Finite Group Databasebeta

← #211564

Subgroups — #211564

Order 512.

755 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22219C211yesmaximal, characteristic, complemented
2#21614C211yesmaximal, characteristic, complemented
3C4×#459C211yesmaximal, characteristic
4#20568C211yesmaximal, characteristic, complemented
5C2×#1697C211yesmaximal, characteristic
6#9747C211yesmaximal, characteristic, complemented
7#9617C211yesmaximal, characteristic, complemented
8#9576C211yesmaximal, characteristic, complemented
9#9571C211yesmaximal, characteristic, complemented
10#9229C211yesmaximal, characteristic
11#9223C211yesmaximal, characteristic, complemented
12#8650C211yesmaximal, characteristic, complemented
13#8278C211yesmaximal, characteristic, complemented
14#8234C211yesmaximal, characteristic, complemented
15C4×#333C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×C4×D8—21no—
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1982C2211yescharacteristic
28#1957C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1926—21no—
32#1926—21no—
33#1925—21no—
34#1919—21no—
35#1879—21no—
36#1864—21no—
37#1864—21no—
38#1862—21no—
39#1844C2211yescharacteristic
40#1802C2211yescharacteristic
41#1802C2211yescharacteristic
42#1776—21no—
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48#1762—21no—
49C4×#102—21no—
50#1753—21no—
51#1753—21no—
52#1749—21no—
53#1744C2211yescharacteristic
54#1728—21no—
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—
87C4×D8—44no—
88#436—22no—
89#436—22no—
90#436—22no—
91#436C2×C414yescomplemented
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
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×#102—21no—
112C2×#102—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—21no—
123#390—21no—
124#390—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C43C2311yescharacteristic
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#336D411yescharacteristic
155#336D411yescharacteristic
156#335—22no—
157#335C2×C414yescomplemented
158#333—22no—
159#333C2×C414yescomplemented
160#331—22no—
161#331C2×C414yescomplemented
162C22×Q8—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C2×D411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—41no—
171C2×D8—24no—
172C2×D8—42no—
173C2×D8—42no—
174C2×M16—24no—
175C2×M16—24no—
176C2×M16—81no—
177C2×M16—81no—
178C2×M16—81no—
179C2×M16—81no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C22×C412yes—
194#127—21no—
195#127—21no—
196#127—21no—
197#127C22×C412yes—
198C4×Q8—82no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—81no—
208C4×D4—81no—
209C4×D4—82no—
210C4×D4C22×C412yes—
211C4×D4#4112yes—
212C2×C4⋊C4—21no—
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—41no—
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—41no—
234C2×C42—81no—
235C2×C42—81no—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42C2411yescharacteristic
243#107—42no—
244#107—42no—
245#106—24no—
246#106—81no—
247#106—81no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8C22×C412yes—
257C4⋊C8C22×C412yes—
258C4⋊C8C22×C412yes—
259C4⋊C8#4112yes—
260#102—24no—
261#102—24no—
262#102—24no—
263#102—24no—
264#102—42no—
265#102—42no—
266#102—42no—
267#102—42no—
268#102—81no—
269#102—81no—
270C22⋊C8—24no—
271C22⋊C8—24no—
272#97—42no—
273#97—81no—
274#97—81no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—42no—
278C4×C8—81no—
279C4×C8C22×C412yes—
280C4×C8#4112yes—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—41no—
285#95—41no—
286C24—21no—
287C24—41no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—81no—
291C2×Q8—81no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
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—81no—
312C2×D4—81no—
313C2×D4C4×D412yes—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4111yescharacteristic
342D8—48no—
343M16—84no—
344M16—84no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—41no—
360C2×C8—41no—
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×C8C4×D412yes—
374C2×C8C4×D412yes—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—81no—
393C4⋊C4—81no—
394C4⋊C4—82no—
395C4⋊C4C4×D412yes—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—81no—
414C42—81no—
415C42—82no—
416C42AGL(1,ℤ/8)12yes—
417C42#13712yes—
418C42C23×C411yescharacteristic
419C42C2×#4111yescharacteristic
420C42C2×#4111yescharacteristic
421C42C2×#4111yescharacteristic
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23C2×#12111yescharacteristic
437Q8—82no—
438Q8—82no—
439D4—22no—
440D4—22no—
441D4—41no—
442D4—41no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—82no—
448D4—82no—
449D4—81no—
450D4—81no—
451D4—81no—
452D4—81no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—164no—
490C2×C4#46812yes—
491C2×C4#46812yes—
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C2×C4×D411yescharacteristic
494C2×C4C4×#4111yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4#52911yescharacteristic
497C2×C4#53211yescharacteristic
498C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
499C2×C4C2×#13711yescharacteristic
500C2×C4#57511yescharacteristic
501C2×C4#57511yescharacteristic
502C2×C4#57611yescharacteristic
503C2×C4#57611yescharacteristic
504C2×C4#57711yescharacteristic
505C2×C4#57811yescharacteristic
506C8—22no—
507C8—22no—
508C8—41no—
509C8—42no—
510C8—41no—
511C8—42no—
512C8—42no—
513C8—42no—
514C8—81no—
515C8—81no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22—161no—
530C22C4×#12111yescharacteristic
531C22#304111yescharacteristic
532C22#304611yescharacteristic
533C22C2×#46811yescharacteristic
534C22#305411yescharacteristic
535C22#306811yescharacteristic
536C22#306811yescharacteristic
537C4—21no—
538C4—21no—
539C4—22no—
540C4—22no—
541C4—42no—
542C4—41no—
543C4—41no—
544C4—82no—
545C4—81no—
546C4—81no—
547C4—161no—
548C4—161no—
549C4C4×AGL(1,ℤ/8)11yescharacteristic
550C4C4×#13711yescharacteristic
551C4#294611yescharacteristic
552C4#295111yescharacteristic
553C4#311811yescharacteristic
554C4#312411yescharacteristic
555C4#317111yescharacteristic
556C4#317211yescharacteristic
557C2—82no—
558C2—162no—
559C2#1942011yescharacteristic
560C2#1942511yescharacteristic
561C2#1945011yescharacteristic
562C2#1945511yescharacteristic
563C2#2150911yescharacteristic
564C2C4×#46811yescharacteristic
565C2#2155511yescharacteristic
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. #22219C2semidirect
2. #21614C2semidirect
4. #20568C2semidirect
6. #9747C2semidirect
7. #9617C2semidirect
8. #9576C2semidirect
9. #9571C2semidirect
11. #9223C2semidirect
12. #8650C2semidirect
13. #8278C2semidirect
14. #8234C2semidirect
15. C4×#333C2semidirect
55. #1697C4semidirect
82. #486C2×C4semidirect
84. #476C2×C4semidirect
91. #436C2×C4semidirect
157. #335C2×C4semidirect
159. #333C2×C4semidirect
161. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222192
#216142
C4×#4592
#205682
C2×#16972
#97472
#96172
#95762
#95712
#92292
#92232
#86502
#82782
#82342
C4×#3332