Finite Group Databasebeta

← #211462

Subgroups — #211462

Order 512.

746 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#486C211yesmaximal, characteristic
2#21506C211yesmaximal, characteristic, complemented
3#21505C211yesmaximal, characteristic
4C4×#446C211yesmaximal, characteristic, complemented
5C2×#1702C211yesmaximal, characteristic
6#9746C211yesmaximal, characteristic
7#9622C211yesmaximal, characteristic, complemented
8#9574C211yesmaximal, characteristic, complemented
9#9572C211yesmaximal, characteristic
10#9230C211yesmaximal, characteristic
11#9226C211yesmaximal, characteristic, complemented
12#8657C211yesmaximal, characteristic, complemented
13#8273C211yesmaximal, characteristic, complemented
14#8236C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic
16C2×#486C2211yescharacteristic
17C2×#462C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#441—21no—
22C2×#440—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#127C2211yescharacteristic
25C42×Q8C2211yescharacteristic
26#1990C2211yescharacteristic
27#1982C2211yescharacteristic
28#1981C2211yescharacteristic
29#1957C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1934—21no—
33#1934—21no—
34#1927—21no—
35#1925—21no—
36#1880—21no—
37#1878—21no—
38#1834C2211yescharacteristic
39#1803C2211yescharacteristic
40#1803C2211yescharacteristic
41#1771—21no—
42#1771—21no—
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45#1767C2211yescharacteristic
46#1767C2211yescharacteristic
47C4×C4⋊C8—21no—
48C4×C4⋊C8C2211yescharacteristic
49#1763—21no—
50#1762—21no—
51C4×#102—21no—
52C4×#102—21no—
53C4×#97—21no—
54C42×C8C2211yescharacteristic
55#1702C418yescomplemented
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×Q16—21no—
69C22×SD16—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79#486—22no—
80#486C2×C414yes—
81#462—22no—
82#462C2×C414yescomplemented
83#460—22no—
84#460C2×C414yes—
85#446—22no—
86#446—22no—
87#446—22no—
88#446C2×C414yescomplemented
89#441—44no—
90#440—44no—
91C2×#107—21no—
92C2×#106—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×#97—21no—
119C2×C4×C8—21no—
120C2×C4×C8D411yescharacteristic
121C2×C4×C8D411yescharacteristic
122C2×C4×C8D411yescharacteristic
123C2×C4×C8D411yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391—21no—
127#390—21no—
128#390—21no—
129#390—21no—
130#389—21no—
131#389—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C43—21no—
143C43C2311yescharacteristic
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#334—22no—
158#334C2×C414yescomplemented
159#332—22no—
160#332C2×C414yescomplemented
161#331—22no—
162#331C2×C414yes—
163C22×Q8—21no—
164C22×Q8C2×D411yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C23×C4—41no—
169C2×Q16—24no—
170C2×Q16—42no—
171C2×Q16—42no—
172C2×SD16—24no—
173C2×SD16—42no—
174C2×SD16—42no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22×C412yes—
189#127—21no—
190#127—21no—
191#127—21no—
192#127C22×C412yes—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8C22×C412yes—
202C4×Q8#4112yes—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—82no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—41no—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42C2411yescharacteristic
240#107—42no—
241#106—42no—
242C4⋊C8—24no—
243C4⋊C8—24no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8C22×C412yes—
253C4⋊C8C22×C412yes—
254C4⋊C8C22×C412yes—
255C4⋊C8#4112yes—
256#103—24no—
257#103—42no—
258#103—42no—
259#103—42no—
260#103—42no—
261#102—24no—
262#102—24no—
263#102—24no—
264#102—42no—
265#102—42no—
266#102—42no—
267#102—42no—
268#97—24no—
269#97—24no—
270#97—42no—
271#97—42no—
272#97—42no—
273#97—42no—
274C4×C8—22no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—41no—
278C4×C8—41no—
279C4×C8—41no—
280C4×C8—42no—
281C4×C8C22×C412yes—
282C4×C8#4112yes—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—41no—
287#95—41no—
288C24—41no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8C4×D412yes—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—81no—
316C2×D4—81no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334Q16—48no—
335SD16—48no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
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×C8C4×D412yes—
364C2×C8C4×D412yes—
365C2×C8#13712yes—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C4⋊C4—82no—
390C4⋊C4C4×D412yes—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C42—22no—
395C42—22no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—22no—
403C42—41no—
404C42—42no—
405C42—42no—
406C42—41no—
407C42—41no—
408C42—42no—
409C42—42no—
410C42—82no—
411C42AGL(1,ℤ/8)12yes—
412C42AGL(1,ℤ/8)12yes—
413C42C23×C411yescharacteristic
414C42C2×#4111yescharacteristic
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C23—42no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23C2×#12111yescharacteristic
425Q8—22no—
426Q8—22no—
427Q8—41no—
428Q8—41no—
429Q8—42no—
430Q8—42no—
431Q8—81no—
432Q8—81no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—82no—
438D4—82no—
439D4—81no—
440D4—81no—
441D4—81no—
442D4—81no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—164no—
476C2×C4#46912yes—
477C2×C4#46912yes—
478C2×C4#47112yes—
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C4×#4111yescharacteristic
482C2×C4#52511yescharacteristic
483C2×C4#52911yescharacteristic
484C2×C4#53211yescharacteristic
485C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
486C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
487C2×C4C2×#13711yescharacteristic
488C2×C4#57511yescharacteristic
489C2×C4#57511yescharacteristic
490C2×C4#57511yescharacteristic
491C2×C4#57511yescharacteristic
492C2×C4#57511yescharacteristic
493C8—22no—
494C8—22no—
495C8—22no—
496C8—41no—
497C8—42no—
498C8—42no—
499C8—42no—
500C8—42no—
501C8—81no—
502C8—81no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22—161no—
510C22C4×#12111yescharacteristic
511C22#303711yescharacteristic
512C22#304011yescharacteristic
513C22C2×#46911yescharacteristic
514C22#305411yescharacteristic
515C22C2×#47111yescharacteristic
516C22#306711yescharacteristic
517C4—21no—
518C4—21no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—42no—
523C4—42no—
524C4—41no—
525C4—82no—
526C4—81no—
527C4—81no—
528C4—161no—
529C4—161no—
530C4#294311yescharacteristic
531C4C4×AGL(1,ℤ/8)11yescharacteristic
532C4C4×AGL(1,ℤ/8)11yescharacteristic
533C4C4×#13711yescharacteristic
534C4#311111yescharacteristic
535C4#311711yescharacteristic
536C4#317111yescharacteristic
537C4#317111yescharacteristic
538C2—162no—
539C2#1928611yescharacteristic
540C2#1928811yescharacteristic
541C2#1930111yescharacteristic
542C2#1930311yescharacteristic
543C2#2150811yescharacteristic
544C2C4×#46911yescharacteristic
545C2C4×#47111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21506C2semidirect
4. C4×#446C2semidirect
7. #9622C2semidirect
8. #9574C2semidirect
11. #9226C2semidirect
12. #8657C2semidirect
13. #8273C2semidirect
14. #8236C2semidirect
55. #1702C4semidirect
82. #462C2×C4semidirect
88. #446C2×C4semidirect
158. #334C2×C4semidirect
160. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4×#4862
#215062
#215052
C4×#4462
C2×#17022
#97462
#96222
#95742
#95722
#92302
#92262
#86572
#82732
#82362
#82342