Finite Group Databasebeta

← #6494999

Subgroups — #6494999

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2868C211yesmaximal, characteristic
2#36640C211yesmaximal, characteristic
3#33316C212yesmaximal, complemented
4#33297C212yesmaximal, complemented
5#30305C212yesmaximal, complemented
6#30304C212yesmaximal, complemented
7#29213C214yesmaximal, complemented
8#28694C214yesmaximal, complemented
9#27953C214yesmaximal
10#25816C214yesmaximal
11#21982C212yesmaximal
12#21981C212yesmaximal
13#20463C211yesmaximal, characteristic
14C22×C4⋊Q8—21no—
15C22×C4⋊Q8—21no—
16C22×C4⋊Q8C2212yes—
17C2×#501C2218yes—
18C2×#431C2212yes—
19#2868—21no—
20#2868—21no—
21#2868—21no—
22#2868—21no—
23#2868C2212yes—
24#2868C2212yes—
25#2868C2212yes—
26#2868C2212yes—
27#2864—21no—
28#2864—21no—
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2851—21no—
34#2851—21no—
35#2851C2212yes—
36#2851C2212yes—
37C4×#125—21no—
38C4×#125—21no—
39C4×#125C2212yes—
40C4×#125C2212yes—
41#2296—21no—
42#2296—21no—
43#2285C2211yescharacteristic
44#2285C2211yescharacteristic
45C2×#384—21no—
46C2×#384—21no—
47C2×#384C2212yes—
48C2×#384C2212yes—
49C2×C43C2211yescharacteristic
50#2084C2218yes—
51#2083C2218yes—
52#2058C2218yes—
53#2057C2218yes—
54#1993C2218yes—
55#1992C2218yes—
56#1948C2218yes—
57#1947C2218yes—
58#1853C2218yes—
59#1852C2218yes—
60#1847C2218yes—
61#1845C2214yes—
62#1844C2214yes—
63#1834C2214yes—
64#1834C2214yes—
65#1767C2212yes—
66#1767C2212yes—
67#1749C2212yes—
68#1749C2212yes—
69#1744C2212yes—
70C4×#97C2212yes—
71C23×Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—48no—
83C2×C4⋊Q8—48no—
84C2×C4⋊Q8—48no—
85C2×C4⋊Q8C2314yes—
86C2×C4⋊Q8C2314yes—
87C2×#125—21no—
88C2×#125—21no—
89C2×#125C2312yes—
90C2×#125C2312yes—
91C2×#117—21no—
92C2×#117—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2314yes—
99C22×C42D411yescharacteristic
100C22×C42D411yescharacteristic
101C22×C42D411yescharacteristic
102C22×C42D411yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#501—416no—
107#431Q8116yescomplemented
108C2×#107C23116yes—
109C2×#106C23116yes—
110C2×C4⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×#97C2314yes—
113C2×#97C2314yes—
114C2×C4×C8C2314yes—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389C2312yes—
128#389C2312yes—
129#389C2312yes—
130#389C2312yes—
131#387—22no—
132#387—22no—
133#387—22no—
134#387—22no—
135#387—22no—
136#387—22no—
137#387—22no—
138#387—22no—
139#385—22no—
140#385—22no—
141#385—22no—
142#385—22no—
143#384—22no—
144#384—22no—
145#384—22no—
146#384—22no—
147#384—22no—
148#384—22no—
149#384—22no—
150#384—22no—
151#384—22no—
152#384—22no—
153#384—22no—
154#384—22no—
155#384C2314yes—
156#384C2311yescharacteristic
157#384C2311yescharacteristic
158#384C2314yes—
159#384C2311yescharacteristic
160#384C2311yescharacteristic
161#384C2314yes—
162#384C2314yes—
163#382—21no—
164#382—21no—
165#382—21no—
166#382—21no—
167#382C2312yes—
168#382C2312yes—
169#380—21no—
170#380—21no—
171#380C2312yes—
172#380C2312yes—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C43D411yescharacteristic
182C43D411yescharacteristic
183C43D411yescharacteristic
184C43D411yescharacteristic
185C43C2311yescharacteristic
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188C43C2311yescharacteristic
189#340C23116yes—
190#340C23116yes—
191#336C2314yes—
192#336C2314yes—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—48no—
196C22×Q8—48no—
