215032
C4×#4362
C2×#16972
#97472
#96192
#95752
#95712
#92292
#92232
#86542
#82742
#82352
#82342