Finite Group Databasebeta

← #278024

Subgroups — #278024

Order 512.

676 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33207C211yesmaximal, characteristic
2#24702C211yesmaximal, characteristic
3#24652C211yesmaximal, characteristic
4C8×C22⋊Q8C211yesmaximal, characteristic
5C2×#1672C211yesmaximal, characteristic, complemented
6C2×#1564C211yesmaximal, characteristic
7#10594C211yesmaximal, characteristic, complemented
8#10593C211yesmaximal, characteristic, complemented
9#10483C211yesmaximal, characteristic, complemented
10#10478C211yesmaximal, characteristic, complemented
11#9729C211yesmaximal, characteristic, complemented
12#9611C211yesmaximal, characteristic, complemented
13#9132C211yesmaximal, characteristic
14#9128C211yesmaximal, characteristic
15#8643C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#462C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26#2076—21no—
27#2036—21no—
28#2036—21no—
29#2001—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1987C2211yescharacteristic
33#1984C2211yescharacteristic
34#1981C2211yescharacteristic
35#1969—21no—
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1942—21no—
39#1941—21no—
40#1941—21no—
41#1939C2211yescharacteristic
42#1933—21no—
43#1933—21no—
44#1916—21no—
45#1850—21no—
46#1844C2211yescharacteristic
47#1807—21no—
48#1807—21no—
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52C8×C4⋊C4—21no—
53C8×C4⋊C4C2211yescharacteristic
54C8×C22⋊C4—21no—
55#1672—24no—
56#1564—24no—
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×#334C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×Q16—21no—
70C23×C8D411yescharacteristic
71C23×C8D411yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8D411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#498—22no—
80#498—22no—
81#477—22no—
82#477—41no—
83#462—22no—
84#462—41no—
85C8×Q8—22no—
86C8×Q8—22no—
87#431—22no—
88#431—22no—
89#431—41no—
90#431—41no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×#106—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8D411yescharacteristic
122C2×C4×C8D411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391—21no—
126#391—21no—
127#391D411yescharacteristic
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389D411yescharacteristic
133#387—21no—
134#385—21no—
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#380—21no—
138#380—21no—
139#380—21no—
140#380—21no—
141#380D411yescharacteristic
142C4×C4⋊C4—21no—
143C4×C4⋊C4C2311yescharacteristic
144C4×C22⋊C4—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#335—22no—
161#335—22no—
162#334—22no—
163#334D414yes—
164#333—22no—
165#333—41no—
166#332—22no—
167#332—41no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174C2×Q16—42no—
175C2×Q16—42no—
176C2×Q16—81no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—41no—
188C22×C8Q1614yescomplemented
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—81no—
204C22⋊Q8—81no—
205C22⋊Q8—82no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
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—21no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42C2411yescharacteristic
242#107—42no—
243#107—42no—
244#107—81no—
245#106—42no—
246#106—42no—
247#106—42no—
248#106—42no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8C2×D412yes—
262C4⋊C8C2×D412yes—
263#103—42no—
264#103—42no—
265#103—42no—
266#103—42no—
267#103—81no—
268#103—81no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—42no—
277C4×C8—42no—
278C4×C8—42no—
279C4×C8—42no—
280C4×C8C2×D412yes—
281C4×C8#4112yes—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—41no—
288#95—41no—
289#95—41no—
290#95—41no—
291C24#12111yescharacteristic
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—81no—
306C2×Q8—81no—
307C2×Q8#13512yes—
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—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4C2×Q1612yes—
334C22×C4#13712yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338Q16—84no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—28no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—44no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
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#12112yes—
367C2×C8#12112yes—
368C2×C8C2×Q1612yes—
369C2×C8C2×Q1612yes—
370C2×C8#13512yes—
371C4⋊C4—24no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
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—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4#13512yes—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C42H(5,𝔽2)11yescharacteristic
418C422−1+411yescharacteristic
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23#46212yes—
426C23C2×#12111yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432Q8—81no—
433Q8—81no—
434Q8—81no—
435Q8—161no—
436Q8—161no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—42no—
453C2×C4—42no—
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—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4#46212yes—
481C2×C4#47112yes—
482C2×C4C2×#12111yescharacteristic
483C2×C4#53711yescharacteristic
484C2×C4D4211yescharacteristic
485C2×C4#54711yescharacteristic
486C2×C4#54811yescharacteristic
487C2×C4#55311yescharacteristic
488C2×C4C22×Q1611yescharacteristic
489C2×C4C2×#13511yescharacteristic
490C2×C4C2×#13511yescharacteristic
491C2×C4C2×#13511yescharacteristic
492C2×C4C2×#13711yescharacteristic
493C2×C4#57511yescharacteristic
494C2×C4#57711yescharacteristic
495C2×C4#57711yescharacteristic
496C8—22no—
497C8—22no—
498C8—42no—
499C8—42no—
500C8—41no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C8#46714yes—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22#241611yescharacteristic
514C22C2×#46211yescharacteristic
515C22#303511yescharacteristic
516C22#305011yescharacteristic
517C22#305711yescharacteristic
518C22C2×#47111yescharacteristic
519C22#306711yescharacteristic
520C4—22no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—42no—
525C4—41no—
526C4—42no—
527C4—81no—
528C4—81no—
529C4—161no—
530C4—161no—
531C4—161no—
532C4—161no—
533C4C2×#46711yescharacteristic
534C4#305611yescharacteristic
535C4#324811yescharacteristic
536C4#324911yescharacteristic
537C4#328211yescharacteristic
538C4#328411yescharacteristic
539C4#333711yescharacteristic
540C4#334011yescharacteristic
541C2—42no—
542C2#1911811yescharacteristic
543C2#1912711yescharacteristic
544C2#1920711yescharacteristic
545C2#1921811yescharacteristic
546C2#2475211yescharacteristic
547C2#2477711yescharacteristic
548C2#2480211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#1672C2semidirect
7. #10594C2semidirect
8. #10593C2semidirect
9. #10483C2semidirect
10. #10478C2semidirect
11. #9729C2semidirect
12. #9611C2semidirect
15. #8643C2semidirect
188. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332072
#247022
#246522
C8×C22⋊Q82
C2×#16722
C2×#15642
#105942
#105932
#104832
#104782
#97292
#96112
#91322
#91282
#86432