197C22×Q8—48no—
198C23×C4—24no—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2411yescharacteristic
206C22×C8C2418yes—
207C4⋊Q8—216no—
208C4⋊Q8—48no—
209C4⋊Q8—48no—
210C4⋊Q8—48no—
211C4⋊Q8—48no—
212C4⋊Q8—48no—
213C4⋊Q8—48no—
214C4⋊Q8—48no—
215C4⋊Q8—48no—
216C4⋊Q8—48no—
217C4⋊Q8—48no—
218C4⋊Q8—48no—
219#125—28no—
220#125—28no—
221#125—44no—
222#125—44no—
223#117—44no—
224#117—44no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4C2418yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×Q818yes—
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#107—416no—
303#106—416no—
304C4⋊C8C2×Q8116yescomplemented
305C22⋊C8C2×Q8116yes—
306#97—28no—
307#97—28no—
308C4×C8C2×Q8116yescomplemented
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321C24C22×D411yescharacteristic
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—832no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4C4⋊Q814yes—
354C22×C4C4⋊Q814yes—
355C22×C4AGL(1,ℤ/8)14yes—
356C22×C4#13714yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×Q814yes—
371C22×C4C2511yescharacteristic
372C2×C8—28no—
373C2×C8—28no—
374C2×C8C22×Q8116yes—
375C4⋊C4—24no—
376C4⋊C4—216no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—44no—
430C42—44no—
431C42C4⋊Q814yes—
432C42C4⋊Q814yes—
433C42AGL(1,ℤ/8)14yes—
434C42#13714yes—
435C42C22×Q814yes—
436C42C22×Q814yes—
437C42H(5,𝔽2)14yes—
438C422−1+414yes—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23C2×C4⋊Q812yes—
444C23C2×AGL(1,ℤ/8)12yes—
445C23C2×#13712yes—
446C23C23×D411yescharacteristic
447Q8—816no—
448Q8—816no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—816no—
491C2×C4#501116yescomplemented
492C2×C4C2×C4⋊Q812yes—
493C2×C4C2×C4⋊Q812yes—
494C2×C4C2×C4⋊Q812yes—
495C2×C4C2×C4⋊Q812yes—
496C2×C4C2×C4⋊Q812yes—
497C2×C4#53512yes—
498C2×C4#53512yes—
499C2×C4#53612yes—
500C2×C4#53612yes—
501C2×C4C2×AGL(1,ℤ/8)12yes—
502C2×C4C2×AGL(1,ℤ/8)12yes—
503C2×C4C2×AGL(1,ℤ/8)12yes—
504C2×C4C2×AGL(1,ℤ/8)12yes—
505C2×C4C2×AGL(1,ℤ/8)12yes—
506C2×C4C2×#13712yes—
507C2×C4C2×#13712yes—
508C2×C4C2×#13712yes—
509C2×C4C2×#13712yes—
510C2×C4C2×#13712yes—
511C2×C4C23×Q812yes—
512C2×C4C2×H(5,𝔽2)12yes—
513C2×C4C2×2−1+412yes—
514C8—216no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22C2×#50114yes—
522C22C22×C4⋊Q811yescharacteristic
523C22C2×#53511yescharacteristic
524C22C2×#53611yescharacteristic
525C22C22×AGL(1,ℤ/8)11yescharacteristic
526C22C22×AGL(1,ℤ/8)11yescharacteristic
527C22C22×#13711yescharacteristic
528C22C22×#13711yescharacteristic
529C4—22no—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—82no—
540C4—88no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4C2×#50114yes—
545C4C2×#50114yes—
546C4#316514yes—
547C4#350314yes—
548C2—24no—
549C2#3528111yescharacteristic
550C2C22×#50111yescharacteristic
551C2C2×#316511yescharacteristic
552C2#6079811yescharacteristic
553C2#6080011yescharacteristic
554C2#6086011yescharacteristic
555C2#6086011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33316C2semidirect
4. #33297C2semidirect
5. #30305C2semidirect
6. #30304C2semidirect
7. #29213C2semidirect
8. #28694C2semidirect
107. #431Q8semidirect
304. C4⋊C8C2×Q8semidirect
308. C4×C8C2×Q8semidirect
491. C2×C4#501semidirect

Maximal subgroups

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

subgroupindex
C2×#28682
#366402
#333162
#332972
#303052
#303042
#292132
#286942
#279532
#258162
#219822
#219812
#204632