Finite Group Databasebeta

← #278951

Subgroups — #278951

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29877C211yesmaximal, characteristic
2#24702C211yesmaximal, characteristic
3#24660C211yesmaximal, characteristic
4#22098C211yesmaximal, characteristic
5C2×#1672C211yesmaximal, characteristic, complemented
6C2×#1572C211yesmaximal, characteristic
7#10601C211yesmaximal, characteristic, complemented
8#10594C211yesmaximal, characteristic, complemented
9#10483C211yesmaximal, characteristic, complemented
10#10478C211yesmaximal, characteristic, complemented
11#9729C211yesmaximal, characteristic, complemented
12#9613C211yesmaximal, characteristic, complemented
13#9136C211yesmaximal, characteristic
14#9132C211yesmaximal, characteristic
15#8647C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#462C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26#2088—21no—
27#2088—21no—
28#2077—21no—
29#2077—21no—
30#2076—21no—
31#2036—21no—
32#2035—21no—
33#2001—21no—
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1987C2211yescharacteristic
37#1984C2211yescharacteristic
38#1981C2211yescharacteristic
39#1971—21no—
40#1948C2211yescharacteristic
41#1947C2211yescharacteristic
42#1942—21no—
43#1939—21no—
44#1939C2211yescharacteristic
45#1935—21no—
46#1933—21no—
47#1923—21no—
48#1923—21no—
49#1916—21no—
50#1834C2211yescharacteristic
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1797C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1672—24no—
56#1572—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—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#498—22no—
78#498—22no—
79#477—22no—
80#477—41no—
81#462—22no—
82#462—41no—
83C8×Q8—22no—
84C8×Q8—22no—
85#432—22no—
86#432—41no—
87#431—22no—
88#431—22no—
89#431—22no—
90#431—41no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8D411yescharacteristic
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×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#391—21no—
125#391D411yescharacteristic
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389D411yescharacteristic
131#387—21no—
132#385—21no—
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#380—21no—
136#380—21no—
137#380—21no—
138#380—21no—
139#380D411yescharacteristic
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
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#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—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—81no—
198C22⋊Q8—81no—
199C22⋊Q8—82no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C2×C4⋊C4—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⋊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—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42C2411yescharacteristic
236#107—42no—
237#107—42no—
238#106—42no—
239#106—42no—
240#106—81no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—21no—
244C4⋊C8—21no—
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⋊C8—42no—
254C4⋊C8—81no—
255C4⋊C8C2×D412yes—
256C4⋊C8C2×D412yes—
257#103—42no—
258#103—42no—
259#103—42no—
260#103—42no—
261#103—81no—
262#103—81no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—42no—
266C22⋊C8—42no—
267C22⋊C8—81no—
268#97—42no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—42no—
272C4×C8—42no—
273C4×C8—81no—
274C4×C8C2×D412yes—
275C4×C8#4112yes—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—41no—
282#95—41no—
283#95—41no—
284#95—41no—
285C24#12111yescharacteristic
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—81no—
300C2×Q8—81no—
301C2×Q8#13512yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
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—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4#12111yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332Q16—84no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
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#12112yes—
362C2×C8#12112yes—
363C4⋊C4—24no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C4⋊C4—81no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4#13512yes—
393C22⋊C4—24no—
394C22⋊C4—81no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42C22×D411yescharacteristic
408C42C2×#4111yescharacteristic
409C42H(5,𝔽2)11yescharacteristic
410C422−1+411yescharacteristic
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23C2×#12111yescharacteristic
419Q8—22no—
420Q8—22no—
421Q8—41no—
422Q8—41no—
423Q8—81no—
424Q8—81no—
425Q8—81no—
426Q8—81no—
427Q8—161no—
428Q8—161no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
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—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4C2×#12111yescharacteristic
475C2×C4#53711yescharacteristic
476C2×C4D4211yescharacteristic
477C2×C4#54711yescharacteristic
478C2×C4#54811yescharacteristic
479C2×C4#55311yescharacteristic
480C2×C4C2×#13511yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4#57711yescharacteristic
483C2×C4#57711yescharacteristic
484C2×C4#57711yescharacteristic
485C2×C4#57811yescharacteristic
486C2×C4#57811yescharacteristic
487C2×C4#57811yescharacteristic
488C8—22no—
489C8—22no—
490C8—42no—
491C8—42no—
492C8—41no—
493C8—41no—
494C8—81no—
495C8—81no—
496C8—81no—
497C8—81no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22#241611yescharacteristic
506C22#304411yescharacteristic
507C22#304611yescharacteristic
508C22#305011yescharacteristic
509C22#305711yescharacteristic
510C22#306811yescharacteristic
511C22#306911yescharacteristic
512C4—21no—
513C4—21no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—41no—
519C4—81no—
520C4—81no—
521C4—161no—
522C4—161no—
523C4—161no—
524C4—161no—
525C4#305911yescharacteristic
526C4#306311yescharacteristic
527C4#325811yescharacteristic
528C4#326111yescharacteristic
529C4#328211yescharacteristic
530C4#328411yescharacteristic
531C4#333711yescharacteristic
532C4#334011yescharacteristic
533C2—81no—
534C2#1940711yescharacteristic
535C2#1941911yescharacteristic
536C2#1952911yescharacteristic
537C2#1954511yescharacteristic
538C2#2476411yescharacteristic
539C2#2477711yescharacteristic
540C2#2480511yescharacteristic
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. #10601C2semidirect
8. #10594C2semidirect
9. #10483C2semidirect
10. #10478C2semidirect
11. #9729C2semidirect
12. #9613C2semidirect
15. #8647C2semidirect

Maximal subgroups

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

subgroupindex
#298772
#247022
#246602
#220982
C2×#16722
C2×#15722
#106012
#105942
#104832
#104782
#97292
#96132
#91362
#91322
#86472