Finite Group Databasebeta

← #227032

Subgroups — #227032

Order 512.

729 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23096C211yesmaximal, characteristic, complemented
2#23062C211yesmaximal, characteristic, complemented
3#22543C211yesmaximal, characteristic
4#22543C211yesmaximal, characteristic
5C2×#1640C211yesmaximal, characteristic
6#9822C211yesmaximal, characteristic
7#9564C211yesmaximal, characteristic
8#9563C211yesmaximal, characteristic
9#9220C211yesmaximal, characteristic, complemented
10#9218C211yesmaximal, characteristic, complemented
11#9062C211yesmaximal, characteristic
12#8895C211yesmaximal, characteristic, complemented
13#8891C211yesmaximal, characteristic, complemented
14C8×#103C211yesmaximal, characteristic, complemented
15C8×#103C211yesmaximal, characteristic, complemented
16C2×#559C2211yescharacteristic
17C2×#486C2211yescharacteristic
18C2×#475C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22#2365C2211yescharacteristic
23#2355C2211yescharacteristic
24#2355C2211yescharacteristic
25#1990C2211yescharacteristic
26#1983—21no—
27#1983—21no—
28#1982C2211yescharacteristic
29#1982C2211yescharacteristic
30#1969—21no—
31#1958—21no—
32#1958—21no—
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947C2211yescharacteristic
38#1933—21no—
39#1928—21no—
40#1927—21no—
41#1919—21no—
42#1878—21no—
43#1834C2211yescharacteristic
44#1804—21no—
45#1804—21no—
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48C4×#106—21no—
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C4×#103—21no—
52C4×#103C2211yescharacteristic
53C4×#103C2211yescharacteristic
54C4×#102—21no—
55C4×#102—21no—
56#1640C418yescomplemented
57C2×#332C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×Q16—21no—
69#559—22no—
70#559—22no—
71#559—41no—
72#559C2×C414yes—
73C2×C4⋊Q8C2311yescharacteristic
74C2×#127C2311yescharacteristic
75C2×#124—21no—
76C2×#124—21no—
77C2×#123—41no—
78C2×C4×Q8—21no—
79C2×C4×Q8C2311yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83#486—41no—
84#486C2×C414yes—
85#475—22no—
86#475C2×C414yes—
87#462—22no—
88#462C2×C414yes—
89C8×Q8—22no—
90C8×Q8—22no—
91C8×Q8—41no—
92C8×Q8—41no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103D411yescharacteristic
109C2×#103D411yescharacteristic
110C2×#103D411yescharacteristic
111C2×#103D411yescharacteristic
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—41no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#391—21no—
129#391—21no—
130#391—21no—
131#389—21no—
132#387—41no—
133#387—41no—
134#387—41no—
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#382—21no—
138#382—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—41no—
153#340D411yescharacteristic
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#336—21no—
158#336—21no—
159#332—41no—
160#331C2×C414yes—
161Q8⋊C8C2×C414yes—
162Q8⋊C8C2×C414yes—
163C82—41no—
164C22×Q8—21no—
165C22×Q8C2×D411yescharacteristic
166C22×Q8C2×D411yescharacteristic
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C23×C4—41no—
171C2×Q16—24no—
172C2×Q16—81no—
173C2×Q16—81no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22×C412yes—
187#127—21no—
188#127—21no—
189#127—21no—
190#127C22×C412yes—
191#124—24no—
192#124—24no—
193#124—42no—
194#124—42no—
195#123—44no—
196#123—44no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—42no—
202C4×Q8—81no—
203C4×Q8C22×C412yes—
204C4×Q8C22×C412yes—
205C4×Q8#4112yes—
206C4×Q8#4112yes—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—42no—
210C4×D4—42no—
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—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42C2411yescharacteristic
239#107—81no—
240#106—24no—
241#106—42no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8C22×C412yes—
251C4⋊C8C22×C412yes—
252#103—24no—
253#103—22no—
254#103—22no—
255#103—24no—
256#103—22no—
257#103—22no—
258#103—42no—
259#103—81no—
260#103—81no—
261#103—81no—
262#102—44no—
263#102—42no—
264#102—42no—
265#102—42no—
266#102—42no—
267#102—81no—
268#102—81no—
269#102—81no—
270#102—81no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—42no—
277C4×C8—42no—
278C4×C8—42no—
279C4×C8—81no—
280C4×C8C22×C412yes—
281#95—21no—
282#95—21no—
283#95—41no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288#95—41no—
289#95—41no—
290#95—41no—
291C24—41no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—22no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8C4×D412yes—
309C2×Q8C4×D412yes—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—81no—
325C2×D4—81no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—44no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—161no—
339C22×C4—161no—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343Q16—84no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8#13512yes—
370C2×C8#13512yes—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—82no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C4⋊C4C4×D412yes—
399C4⋊C4C4×D412yes—
400C4⋊C4#13512yes—
401C4⋊C4#13512yes—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—82no—
414C42—161no—
415C42—161no—
416C42C23×C411yescharacteristic
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C42H(5,𝔽2)11yescharacteristic
420C23—42no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23D4211yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—22no—
431Q8—22no—
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436Q8—42no—
437Q8—42no—
438Q8—81no—
439Q8—81no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—82no—
445D4—81no—
446D4—81no—
447D4—81no—
448D4—81no—
449D4—161no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
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—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—164no—
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4#51811yescharacteristic
489C2×C4#54611yescharacteristic
490C2×C4#54611yescharacteristic
491C2×C4#55111yescharacteristic
492C2×C4C2×#13511yescharacteristic
493C2×C4C2×#13511yescharacteristic
494C2×C4C2×#13511yescharacteristic
495C2×C4C2×#13511yescharacteristic
496C2×C4#57611yescharacteristic
497C2×C4#57711yescharacteristic
498C2×C4#57711yescharacteristic
499C2×C4#57811yescharacteristic
500C8—22no—
501C8—22no—
502C8—42no—
503C8—42no—
504C8—42no—
505C8—41no—
506C8—41no—
507C8—81no—
508C8—81no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—161no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22#235811yescharacteristic
517C22#328211yescharacteristic
518C22#328411yescharacteristic
519C22#328511yescharacteristic
520C22#328711yescharacteristic
521C22#329011yescharacteristic
522C22#329011yescharacteristic
523C4—21no—
524C4—21no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—81no—
532C4—81no—
533C4—82no—
534C4—161no—
535C4—161no—
536C4C4×#13511yescharacteristic
537C4C4×#13511yescharacteristic
538C4#294611yescharacteristic
539C4#294711yescharacteristic
540C4#330011yescharacteristic
541C4#330111yescharacteristic
542C4#330411yescharacteristic
543C4#330511yescharacteristic
544C2—162no—
545C2#1999911yescharacteristic
546C2#2000011yescharacteristic
547C2#2001511yescharacteristic
548C2#2001811yescharacteristic
549C2#2287311yescharacteristic
550C2#2287711yescharacteristic
551C2#2289611yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23096C2semidirect
2. #23062C2semidirect
9. #9220C2semidirect
10. #9218C2semidirect
12. #8895C2semidirect
13. #8891C2semidirect
14. C8×#103C2semidirect
15. C8×#103C2semidirect
56. #1640C4semidirect

Maximal subgroups

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

subgroupindex
#230962
#230622
#225432
C2×#16402
#98222
#95642
#95632
#92202
#92182
#90622
#88952
#88912
C8×#1032