Finite Group Databasebeta

← #6436924

Subgroups — #6436924

Order 512.

1893 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#29943C214yesmaximal, complemented
4#29927C214yesmaximal, complemented
5#29908C212yesmaximal, complemented
6#29906C212yesmaximal, complemented
7#29895C212yesmaximal, complemented
8#29892C212yesmaximal
9#23763C214yesmaximal, complemented
10#23761C214yesmaximal, complemented
11#21391C212yesmaximal, complemented
12#21363C212yesmaximal, complemented
13#20715C211yesmaximal, characteristic, complemented
14C2×#529C2211yescharacteristic
15C22×#117—21no—
16C23×C42C2211yescharacteristic
17C2×#432C2212yes—
18#2855C2211yescharacteristic
19#2854C2211yescharacteristic
20#2853—21no—
21#2852—22no—
22#2851C2212yes—
23#2850C2212yes—
24#2849—21no—
25#2848C2211yescharacteristic
26#2847C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
28C4×#123—21no—
29C4×#121C2212yes—
30C4×#120—21no—
31#2290—22no—
32#2285—22no—
33#2285—22no—
34#2285C2214yes—
35#2285C2214yes—
36C2×#384C2212yes—
37#2283C2211yescharacteristic
38C2×#382—21no—
39C2×#380—21no—
40C2×#380C2212yes—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4C2212yes—
43#2099C2214yescomplemented
44#2099C2214yescomplemented
45#2096C2214yescomplemented
46#2093C2214yescomplemented
47#2084C2214yescomplemented
48#2083C2214yescomplemented
49#2077C2218yes—
50#2076C2218yes—
51#2072C2214yescomplemented
52#2071C2214yescomplemented
53#2036C2218yes—
54#2034C2218yes—
55#1989C2214yescomplemented
56#1983C2214yescomplemented
57#1949C2218yescomplemented
58#1874C2214yes—
59#1872C2214yes—
60#1853C2214yes—
61#1852C2214yes—
62#1817C2212yes—
63#1797C2214yes—
64#1779C2212yes—
65#1776C2218yescomplemented
66#1763C2214yescomplemented
67#1762C2214yescomplemented
68#1749C2214yes—
69C24×C4D411yescharacteristic
70C24×C4D411yescharacteristic
71C24×C4C2311yescharacteristic
72#529—28no—
73C2×#126—21no—
74C2×#125C2311yescharacteristic
75C2×#124C2311yescharacteristic
76C2×#123—21no—
77C2×#123—21no—
78C2×C22⋊Q8C2311yescharacteristic
79C2×C22⋊Q8C2312yes—
80C2×#121C2311yescharacteristic
81C2×#121C2312yes—
82C2×#120—21no—
83C2×C4×Q8C2311yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—22no—
86C2×C4×D4C2311yescharacteristic
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—21no—
90C2×#117—48no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4—22no—
95C22×C42—28no—
96C22×C42—22no—
97C22×C42D412yes—
98C22×C42D411yescharacteristic
99C22×C42D411yescharacteristic
100C22×C42D412yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105#432—28no—
106C2×#107C2318yes—
107C2×#106C2318yes—
108C2×C4⋊C8C2314yes—
109C2×#103C2318yes—
110C2×#102C2318yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2318yes—
113C2×#97C2314yes—
114#391C2312yes—
115#390C2312yes—
116#389C2312yes—
117#389C2312yes—
118#388—22no—
119#388—22no—
120#388—22no—
121#388—22no—
122#387—22no—
123#387—22no—
124#387—22no—
125#387—22no—
126#386—22no—
127#386—21no—
128#386—22no—
129#386—22no—
130#386—22no—
131#386—21no—
132#385—22no—
133#385—21no—
134#385—22no—
135#385—22no—
136#385—22no—
137#385—21no—
138#384—22no—
139#384—22no—
140#384C2314yes—
141#384C2314yes—
142#383—21no—
143#382—24no—
144#382—24no—
145#382C2312yes—
146#381—21no—
147#381—21no—
148#380—24no—
149#380—24no—
150#380—24no—
151#380C2312yes—
152#380C2318yes—
153#380C2312yes—
154#380C2311yescharacteristic
155#380C2311yescharacteristic
156#379—21no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4C2314yes—
163C4×C4⋊C4C2314yes—
164C4×C4⋊C4C2312yes—
165C4×C22⋊C4—21no—
166C4×C22⋊C4—22no—
167C4×C22⋊C4—22no—
168#376—21no—
169C2×#95—22no—
170C2×#95—22no—
171#340C2318yes—
172#340C2318yes—
173#340C2318yes—
174#340C2318yes—
175#336C2314yes—
176#336C2314yes—
177C25C2×D411yescharacteristic
178C22×Q8C2411yescharacteristic
179C22×D4—21no—
180C22×D4—22no—
181C22×D4C2411yescharacteristic
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—21no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C22×C8C2418yes—
204#126—44no—
205#125—24no—
206#124—24no—
207#123—44no—
208#123—44no—
209C22⋊Q8—24no—
210C22⋊Q8—44no—
211#121—24no—
212#121—44no—
213#120—44no—
214C4×Q8—24no—
