Finite Group Databasebeta

← #225912

Subgroups — #225912

Order 512.

716 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22721C211yesmaximal, characteristic, complemented
2D4×C4⋊C8C211yesmaximal, characteristic
3#21117C211yesmaximal, characteristic
4C2×#1585C211yesmaximal, characteristic
5C2×SD16⋊C8C211yesmaximal, characteristic
6#9600C211yesmaximal, characteristic, complemented
7#9596C211yesmaximal, characteristic, complemented
8#8848C211yesmaximal, characteristic, complemented
9#8844C211yesmaximal, characteristic, complemented
10#8843C211yesmaximal, characteristic
11#8843C211yesmaximal, characteristic
12#8810C211yesmaximal, characteristic, complemented
13#8568C211yesmaximal, characteristic, complemented
14#8563C211yesmaximal, characteristic
15#8265C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C22×C4⋊C8—21no—
22#2295C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1940—21no—
25#1939—21no—
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1927C411yescharacteristic
31#1927C411yescharacteristic
32#1920C411yescharacteristic
33#1920C411yescharacteristic
34#1877C411yescharacteristic
35#1877C411yescharacteristic
36#1864C411yescharacteristic
37#1864C411yescharacteristic
38#1850—21no—
39#1850—21no—
40#1843—21no—
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1797—21no—
44C4×#106C2211yescharacteristic
45#1768—21no—
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51#1585—24no—
52SD16⋊C8—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C23×C8—21no—
71C23×C8—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C22×C42—21no—
77#446—22no—
78#446—41no—
79C8×Q8—22no—
80C8×Q8—41no—
81C4×SD16D414yes—
82C4×SD16Q814yes—
83C8×D4—22no—
84C8×D4—22no—
85C8×D4—41no—
86C8×D4—41no—
87C2×#106C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—44no—
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#387—21no—
129#387—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138#376—21no—
139C43C2311yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336C2×C411yescharacteristic
152#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154#336C2×C411yescharacteristic
155#336D411yescharacteristic
156#336D411yescharacteristic
157#335—22no—
158#335—22no—
159#335—41no—
160#335—41no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165#322—22no—
166#322—22no—
167C22×Q8C22×C411yescharacteristic
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C2×SD16—22no—
173C2×SD16—22no—
174C2×SD16C4⋊C414yes—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—81no—
186C22×C8—81no—
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8C2×D412yes—
198C4×Q8C2×Q812yes—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—41no—
205C4×D4C2×D412yes—
206C4×D4C2×Q812yes—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#106—22no—
234#106—41no—
235#106C4⋊C414yes—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253C4⋊C8—81no—
254C4⋊C8—81no—
255C4⋊C8—81no—
256C4⋊C8—81no—
257#103—22no—
258#103—41no—
259#103C4⋊C414yes—
260#102—22no—
261#102—22no—
262#102C4⋊C414yes—
263C22⋊C8—42no—
264C22⋊C8—42no—
265#97—42no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—42no—
277C4×C8C2×D412yes—
278C4×C8C2×Q812yes—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289C24—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8C2×C4⋊C412yes—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4C2×C4⋊C412yes—
300C2×D4#13112yes—
301C2×D4#13112yes—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—81no—
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328SD16—28no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
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—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—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—84no—
370C2×C8—84no—
371C2×C8C2×C4⋊C412yes—
372C2×C8C2×C4⋊C412yes—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—42no—
385C4⋊C4—81no—
386C4⋊C4C2×C4⋊C412yes—
387C4⋊C4C2×C4⋊C412yes—
388C4⋊C4#13112yes—
389C22⋊C4—24no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—42no—
393C42—22no—
394C42—22no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42—42no—
409C42AGL(1,ℤ/8)12yes—
410C42#13712yes—
411C42C22×D411yescharacteristic
412C42C22×Q811yescharacteristic
413C42H(5,𝔽2)11yescharacteristic
414C422−1+411yescharacteristic
415C23—22no—
416C23—22no—
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23C2×C4×D411yescharacteristic
423Q8—22no—
424Q8—22no—
425Q8—41no—
426Q8—41no—
427D4—22no—
428D4—22no—
429D4—22no—
430D4#42414yes—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—48no—
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—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
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—84no—
470C2×C4#44012yes—
471C2×C4#44012yes—
472C2×C4C22×C4⋊C411yescharacteristic
473C2×C4#52011yescharacteristic
474C2×C4D4211yescharacteristic
475C2×C4#54711yescharacteristic
476C2×C4D4×Q811yescharacteristic
477C2×C4#55411yescharacteristic
478C2×C4C2×#13111yescharacteristic
479C2×C4C2×#13111yescharacteristic
480C2×C4C2×#13111yescharacteristic
481C2×C4#56811yescharacteristic
482C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
483C2×C4C2×#13711yescharacteristic
484C2×C4#57611yescharacteristic
485C2×C4#57811yescharacteristic
486C8—22no—
487C8—22no—
488C8—42no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—81no—
496C8—81no—
497C8—81no—
498C8—81no—
499C8—81no—
500C22—24no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22D4×C4⋊C411yescharacteristic
508C22#292911yescharacteristic
509C22#293311yescharacteristic
510C22C2×#44011yescharacteristic
511C22#295111yescharacteristic
512C22#295511yescharacteristic
513C22#295611yescharacteristic
514C4—22no—
515C4—21no—
516C4—21no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4C2×#42411yescharacteristic
528C4#290611yescharacteristic
529C4#299211yescharacteristic
530C4#299411yescharacteristic
531C4#327411yescharacteristic
532C4#327611yescharacteristic
533C4#338511yescharacteristic
534C4#338611yescharacteristic
535C2—44no—
536C2#1790511yescharacteristic
537C2#1792611yescharacteristic
538C2#1800011yescharacteristic
539C2#1802711yescharacteristic
540C2#2264711yescharacteristic
541C2#2267111yescharacteristic
542C2#2268711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22721C2semidirect
6. #9600C2semidirect
7. #9596C2semidirect
8. #8848C2semidirect
9. #8844C2semidirect
12. #8810C2semidirect
13. #8568C2semidirect
15. #8265C2semidirect

Maximal subgroups

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

subgroupindex
#227212
D4×C4⋊C82
#211172
C2×#15852
C2×SD16⋊C82
#96002
#95962
#88482
#88442
#88432
#88102
#85682
#85632
#82652