Finite Group Databasebeta

← #211033

Subgroups — #211033

Order 512.

692 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23699C211yesmaximal, characteristic
2#22102C211yesmaximal, characteristic
3#22100C211yesmaximal, characteristic
4C2×#1587C211yesmaximal, characteristic, complemented
5C2×#1450C211yesmaximal, characteristic
6#9656C211yesmaximal, characteristic, complemented
7#9593C211yesmaximal, characteristic, complemented
8#9559C211yesmaximal, characteristic, complemented
9#9559C211yesmaximal, characteristic, complemented
10#9071C211yesmaximal, characteristic
11#9071C211yesmaximal, characteristic
12#8600C211yesmaximal, characteristic
13#8289C211yesmaximal, characteristic, complemented
14#8207C211yesmaximal, characteristic, complemented
15C8×#103C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C4×C22⋊Q8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24#2086C411yescharacteristic
25#2075C411yescharacteristic
26#2066C411yescharacteristic
27#2066C411yescharacteristic
28#2056C411yescharacteristic
29#2035C411yescharacteristic
30#2035C411yescharacteristic
31#2001C411yescharacteristic
32#1984C2211yescharacteristic
33#1984C2211yescharacteristic
34#1940—21no—
35#1939—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1924—21no—
39#1924—21no—
40#1923—21no—
41#1923—21no—
42#1917—21no—
43#1916—21no—
44#1797C2211yescharacteristic
45#1797C2211yescharacteristic
46C4×#106C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#103C2211yescharacteristic
50C4×#103C2211yescharacteristic
51#1587—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58#1450—24no—
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61C2×C82C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C23×C8—21no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#446—22no—
77#446—22no—
78C8×Q8—22no—
79C8×Q8—22no—
80C4×Q16—22no—
81C4×Q16—22no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431—22no—
86C2×#107C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C411yescharacteristic
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8C2×C411yescharacteristic
110C2×C22⋊C8C2×C411yescharacteristic
111C2×C22⋊C8D411yescharacteristic
112C2×C22⋊C8D411yescharacteristic
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#387—21no—
131#387—21no—
132#385—21no—
133#385—21no—
134#380C2×C411yescharacteristic
135#380C2×C411yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#335—22no—
159#334—22no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164#322—22no—
165#322D414yes—
166C82—22no—
167C82—22no—
168C22×Q8C22×C411yescharacteristic
169C22×Q8C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173C2×Q16—22no—
174C2×Q16—22no—
175C2×Q16—41no—
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—24no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—22no—
234#106—22no—
235#106—41no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—81no—
251C4⋊C8—81no—
252#103—22no—
253#103—22no—
254#103—22no—
255#103—22no—
256#103—41no—
257#103—41no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—81no—
263C22⋊C8—81no—
264#97—42no—
265#97—81no—
266C4×C8—24no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—42no—
274C4×C8—81no—
275C4×C8C2×D412yes—
276C4×C8C2×D412yes—
277C4×C8C2×D412yes—
278C4×C8#4112yes—
279C4×C8#4112yes—
280C4×C8#4112yes—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291C24C4×D411yescharacteristic
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8#13112yes—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334Q16—44no—
335C2×C8—216no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
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×C8C4×D412yes—
375C2×C8C4×D412yes—
376C2×C8#12112yes—
377C2×C8#12112yes—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4#13112yes—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42C22×D411yescharacteristic
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C42C2×#4111yescharacteristic
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23#44312yes—
427C23C2×C4×D411yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—41no—
431Q8—41no—
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4#44312yes—
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C4×#4111yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4#52511yescharacteristic
485C2×C4#52911yescharacteristic
486C2×C4#53211yescharacteristic
487C2×C4C2×#13111yescharacteristic
488C2×C4C2×#13111yescharacteristic
489C2×C4#56811yescharacteristic
490C2×C4#56811yescharacteristic
491C2×C4#57711yescharacteristic
492C2×C4#57711yescharacteristic
493C2×C4#57811yescharacteristic
494C2×C4#57811yescharacteristic
495C8—24no—
496C8—24no—
497C8—22no—
498C8—22no—
499C8—22no—
500C8—22no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—81no—
506C8—81no—
507C8—81no—
508C8—81no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22C4×#12111yescharacteristic
517C22#293311yescharacteristic
518C22#293311yescharacteristic
519C22#294711yescharacteristic
520C22#294711yescharacteristic
521C22C2×#44311yescharacteristic
522C22C2×#44311yescharacteristic
523C4—21no—
524C4—21no—
525C4—22no—
526C4—41no—
527C4—42no—
528C4—41no—
529C4—42no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4#293311yescharacteristic
537C4#293511yescharacteristic
538C4#296611yescharacteristic
539C4#296711yescharacteristic
540C4#306311yescharacteristic
541C4#306311yescharacteristic
542C4#317311yescharacteristic
543C4#317711yescharacteristic
544C2—42no—
545C2#1801611yescharacteristic
546C2#1801711yescharacteristic
547C2#1801711yescharacteristic
548C2#1801911yescharacteristic
549C2#2149611yescharacteristic
550C2#2154211yescharacteristic
551C2#2154411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1587C2semidirect
6. #9656C2semidirect
7. #9593C2semidirect
8. #9559C2semidirect
9. #9559C2semidirect
13. #8289C2semidirect
14. #8207C2semidirect
15. C8×#103C2semidirect

Maximal subgroups

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

subgroupindex
#236992
#221022
#221002
C2×#15872
C2×#14502
#96562
#95932
#95592
#90712
#86002
#82892
#82072
C8×#1032