Finite Group Databasebeta

← #211304

Subgroups — #211304

Order 512.

815 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22269C211yesmaximal, characteristic
2#21505C211yesmaximal, characteristic, complemented
3C4×#460C211yesmaximal, characteristic, complemented
4#20569C211yesmaximal, characteristic, complemented
5C2×#1668C211yesmaximal, characteristic
6#9734C211yesmaximal, characteristic
7#9616C211yesmaximal, characteristic, complemented
8#9564C211yesmaximal, characteristic, complemented
9#9563C211yesmaximal, characteristic, complemented
10#9222C211yesmaximal, characteristic, complemented
11#9222C211yesmaximal, characteristic, complemented
12#8649C211yesmaximal, characteristic, complemented
13#8278C211yesmaximal, characteristic, complemented
14#8234C211yesmaximal, characteristic, complemented
15C4×#331C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#460C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#440—21no—
22C2×C4×SD16—21no—
23C4×#127C2211yescharacteristic
24C4×#127C2211yescharacteristic
25C42×Q8C2211yescharacteristic
26#1982C2211yescharacteristic
27#1982C2211yescharacteristic
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1926—21no—
33#1925—21no—
34#1880—21no—
35#1880—21no—
36#1879—21no—
37#1879—21no—
38#1879—21no—
39#1879—21no—
40#1834C2211yescharacteristic
41#1803C2211yescharacteristic
42#1803C2211yescharacteristic
43C8×C4⋊C4—21no—
44C8×C4⋊C4—21no—
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1767C2211yescharacteristic
48C4×C4⋊C8—21no—
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51#1762—21no—
52C4×#102—21no—
53#1744—21no—
54#1744C2211yescharacteristic
55#1668C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70C22×D8—21no—
71C22×D8—21no—
72C2×#127C2311yescharacteristic
73C2×#127C2311yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83#493—22no—
84#493C2×C414yes—
85#460—22no—
86#460—22no—
87#460C2×C414yescomplemented
88#460C2×C414yescomplemented
89C8×Q8—22no—
90C8×Q8—22no—
91C8×Q8—22no—
92C8×Q8C2×C414yescomplemented
93#440—44no—
94C4×SD16—44no—
95C2×#107—21no—
96C2×#106—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#390—21no—
127#390—21no—
128#390—21no—
129#390—21no—
130#390—21no—
131#390—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C43—21no—
141C43C2311yescharacteristic
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336D411yescharacteristic
154#336D411yescharacteristic
155#336D411yescharacteristic
156#336D411yescharacteristic
157#335—22no—
158#335C2×C414yescomplemented
159#331—22no—
160#331—22no—
161#331C2×C414yescomplemented
162#331C2×C414yescomplemented
163C22×Q8C2×D411yescharacteristic
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C23×C4—41no—
171C23×C4—41no—
172C2×SD16—24no—
173C2×SD16—24no—
174C2×SD16—42no—
175C2×SD16—42no—
176C2×SD16—42no—
177C2×SD16—42no—
178C2×D8—24no—
179C2×D8—24no—
180C2×D8—81no—
181C2×D8—81no—
182C2×D8—81no—
183C2×D8—81no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194#127—21no—
195#127—21no—
196#127—21no—
197#127—21no—
198#127—21no—
199#127—21no—
200#127C22×C412yes—
201#127C22×C412yes—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8D814yescomplemented
208C4×Q8C22×C412yes—
209C4×Q8#4112yes—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—81no—
215C4×D4—81no—
216C4×D4—81no—
217C4×D4—81no—
218C4×D4—82no—
219C4×D4—82no—
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—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42C2411yescharacteristic
251#107—42no—
252#106—42no—
253C4⋊C8—24no—
254C4⋊C8—24no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8C22×C412yes—
264C4⋊C8C22×C412yes—
265C4⋊C8C22×C412yes—
266C4⋊C8#4112yes—
267#103—42no—
268#103—42no—
269#102—24no—
270#102—24no—
271#102—42no—
272#102—42no—
273#102—42no—
274#102—42no—
275#102—81no—
276#102—81no—
277#102—81no—
278#102—81no—
279#97—42no—
280C4×C8—24no—
281C4×C8—24no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—42no—
285C4×C8—81no—
