Finite Group Databasebeta

← #225483

Subgroups — #225483

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22721C211yesmaximal, characteristic, complemented
2#22651C211yesmaximal, characteristic
3#21274C211yesmaximal, characteristic
4C2×#1588C211yesmaximal, characteristic
5C2×#1585C211yesmaximal, characteristic
6#9600C211yesmaximal, characteristic, complemented
7#9600C211yesmaximal, characteristic, complemented
8#8848C211yesmaximal, characteristic, complemented
9#8847C211yesmaximal, characteristic
10#8844C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic
12#8806C211yesmaximal, characteristic, complemented
13#8568C211yesmaximal, characteristic, complemented
14#8567C211yesmaximal, characteristic
15#8261C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#424—21no—
22#2295C2211yescharacteristic
23#2293C2211yescharacteristic
24#1968C411yescharacteristic
25#1968C411yescharacteristic
26#1941C411yescharacteristic
27#1941C411yescharacteristic
28#1940—21no—
29#1939—21no—
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1934C411yescharacteristic
35#1934C411yescharacteristic
36#1933C411yescharacteristic
37#1933C411yescharacteristic
38#1851—21no—
39#1850—21no—
40#1843—21no—
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1798—21no—
44C4×#106C2211yescharacteristic
45#1768—21no—
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#102C2211yescharacteristic
50C42×C8C2211yescharacteristic
51#1588—24no—
52#1585—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
70C22×M16—21no—
71C23×C8—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75#446—22no—
76#446—41no—
77C8×Q8—22no—
78C8×Q8—41no—
79C4×SD16D414yes—
80C4×SD16Q814yes—
81#436—22no—
82#436—41no—
83C8×D4—22no—
84C8×D4—41no—
85C2×#106C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88#424—44no—
89C2×C4⋊C8—21no—
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⋊C8D411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2×C411yescharacteristic
117C2×C4×C8C2×C411yescharacteristic
118C2×C4×C8C2×C411yescharacteristic
119C2×C4×C8C2×C411yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#388—21no—
135#388—21no—
136#387—21no—
137#387—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142#376—21no—
143C43C2311yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
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—
170C2×SD16—41no—
171C2×SD16—41no—
172C2×SD16C4⋊C414yes—
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—81no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8—81no—
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8C2×D412yes—
196C4×Q8C2×Q812yes—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4C2×D412yes—
200C4×D4C2×Q812yes—
201#117—42no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#106—41no—
230#106—41no—
231#106C4⋊C414yes—
232C4⋊C8—24no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—81no—
250C4⋊C8—81no—
251C4⋊C8—81no—
252C4⋊C8—81no—
253#103—22no—
254#103—41no—
255#103C4⋊C414yes—
256#102—22no—
257#102—41no—
258#102C4⋊C414yes—
259C22⋊C8—42no—
260C22⋊C8—42no—
261#97—42no—
262C4×C8—22no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—41no—
274C4×C8—41no—
275C4×C8—41no—
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×D4—21no—
300C2×D4—21no—
301C2×D4C2×C4⋊C412yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322SD16—44no—
323M16—84no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
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—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—41no—
362C2×C8—41no—
363C2×C8—84no—
364C2×C8C2×C4⋊C412yes—
365C2×C8C2×C4⋊C412yes—
366C2×C8#13112yes—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—81no—
383C4⋊C4C2×C4⋊C412yes—
384C4⋊C4C2×C4⋊C412yes—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—81no—
391C42—22no—
392C42—22no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—22no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42AGL(1,ℤ/8)12yes—
406C42#13712yes—
407C42C22×D411yescharacteristic
408C42C22×Q811yescharacteristic
409C42H(5,𝔽2)11yescharacteristic
410C422−1+411yescharacteristic
411C23—22no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23C2×C4×D411yescharacteristic
419Q8—22no—
420Q8—22no—
421Q8—41no—
422Q8—41no—
423D4—22no—
424D4—22no—
425D4—41no—
426D4—41no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
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—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4#44012yes—
464C2×C4#44012yes—
465C2×C4#44312yes—
466C2×C4C22×C4⋊C411yescharacteristic
467C2×C4#52011yescharacteristic
468C2×C4D4211yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4D4×Q811yescharacteristic
471C2×C4#55411yescharacteristic
472C2×C4C2×#13111yescharacteristic
473C2×C4#56811yescharacteristic
474C2×C4#56811yescharacteristic
475C2×C4#56811yescharacteristic
476C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C2×C4#57611yescharacteristic
479C2×C4#57811yescharacteristic
480C8—22no—
481C8—22no—
482C8—22no—
483C8—42no—
484C8—42no—
485C8—41no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—81no—
491C8—81no—
492C8—81no—
493C8—81no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22D4×C4⋊C411yescharacteristic
502C22#293211yescharacteristic
503C22#293511yescharacteristic
504C22C2×#44011yescharacteristic
505C22#295111yescharacteristic
506C22C2×#44311yescharacteristic
507C22#295711yescharacteristic
508C4—22no—
509C4—22no—
510C4—21no—
511C4—21no—
512C4—22no—
513C4—41no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—81no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4#290511yescharacteristic
522C4#290611yescharacteristic
523C4#299211yescharacteristic
524C4#299511yescharacteristic
525C4#327411yescharacteristic
526C4#327611yescharacteristic
527C4#338511yescharacteristic
528C4#338611yescharacteristic
529C2—82no—
530C2#1797911yescharacteristic
531C2#1799211yescharacteristic
532C2#1807111yescharacteristic
533C2#1808711yescharacteristic
534C2#2265211yescharacteristic
535C2#2267111yescharacteristic
536C2#2268811yescharacteristic
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. #9600C2semidirect
8. #8848C2semidirect
10. #8844C2semidirect
12. #8806C2semidirect
13. #8568C2semidirect
15. #8261C2semidirect

Maximal subgroups

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

subgroupindex
#227212
#226512
#212742
C2×#15882
C2×#15852
#96002
#88482
#88472
#88442
#88432
#88062
#85682
#85672
#82612