Finite Group Databasebeta

← #206697

Subgroups — #206697

Order 512.

631 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22012C211yesmaximal, characteristic, complemented
2#22009C211yesmaximal, characteristic, complemented
3#21118C211yesmaximal, characteristic
4#9665C211yesmaximal, characteristic
5#9657C211yesmaximal, characteristic
6#9600C211yesmaximal, characteristic, complemented
7#9593C211yesmaximal, characteristic, complemented
8#9560C211yesmaximal, characteristic, complemented
9#9560C211yesmaximal, characteristic, complemented
10#9554C211yesmaximal, characteristic, complemented
11#9554C211yesmaximal, characteristic, complemented
12#8279C211yesmaximal, characteristic, complemented
13#8177C211yesmaximal, characteristic, complemented
14C4×D4⋊C8C211yesmaximal, characteristic, complemented
15#7841C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17#2293C2211yescharacteristic
18C42×D4C2211yescharacteristic
19#1954C2211yescharacteristic
20#1954C2211yescharacteristic
21#1954C2211yescharacteristic
22#1954C2211yescharacteristic
23#1939C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1926C411yescharacteristic
28#1926C411yescharacteristic
29#1924—21no—
30#1924—21no—
31#1923—21no—
32#1923—21no—
33#1921C411yescharacteristic
34#1921C411yescharacteristic
35#1879C411yescharacteristic
36#1879C411yescharacteristic
37#1864C411yescharacteristic
38#1864C411yescharacteristic
39#1818—21no—
40#1817—21no—
41#1798—21no—
42#1797—21no—
43C4×#106C2211yescharacteristic
44#1767C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59C2×C82C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C22×C42—21no—
74C4×D8—22no—
75C4×D8—22no—
76C2×#107C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#102C2×C411yescharacteristic
88C2×#102C2×C411yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#390C2×C411yescharacteristic
112#390C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#388—21no—
116#388—21no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C43C2311yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336C2×C411yescharacteristic
139#336C2×C411yescharacteristic
140#336C2×C411yescharacteristic
141#336C2×C411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#336D411yescharacteristic
146#335—22no—
147#334—22no—
148D4⋊C8—22no—
149D4⋊C8—22no—
150D4⋊C8—22no—
151D4⋊C8—22no—
152#322—22no—
153#322—22no—
154C82—22no—
155C82—22no—
156C22×D4C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C2×D8—22no—
163C2×D8—22no—
164C2×D8—41no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—41no—
190C4×D4—41no—
191#117—81no—
192#117—81no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—81no—
214C2×C42—81no—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—22no—
223#106—22no—
224#106—41no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8—81no—
237#102—22no—
238#102—22no—
239#102—22no—
240#102—22no—
241#102—41no—
242#102—41no—
243C22⋊C8—81no—
244C22⋊C8—81no—
245C22⋊C8—81no—
246C22⋊C8—81no—
247C22⋊C8—81no—
248C22⋊C8—81no—
249C22⋊C8—81no—
250C22⋊C8—81no—
251#97—81no—
252C4×C8—24no—
253C4×C8—24no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—81no—
263C4×C8#4112yes—
264C4×C8#4112yes—
265C4×C8#4112yes—
266C4×C8#4112yes—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273C24—21no—
274C24—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4#13112yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318D8—44no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
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#11712yes—
360C2×C8#11712yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4#13112yes—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—81no—
384C22⋊C4—81no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—22no—
394C42—22no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—81no—
400C42—81no—
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C23—22no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23C2×C4×D411yescharacteristic
420D4—22no—
421D4—22no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—41no—
428C2×C4—21no—
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—21no—
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—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—84no—
459C2×C4—84no—
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#44312yes—
469C2×C4#44312yes—
470C2×C4C2×#11711yescharacteristic
471C2×C4C4×#4111yescharacteristic
472C2×C4#52911yescharacteristic
473C2×C4#52911yescharacteristic
474C2×C4#53211yescharacteristic
475C2×C4#53211yescharacteristic
476C2×C4C2×#13111yescharacteristic
477C2×C4C2×#13111yescharacteristic
478C2×C4#56811yescharacteristic
479C2×C4#56811yescharacteristic
480C2×C4#57611yescharacteristic
481C2×C4#57611yescharacteristic
482C2×C4#57711yescharacteristic
483C2×C4#57711yescharacteristic
484C8—24no—
485C8—24no—
486C8—22no—
487C8—22no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—81no—
495C8—81no—
496C8—81no—
497C8—81no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22#229311yescharacteristic
513C22#293311yescharacteristic
514C22#293311yescharacteristic
515C22#294611yescharacteristic
516C22#294611yescharacteristic
517C22C2×#44311yescharacteristic
518C22C2×#44311yescharacteristic
519C4—21no—
520C4—21no—
521C4—22no—
522C4—22no—
523C4—42no—
524C4—41no—
525C4—41no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4#292411yescharacteristic
532C4#292511yescharacteristic
533C4#296611yescharacteristic
534C4#296711yescharacteristic
535C4#317311yescharacteristic
536C4#317611yescharacteristic
537C4#317911yescharacteristic
538C4#317911yescharacteristic
539C2—82no—
540C2—161no—
541C2#1801411yescharacteristic
542C2#1801411yescharacteristic
543C2#1801511yescharacteristic
544C2#1801811yescharacteristic
545C2#2127311yescharacteristic
546C2#2129011yescharacteristic
547C2#2129211yescharacteristic
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 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22012C2semidirect
2. #22009C2semidirect
6. #9600C2semidirect
7. #9593C2semidirect
8. #9560C2semidirect
9. #9560C2semidirect
10. #9554C2semidirect
11. #9554C2semidirect
12. #8279C2semidirect
13. #8177C2semidirect
14. C4×D4⋊C8C2semidirect
15. #7841C2semidirect

Maximal subgroups

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

subgroupindex
#220122
#220092
#211182
#96652
#96572
#96002
#95932
#95602
#95542
#82792
#81772
C4×D4⋊C82
#78412