Finite Group Databasebeta

← #229654

Subgroups — #229654

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23205C211yesmaximal, characteristic, complemented
2D4×C4⋊C8C211yesmaximal, characteristic, complemented
3C4×#463C211yesmaximal, characteristic
4C2×#1586C211yesmaximal, characteristic
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#9646C211yesmaximal, characteristic, complemented
7#9605C211yesmaximal, characteristic, complemented
8#9559C211yesmaximal, characteristic, complemented
9#9554C211yesmaximal, characteristic, complemented
10#8947C211yesmaximal, characteristic
11#8945C211yesmaximal, characteristic, complemented
12#8849C211yesmaximal, characteristic, complemented
13#8846C211yesmaximal, characteristic, complemented
14#8262C211yesmaximal, characteristic, complemented
15#8172C211yesmaximal, characteristic, complemented
16C2×#463C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C22×C4⋊C8—21no—
24C4×#124C2211yescharacteristic
25C42×D4C2211yescharacteristic
26#1984C2211yescharacteristic
27#1983C411yescharacteristic
28#1954C2211yescharacteristic
29#1949C411yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1928C411yescharacteristic
33#1926C411yescharacteristic
34#1924—21no—
35#1923—21no—
36#1879C411yescharacteristic
37#1877C411yescharacteristic
38#1851—21no—
39#1850—21no—
40#1850—21no—
41#1842C2211yescharacteristic
42#1804C411yescharacteristic
43#1797—21no—
44#1767C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1753—21no—
50C4×#97C2211yescharacteristic
51#1586—24no—
52C8×SD16—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74C2×#124C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C22×C42—21no—
81#463—24no—
82C8×Q8—22no—
83C8×Q8—41no—
84C4×SD16—22no—
85C4×SD16—22no—
86C4×D8—22no—
87C4×D8—22no—
88#436—22no—
89#436—41no—
90C8×D4—22no—
91C8×D4—22no—
92C8×D4—41no—
93C8×D4—41no—
94C2×#107C2×C411yescharacteristic
95C2×#106C2×C411yescharacteristic
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⋊C8C2×C411yescharacteristic
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#391C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#388—21no—
136#388—21no—
137#382C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C43C2311yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336C2×C411yescharacteristic
157#336C2×C411yescharacteristic
158#336D411yescharacteristic
159#336D411yescharacteristic
160#334—22no—
161#334—22no—
162Q8⋊C8—22no—
163Q8⋊C8—41no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—41no—
167D4⋊C8D414yes—
168#322—22no—
169C82—22no—
170C22×Q8C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C22×D4C22×C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C2×SD16—22no—
178C2×SD16—22no—
179C2×SD16M1614yescomplemented
180C2×D8—22no—
181C2×D8—22no—
182C2×D8—22no—
183C2×M16—42no—
184C2×M16—42no—
185C2×M16—81no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—42no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—81no—
197C22×C8—81no—
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206#124—22no—
207#124—41no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—41no—
222C4×D4C2×D412yes—
223C4×D4#4112yes—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42C2411yescharacteristic
251#107—22no—
252#106—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C4⋊C8—81no—
269C4⋊C8C2×D412yes—
270#103—22no—
271#103—22no—
272#103—22no—
273#102—22no—
274#102—22no—
275#102—22no—
276#102—22no—
277#102—22no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—81no—
283#97—24no—
284#97—42no—
285#97—81no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—42no—
298C4×C8C2×D412yes—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306C24—21no—
307C24—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8C2×M1612yes—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4C4×D412yes—
324C2×D4C2×M1612yes—
325C2×D4#13112yes—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—81no—
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4111yescharacteristic
357SD16—28no—
358D8—44no—
359M16—84no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—22no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—84no—
401C2×C8—84no—
402C2×C8C2×M1612yes—
403C2×C8AGL(1,ℤ/8)12yes—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4C4×D412yes—
415C4⋊C4#13112yes—
416C22⋊C4—24no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—42no—
425C22⋊C4—81no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—22no—
435C42—41no—
436C42—42no—
437C42—42no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42C22×D411yescharacteristic
443C42C2×#4111yescharacteristic
444C42H(5,𝔽2)11yescharacteristic
445C422−1+411yescharacteristic
446C23—22no—
447C23—22no—
448C23—22no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23C2×C4×D411yescharacteristic
461Q8—22no—
462Q8—22no—
463Q8—41no—
464Q8—41no—
465D4—22no—
466D4—22no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4#43514yes—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—48no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—84no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4#44312yes—
515C2×C4C2×C4×D411yescharacteristic
516C2×C4#52011yescharacteristic
517C2×C4D4211yescharacteristic
518C2×C4#54711yescharacteristic
519C2×C4#54811yescharacteristic
520C2×C4#55311yescharacteristic
521C2×C4C22×M1611yescharacteristic
522C2×C4C2×#13111yescharacteristic
523C2×C4C2×#13111yescharacteristic
524C2×C4#56811yescharacteristic
525C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
526C2×C4#57511yescharacteristic
527C2×C4#57711yescharacteristic
528C2×C4#57711yescharacteristic
529C8—24no—
530C8—22no—
531C8—22no—
532C8—42no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—42no—
540C8—81no—
541C8—81no—
542C8—81no—
543C22—24no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#236111yescharacteristic
558C22#292911yescharacteristic
559C22D4×M1611yescharacteristic
560C22C4×AGL(1,ℤ/8)11yescharacteristic
561C22#295211yescharacteristic
562C22C2×#44311yescharacteristic
563C22#295611yescharacteristic
564C4—21no—
565C4—21no—
566C4—22no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4C2×#43511yescharacteristic
577C4#293511yescharacteristic
578C4#299011yescharacteristic
579C4#299211yescharacteristic
580C4#326511yescharacteristic
581C4#326711yescharacteristic
582C4#337311yescharacteristic
583C4#337611yescharacteristic
584C2—44no—
585C2—161no—
586C2#1791011yescharacteristic
587C2#1792711yescharacteristic
588C2#1804811yescharacteristic
589C2#1805711yescharacteristic
590C2#2298711yescharacteristic
591C2#2300711yescharacteristic
592C2#2301511yescharacteristic
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 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23205C2semidirect
2. D4×C4⋊C8C2semidirect
5. C2×C8×SD16C2semidirect
6. #9646C2semidirect
7. #9605C2semidirect
8. #9559C2semidirect
9. #9554C2semidirect
11. #8945C2semidirect
12. #8849C2semidirect
13. #8846C2semidirect
14. #8262C2semidirect
15. #8172C2semidirect
179. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
#232052
D4×C4⋊C82
C4×#4632
C2×#15862
C2×C8×SD162
#96462
#96052
#95592
#95542
#89472
#89452
#88492
#88462
#82622
#81722