286C4×C8—81no—
287C4×C8—81no—
288C4×C8C22×C412yes—
289C4×C8#4112yes—
290#95—21no—
291#95—21no—
292#95—21no—
293C24—41no—
294C24—41no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8C4×D412yes—
299C2×Q8C2×D812yes—
300C2×Q8#13512yes—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
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—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—81no—
330C2×D4—81no—
331C2×D4—81no—
332C2×D4—81no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4111yescharacteristic
352SD16—48no—
353SD16—48no—
354D8—84no—
355D8—84no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
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—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—24no—
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—81no—
402C4⋊C4—81no—
403C4⋊C4C4×D412yes—
404C4⋊C4C2×D812yes—
405C4⋊C4#13512yes—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—24no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—81no—
427C42—81no—
428C42C2×D812yes—
429C42#13512yes—
430C42C23×C411yescharacteristic
431C42C2×#4111yescharacteristic
432C42C2×#4111yescharacteristic
433C42C2×#4111yescharacteristic
434C23—42no—
435C23—42no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23C2×#12111yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—22no—
452Q8C4×D814yes—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—82no—
462D4—82no—
463D4—82no—
464D4—82no—
465D4—81no—
466D4—81no—
467D4—81no—
468D4—81no—
469D4—81no—
470D4—81no—
471D4—81no—
472D4—81no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—164no—
504C2×C4—164no—
505C2×C4#46612yes—
506C2×C4#46612yes—
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C4×#4111yescharacteristic
510C2×C4#52511yescharacteristic
511C2×C4#52911yescharacteristic
512C2×C4#53211yescharacteristic
513C2×C4C22×D811yescharacteristic
514C2×C4C2×#13511yescharacteristic
515C2×C4C2×#13511yescharacteristic
516C2×C4C2×#13511yescharacteristic
517C2×C4#57611yescharacteristic
518C2×C4#57711yescharacteristic
519C2×C4#57711yescharacteristic
520C2×C4#57711yescharacteristic
521C8—22no—
522C8—22no—
523C8—41no—
524C8—42no—
525C8—41no—
526C8—42no—
527C8—42no—
528C8—42no—
529C8—81no—
530C8—81no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—161no—
538C22—161no—
539C22—161no—
540C22—161no—
541C22—161no—
542C22—161no—
543C22—161no—
544C22—161no—
545C22C4×#12111yescharacteristic
546C22#304311yescharacteristic
547C22#304511yescharacteristic
548C22C2×#46611yescharacteristic
549C22#305011yescharacteristic
550C22#306811yescharacteristic
551C22#306811yescharacteristic
552C4—24no—
553C4—21no—
554C4—21no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—41no—
559C4—42no—
560C4—41no—
561C4—42no—
562C4—161no—
563C4—161no—
564C4C2×C4×D811yescharacteristic
565C4C4×#13511yescharacteristic
566C4#294611yescharacteristic
567C4#294911yescharacteristic
568C4#311911yescharacteristic
569C4#312611yescharacteristic
570C4#316811yescharacteristic
571C4#316911yescharacteristic
572C2—162no—
573C2—162no—
574C2#1939811yescharacteristic
575C2#1940111yescharacteristic
576C2#1941011yescharacteristic
577C2#1941211yescharacteristic
578C2#2151511yescharacteristic
579C2C4×#46611yescharacteristic
580C2#2155511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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
2. #21505C2semidirect
3. C4×#460C2semidirect
4. #20569C2semidirect
7. #9616C2semidirect
8. #9564C2semidirect
9. #9563C2semidirect
10. #9222C2semidirect
11. #9222C2semidirect
12. #8649C2semidirect
13. #8278C2semidirect
14. #8234C2semidirect
15. C4×#331C2semidirect
55. #1668C4semidirect
87. #460C2×C4semidirect
88. #460C2×C4semidirect
92. C8×Q8C2×C4semidirect
158. #335C2×C4semidirect
161. #331C2×C4semidirect
162. #331C2×C4semidirect
207. C4×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#222692
#215052
C4×#4602
#205692
C2×#16682
#97342
#96162
#95642
#95632
#92222
#86492
#82782
#82342
C4×#3312