Finite Group Databasebeta

← #211440

Subgroups — #211440

Order 512.

766 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22146C211yesmaximal, characteristic, complemented
2#22145C211yesmaximal, characteristic
3C4×#486C211yesmaximal, characteristic, complemented
4#20485C211yesmaximal, characteristic, complemented
5C2×#1701C211yesmaximal, characteristic
6#9746C211yesmaximal, characteristic, complemented
7#9622C211yesmaximal, characteristic, complemented
8#9566C211yesmaximal, characteristic, complemented
9#9564C211yesmaximal, characteristic, complemented
10#9221C211yesmaximal, characteristic
11#9217C211yesmaximal, characteristic, complemented
12#8656C211yesmaximal, characteristic, complemented
13C4×#334C211yesmaximal, characteristic, complemented
14#8237C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#461C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#405—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C2211yescharacteristic
24#2293C2211yescharacteristic
25#1990C2211yescharacteristic
26#1982C2211yescharacteristic
27#1981C2211yescharacteristic
28#1970—21no—
29#1968—21no—
30#1957C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1934—21no—
34#1934—21no—
35#1934—21no—
36#1933—21no—
37#1880—21no—
38#1879—21no—
39#1865—21no—
40#1862—21no—
41#1834C2211yescharacteristic
42#1802C2211yescharacteristic
43#1802C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46#1767C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×#102—21no—
50C4×#102—21no—
51#1753—21no—
52C8×C22⋊C4—21no—
53#1750—21no—
54C42×C8C2211yescharacteristic
55#1701C418yescomplemented
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
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—
70C22×M16—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×D4C2311yescharacteristic
79C2×#117—21no—
80#486—22no—
81#486C2×C414yes—
82#461—22no—
83#461C2×C414yescomplemented
84#459—22no—
85#459C2×C414yes—
86#436—22no—
87#436C2×C414yescomplemented
88C8×D4—22no—
89C8×D4—22no—
90C2×#107—21no—
91C2×#106—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
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×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112#405—44no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8D411yescharacteristic
121C2×C4×C8D411yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391—21no—
125#390—21no—
126#390—21no—
127#390—21no—
128#390D411yescharacteristic
129#390D411yescharacteristic
130#389—21no—
131#389—21no—
132#389D411yescharacteristic
133#389D411yescharacteristic
134#388—21no—
135#388—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142C43C2311yescharacteristic
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#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#334—22no—
157#334C2×C414yescomplemented
158#332—22no—
159#332C2×C414yescomplemented
160#331—22no—
161#331C2×C414yes—
162C22×Q8—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C2×D411yescharacteristic
167C23×C4—21no—
168C23×C4—41no—
169C2×SD16—24no—
170C2×SD16—81no—
171C2×SD16—81no—
172C2×D8—24no—
173C2×D8—81no—
174C2×D8—81no—
175C2×M16—24no—
176C2×M16—42no—
177C2×M16—42no—
178C22×C8—24no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—42no—
186C22×C8—42no—
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—81no—
200C4×Q8—81no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—81no—
206C4×D4—81no—
207C4×D4—82no—
208C4×D4C22×C412yes—
209C4×D4#4112yes—
210#117—42no—
211#117—42no—
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—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C42—21no—
235C2×C42—41no—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42C2411yescharacteristic
243#107—81no—
244#106—81no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8C22×C412yes—
254C4⋊C8C22×C412yes—
255C4⋊C8C22×C412yes—
256C4⋊C8#4112yes—
257#103—24no—
258#103—81no—
259#103—81no—
260#102—24no—
261#102—24no—
262#102—24no—
263#102—42no—
264#102—42no—
265#102—81no—
266#102—81no—
267#102—81no—
268#102—81no—
269C22⋊C8—24no—
270C22⋊C8—24no—
271#97—42no—
272#97—42no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—41no—
279C4×C8—42no—
280C4×C8—81no—
281C4×C8—81no—
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—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
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—41no—
315C2×D4—41no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4C4×D412yes—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341SD16—84no—
342D8—84no—
343M16—48no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—48no—
358C2×C8—41no—
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×C8C4×D412yes—
373C2×C8C4×D412yes—
374C2×C8#13512yes—
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—42no—
385C4⋊C4—42no—
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—81no—
395C4⋊C4—81no—
396C4⋊C4—82no—
397C4⋊C4C4×D412yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—22no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—82no—
418C42#13512yes—
419C42AGL(1,ℤ/8)12yes—
420C42C23×C411yescharacteristic
421C42C2×#4111yescharacteristic
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23C2×#12111yescharacteristic
439Q8—42no—
440Q8—42no—
441Q8—81no—
442Q8—81no—
443D4—22no—
444D4—22no—
445D4—41no—
446D4—41no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—82no—
452D4—82no—
453D4—81no—
454D4—81no—
455D4—81no—
456D4—81no—
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—21no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—164no—
491C2×C4#46312yes—
492C2×C4#46312yes—
493C2×C4#47112yes—
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C4×#4111yescharacteristic
497C2×C4#52511yescharacteristic
498C2×C4#52911yescharacteristic
499C2×C4#53211yescharacteristic
500C2×C4C2×#13511yescharacteristic
501C2×C4C2×#13511yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4#57511yescharacteristic
504C2×C4#57611yescharacteristic
505C2×C4#57611yescharacteristic
506C2×C4#57711yescharacteristic
507C2×C4#57711yescharacteristic
508C8—22no—
509C8—22no—
510C8—22no—
511C8—41no—
512C8—42no—
513C8—42no—
514C8—42no—
515C8—42no—
516C8—81no—
517C8—81no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—161no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22C4×#12111yescharacteristic
533C22C2×#46311yescharacteristic
534C22#303511yescharacteristic
535C22#305811yescharacteristic
536C22#306011yescharacteristic
537C22C2×#47111yescharacteristic
538C22#306711yescharacteristic
539C4—22no—
540C4—22no—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—41no—
545C4—42no—
546C4—81no—
547C4—82no—
548C4—81no—
549C4—161no—
550C4—161no—
551C4C4×#13511yescharacteristic
552C4C4×AGL(1,ℤ/8)11yescharacteristic
553C4#294611yescharacteristic
554C4#294611yescharacteristic
555C4#310211yescharacteristic
556C4#310511yescharacteristic
557C4#317911yescharacteristic
558C4#317911yescharacteristic
559C2—82no—
560C2—162no—
561C2#1919811yescharacteristic
562C2#1920011yescharacteristic
563C2#1921911yescharacteristic
564C2#1922111yescharacteristic
565C2C4×#46311yescharacteristic
566C2#2153611yescharacteristic
567C2C4×#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. #22146C2semidirect
3. C4×#486C2semidirect
4. #20485C2semidirect
6. #9746C2semidirect
7. #9622C2semidirect
8. #9566C2semidirect
9. #9564C2semidirect
11. #9217C2semidirect
12. #8656C2semidirect
13. C4×#334C2semidirect
14. #8237C2semidirect
15. #8234C2semidirect
55. #1701C4semidirect
83. #461C2×C4semidirect
87. #436C2×C4semidirect
157. #334C2×C4semidirect
159. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#221462
#221452
C4×#4862
#204852
C2×#17012
#97462
#96222
#95662
#95642
#92212
#92172
#86562
C4×#3342
#82372
#82342