Finite Group Databasebeta

← #218480

Subgroups — #218480

Order 512.

706 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#124C211yesmaximal, characteristic, complemented
2C4×#463C211yesmaximal, characteristic
3#21264C211yesmaximal, characteristic, complemented
4#9675C211yesmaximal, characteristic, complemented
5#9663C211yesmaximal, characteristic
6#9661C211yesmaximal, characteristic, complemented
7#9603C211yesmaximal, characteristic, complemented
8#9558C211yesmaximal, characteristic, complemented
9#9557C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic
12#8845C211yesmaximal, characteristic, complemented
13#8844C211yesmaximal, characteristic, complemented
14#8565C211yesmaximal, characteristic, complemented
15#8260C211yesmaximal, characteristic, complemented
16C2×#463C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#431—21no—
22C4×#124C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1984C2211yescharacteristic
25#1983C411yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1954C2211yescharacteristic
29#1949C411yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1928C411yescharacteristic
33#1926C411yescharacteristic
34#1923—21no—
35#1923—21no—
36#1923—21no—
37#1923—21no—
38#1879C411yescharacteristic
39#1877C411yescharacteristic
40#1831C2211yescharacteristic
41#1804C411yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C8×C22⋊C4—21no—
48C8×C22⋊C4—21no—
49C4×C22⋊C8—21no—
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×D8C2×C411yescharacteristic
69C23×C8—21no—
70C2×#124C2×C411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C22×C42—21no—
77#463—24no—
78C8×Q8—41no—
79C8×Q8—41no—
80C4×SD16—41no—
81C4×SD16—41no—
82C4×D8—41no—
83C4×D8—41no—
84C8×D4—41no—
85C8×D4—41no—
86#431—44no—
87C2×#107C2×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⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8D411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#388—21no—
129#388—21no—
130#382C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C43C2311yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154#335—41no—
155#334—41no—
156Q8⋊C8—41no—
157Q8⋊C8—41no—
158D4⋊C8—41no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161D4⋊C8—41no—
162#322—41no—
163#322—41no—
164C22×Q8C22×C411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C22×D4C22×C411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C2×SD16—22no—
172C2×SD16—22no—
173C2×SD16—41no—
174C2×D8—22no—
175C2×D8—22no—
176C2×D8—41no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194#124—22no—
195#124M1614yes—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42C2411yescharacteristic
239#107—41no—
240#106—41no—
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—42no—
252C4⋊C8—42no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255#103—22no—
256#103—22no—
257#103—41no—
258#102—22no—
259#102—22no—
260#102—41no—
261#102—41no—
262#102—41no—
263C22⋊C8—42no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266C22⋊C8—42no—
267C22⋊C8—42no—
268C22⋊C8—42no—
269C22⋊C8—42no—
270C4×C8—22no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—41no—
286C4×C8—42no—
287C4×C8—42no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295C24—21no—
296C24—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8C2×M1612yes—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4C2×M1612yes—
314C2×D4#13112yes—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346SD16—44no—
347D8—44no—
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—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—41no—
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×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—84no—
388C2×C8#13512yes—
389C2×C8#13512yes—
390C2×C8AGL(1,ℤ/8)12yes—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4#13112yes—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—42no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—22no—
420C42—41no—
421C42—42no—
422C42—42no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42C2×M1612yes—
428C42#13112yes—
429C42H(5,𝔽2)11yescharacteristic
430C42H(5,𝔽2)11yescharacteristic
431C42H(5,𝔽2)11yescharacteristic
432C422−1+411yescharacteristic
433C23—22no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23C2×C4×D411yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—41no—
451Q8—41no—
452D4—22no—
453D4—22no—
454D4—22no—
455D4—22no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—41no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—48no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—84no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4#43512yes—
500C2×C4#43512yes—
501C2×C4#44412yes—
502C2×C4#51811yescharacteristic
503C2×C4#52011yescharacteristic
504C2×C4#53511yescharacteristic
505C2×C4#53711yescharacteristic
506C2×C4#54011yescharacteristic
507C2×C4#54211yescharacteristic
508C2×C4C22×M1611yescharacteristic
509C2×C4C2×#13111yescharacteristic
510C2×C4C2×#13111yescharacteristic
511C2×C4C2×#13111yescharacteristic
512C2×C4C2×#13511yescharacteristic
513C2×C4C2×#13511yescharacteristic
514C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
515C2×C4#57511yescharacteristic
516C8—22no—
517C8—22no—
518C8—22no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—81no—
525C8—81no—
526C8—81no—
527C8—81no—
528C8—81no—
529C8—81no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22#232411yescharacteristic
545C22C2×#43511yescharacteristic
546C22#292911yescharacteristic
547C22C4×#13511yescharacteristic
548C22C4×AGL(1,ℤ/8)11yescharacteristic
549C22C2×#44411yescharacteristic
550C22#295511yescharacteristic
551C4—21no—
552C4—21no—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—42no—
557C4—41no—
558C4—42no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4#297411yescharacteristic
564C4#297911yescharacteristic
565C4#299011yescharacteristic
566C4#299411yescharacteristic
567C4#320711yescharacteristic
568C4#321111yescharacteristic
569C4#324711yescharacteristic
570C4#325011yescharacteristic
571C2—82no—
572C2—82no—
573C2#1786311yescharacteristic
574C2#1787111yescharacteristic
575C2#1790411yescharacteristic
576C2#1791411yescharacteristic
577C2#2213511yescharacteristic
578C2#2214911yescharacteristic
579C2#2216611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C8×#124C2semidirect
3. #21264C2semidirect
4. #9675C2semidirect
6. #9661C2semidirect
7. #9603C2semidirect
8. #9558C2semidirect
9. #9557C2semidirect
10. #8948C2semidirect
12. #8845C2semidirect
13. #8844C2semidirect
14. #8565C2semidirect
15. #8260C2semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
C4×#4632
#212642
#96752
#96632
#96612
#96032
#95582
#95572
#89482
#89472
#88452
#88442
#85652
#82602