Finite Group Databasebeta

← #6482933

Subgroups — #6482933

Order 512.

1893 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#33099C212yesmaximal, complemented
4#33095C212yesmaximal, complemented
5#29893C212yesmaximal, complemented
6#29890C212yesmaximal
7#27894C214yesmaximal, complemented
8#27880C214yesmaximal, complemented
9#27631C214yesmaximal, complemented
10#27627C214yesmaximal, complemented
11#21987C212yesmaximal, complemented
12#21867C212yesmaximal, 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#2077C2218yes—
44#2075C2218yes—
45#2035C2218yes—
46#2033C2218yes—
47#1992C2214yescomplemented
48#1959C2214yescomplemented
49#1948C2214yescomplemented
50#1947C2214yescomplemented
51#1928C2218yescomplemented
52#1926C2218yescomplemented
53#1921C2218yescomplemented
54#1919C2218yescomplemented
55#1874C2214yes—
56#1872C2214yes—
57#1853C2214yes—
58#1852C2214yes—
59#1847C2214yescomplemented
60#1836C2214yescomplemented
61#1817C2212yes—
62#1803C2214yescomplemented
63#1802C2214yescomplemented
64#1797C2214yes—
65#1779C2212yes—
66#1749C2214yes—
67C24×C4D411yescharacteristic
68C24×C4D411yescharacteristic
69C24×C4C2311yescharacteristic
70#529—28no—
71C2×#126—21no—
72C2×#125C2311yescharacteristic
73C2×#124C2311yescharacteristic
74C2×#123—21no—
75C2×#123—21no—
76C2×C22⋊Q8C2311yescharacteristic
77C2×C22⋊Q8C2312yes—
78C2×#121C2311yescharacteristic
79C2×#121C2312yes—
80C2×#120—21no—
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—22no—
84C2×C4×D4C2311yescharacteristic
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—21no—
88C2×#117—48no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4C2314yes—
92C22×C22⋊C4—22no—
93C22×C42—28no—
94C22×C42—22no—
95C22×C42D412yes—
96C22×C42D411yescharacteristic
97C22×C42D411yescharacteristic
98C22×C42D412yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103#432—28no—
104C2×#107C2318yes—
105C2×#106C2318yes—
106C2×C4⋊C8C2314yes—
107C2×#103C2318yes—
108C2×#102C2318yes—
109C2×C22⋊C8C2314yes—
110C2×C22⋊C8C2318yes—
111C2×#97C2314yes—
112#391C2312yes—
113#390C2312yes—
114#389C2312yes—
115#389C2312yes—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#387—22no—
121#387—22no—
122#387—22no—
123#387—22no—
124#386—22no—
125#386—21no—
126#386—22no—
127#386—22no—
128#386—22no—
129#386—21no—
130#385—22no—
131#385—21no—
132#385—22no—
133#385—22no—
134#385—22no—
135#385—21no—
136#384—22no—
137#384—22no—
138#384C2314yes—
139#384C2314yes—
140#383—21no—
141#382—24no—
142#382—24no—
143#382C2312yes—
144#381—21no—
145#381—21no—
146#380—24no—
147#380—24no—
148#380—24no—
149#380C2312yes—
150#380C2318yes—
151#380C2312yes—
152#380C2311yescharacteristic
153#380C2311yescharacteristic
154#379—21no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4C2318yes—
160C4×C4⋊C4C2312yes—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—22no—
163C4×C22⋊C4—22no—
164#376—21no—
165C2×#95—22no—
166C2×#95—22no—
167#340C2318yes—
168#340C2318yes—
169#340C2318yes—
170#340C2318yes—
171#336C2318yes—
172C25C2×D411yescharacteristic
173C22×Q8C2411yescharacteristic
174C22×D4—21no—
175C22×D4—22no—
176C22×D4C2411yescharacteristic
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—21no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C22×C8C2418yes—
199#126—44no—
200#125—24no—
201#124—24no—
202#123—44no—
203#123—44no—
204C22⋊Q8—24no—
205C22⋊Q8—44no—
206#121—24no—
207#121—44no—
208#120—44no—
209C4×Q8—24no—
210C4×D4—24no—
211C4×D4—48no—
212C4×D4—44no—
213#117—416no—
214#117—416no—
215#117—44no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4C2418yes—
232C2×C4⋊C4C2411yescharacteristic
233C2×C4⋊C4C2411yescharacteristic
234C2×C4⋊C4C2412yes—