215C4×D4—24no—
216C4×D4—48no—
217C4×D4—44no—
218#117—416no—
219#117—416no—
220#117—44no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4C2418yes—
237C2×C4⋊C4C2411yescharacteristic
238C2×C4⋊C4C2411yescharacteristic
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—48no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—48no—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42#4118yes—
276C2×C42C2412yes—
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279#107—48no—
280#106—48no—
281C4⋊C8—28no—
282#103—48no—
283#102—48no—
284C22⋊C8—28no—
285C22⋊C8—48no—
286#97—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301C24—22no—
302C24—22no—
303C24—21no—
304C24—42no—
305C24—42no—
306C24#13714yes—
307C24C22×D412yes—
308C24C22×D411yescharacteristic
309C2×Q8—22no—
310C2×Q8—24no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—42no—
319C22×C4—232no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—28no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—416no—
355C22×C4—416no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4#12314yes—
362C22×C4#12314yes—
363C22×C4AGL(1,ℤ/8)14yes—
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C2×#4114yes—
373C22×C4C2511yescharacteristic
374C2×C8—28no—
375C2×C8—28no—
376C2×C8—28no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—28no—
380C4⋊C4—24no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—416no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—44no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C42—24no—
410C42—24no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—48no—
416C42—48no—
417C42—44no—
418C42—42no—
419C42#12314yes—
420C42#12314yes—
421C42C2×#4114yes—
422C42C2×#4114yes—
423C23—22no—
424C23—22no—
425C23—28no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—48no—
433C23—44no—
434C23—44no—
435C23—48no—
436C23—41no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23C2×#12312yes—
442C23C2×AGL(1,ℤ/8)12yes—
443C23C2×#13714yes—
444C23C2×#13712yes—
445C23C23×D411yescharacteristic
446Q8—44no—
447D4—44no—
448D4—88no—
449D4—84no—
450C2×C4—216no—
451C2×C4—216no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—28no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—44no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—48no—
474C2×C4—44no—
475C2×C4—48no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—48no—
492C2×C4—84no—
493C2×C4—816no—
494C2×C4C2×#12312yes—
495C2×C4C2×#12312yes—
496C2×C4C2×#12312yes—
497C2×C4C2×#12312yes—
498C2×C4C2×#12312yes—
499C2×C4C2×AGL(1,ℤ/8)14yes—
500C2×C4#57514yes—
501C2×C4C22×#4112yes—
502C2×C4#58514yes—
503C8—48no—
504C22—24no—
505C22—24no—
506C22—22no—
507C22—22no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—416no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—82no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22#309214yes—
522C22C22×#12311yescharacteristic
523C22#344912yes—
524C22C22×AGL(1,ℤ/8)11yescharacteristic
525C22C22×#13711yescharacteristic
526C22C2×#57512yes—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—24no—
531C4—24no—
532C4—22no—
533C4—22no—
534C4—42no—
535C4—82no—
536C4—82no—
537C4—88no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4#345514yes—
542C2—24no—
543C2—44no—
544C2—161no—
545C2#3463511yescharacteristic
546C2C2×#309211yescharacteristic
547C2#6051111yescharacteristic
548C2#6096012yes—
549C2#6096512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
3. #29943C2semidirect
4. #29927C2semidirect
5. #29908C2semidirect
6. #29906C2semidirect
7. #29895C2semidirect
9. #23763C2semidirect
10. #23761C2semidirect
11. #21391C2semidirect
12. #21363C2semidirect
13. #20715C2semidirect
43. #2099C22semidirect
44. #2099C22semidirect
45. #2096C22semidirect
46. #2093C22semidirect
47. #2084C22semidirect
48. #2083C22semidirect
51. #2072C22semidirect
52. #2071C22semidirect
55. #1989C22semidirect
56. #1983C22semidirect
57. #1949C22semidirect
65. #1776C22semidirect
66. #1763C22semidirect
67. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#366212
C2×#22852
#299432
#299272
#299082
#299062
#298952
#298922
#237632
#237612
#213912
#213632
#207152