235C2×C4⋊C4C2412yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—48no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—28no—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—28no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—48no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42#4118yes—
270C2×C42C2412yes—
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273#107—48no—
274#106—48no—
275C4⋊C8—28no—
276#103—48no—
277#102—48no—
278C22⋊C8—28no—
279C22⋊C8—48no—
280#97—28no—
281#95—28no—
282#95—28no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—42no—
298C24—42no—
299C24AGL(1,ℤ/8)14yes—
300C24C22×D412yes—
301C24C22×D411yescharacteristic
302C2×Q8—22no—
303C2×Q8—24no—
304C2×D4—22no—
305C2×D4—24no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—42no—
311C2×D4—42no—
312C22×C4—232no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
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—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—28no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—416no—
348C22×C4—416no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4#12314yes—
355C22×C4#12314yes—
356C22×C4#13714yes—
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C2×#4114yes—
366C22×C4C2511yescharacteristic
367C2×C8—28no—
368C2×C8—28no—
369C2×C8—28no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—42no—
393C22⋊C4—416no—
394C22⋊C4—44no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—44no—
402C42—24no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—416no—
409C42—42no—
410C42—44no—
411C42#12314yes—
412C42#12314yes—
413C42C2×#4114yes—
414C42C2×#4114yes—
415C23—22no—
416C23—22no—
417C23—28no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—48no—
425C23—44no—
426C23—44no—
427C23—48no—
428C23—41no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23C2×#12312yes—
434C23C2×AGL(1,ℤ/8)14yes—
435C23C2×AGL(1,ℤ/8)12yes—
436C23C2×#13712yes—
437C23C23×D411yescharacteristic
438Q8—44no—
439D4—44no—
440D4—88no—
441D4—84no—
442C2×C4—216no—
443C2×C4—216no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—816no—
485C2×C4—84no—
486C2×C4C2×#12312yes—
487C2×C4C2×#12312yes—
488C2×C4C2×#12312yes—
489C2×C4C2×#12312yes—
490C2×C4C2×#12312yes—
491C2×C4C2×#13714yes—
492C2×C4#57514yes—
493C2×C4C22×#4112yes—
494C2×C4#58514yes—
495C8—48no—
496C22—24no—
497C22—24no—
498C22—22no—
499C22—22no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22—416no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—82no—
509C22—82no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22#309214yes—
514C22C22×#12311yescharacteristic
515C22#344912yes—
516C22C22×AGL(1,ℤ/8)11yescharacteristic
517C22C22×#13711yescharacteristic
518C22C2×#57512yes—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—24no—
523C4—22no—
524C4—42no—
525C4—42no—
526C4—82no—
527C4—88no—
528C4—82no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4#309314yes—
533C4#349314yes—
534C2—24no—
535C2—44no—
536C2—161no—
537C2#3515711yescharacteristic
538C2C2×#309211yescharacteristic
539C2C2×#309311yescharacteristic
540C2#6095212yes—
541C2#6095812yes—
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. #33099C2semidirect
4. #33095C2semidirect
5. #29893C2semidirect
7. #27894C2semidirect
8. #27880C2semidirect
9. #27631C2semidirect
10. #27627C2semidirect
11. #21987C2semidirect
12. #21867C2semidirect
13. #20715C2semidirect
47. #1992C22semidirect
48. #1959C22semidirect
49. #1948C22semidirect
50. #1947C22semidirect
51. #1928C22semidirect
52. #1926C22semidirect
53. #1921C22semidirect
54. #1919C22semidirect
59. #1847C22semidirect
60. #1836C22semidirect
62. #1803C22semidirect
63. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#366212
C2×#22852
#330992
#330952
#298932
#298902
#278942
#278802
#276312
#276272
#219872
#218672
#207152