Finite Group Databasebeta

← #6471017

Subgroups — #6471017

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37307C211yesmaximal, characteristic, complemented
2#34265C211yesmaximal, characteristic
3#33157C212yesmaximal, complemented
4#32956C212yesmaximal, complemented
5#29971C212yesmaximal, complemented
6#29962C212yesmaximal, complemented
7#26436C214yesmaximal, complemented
8#26423C214yesmaximal, complemented
9#26359C214yesmaximal, complemented
10#26338C214yesmaximal, complemented
11#21937C211yesmaximal, characteristic, complemented
12#21770C212yesmaximal
13#20848C212yesmaximal, complemented
14C2×#535C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#476C2214yes—
18C2×#428C2212yescomplemented
19C22×#106C2212yescomplemented
20C2×#408C2212yes—
21#2857C2212yes—
22#2806—22no—
23#2748—22no—
24#2732—22no—
25#2683—22no—
26#2629C2212yes—
27#2441C2214yescomplemented
28#2440C2214yescomplemented
29#2367—22no—
30#2366—22no—
31#2363C2214yes—
32#2361C2214yes—
33#2321—22no—
34#2311C2212yes—
35#2309C2211yescharacteristic
36C4×#120—22no—
37#2297C2211yescharacteristic
38#2296C2211yescharacteristic
39#2274C2211yescharacteristic
40#2099C2212yescomplemented
41#2099C2212yescomplemented
42#2094C2212yes—
43#2093C2212yes—
44#2088C2214yescomplemented
45#2087C2214yescomplemented
46#2067C2214yes—
47#2065C2214yes—
48#2058C2212yescomplemented
49#2057C2212yescomplemented
50#2025C2212yes—
51#2025C2212yes—
52#1989C2214yescomplemented
53#1948C2214yescomplemented
54#1920C2214yescomplemented
55#1919C2214yescomplemented
56#1895C2214yes—
57#1893C2214yes—
58#1875C2214yes—
59#1874C2214yes—
60#1864C2214yes—
61#1864C2214yes—
62#1854C2214yes—
63#1854C2214yescomplemented
64#1853C2214yes—
65#1853C2214yescomplemented
66#1851C2214yescomplemented
67#1850C2214yescomplemented
68#1825C2214yes—
69#1802C2214yes—
70#1798C2212yes—
71#1797C2212yes—
72#1786C2214yes—
73#1736C2212yescomplemented
74#1735C2212yescomplemented
75C22×#41C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4—21no—
78C22×M16C2312yes—
79C23×C8C2312yes—
80#535—24no—
81#535Q818yescomplemented
82C2×C4⋊Q8C2312yes—
83C2×#127—21no—
84C2×#125C2312yes—
85C2×#124—21no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—24no—
88C2×#121C2314yes—
89C2×#120—22no—
90#520—44no—
91C2×C4×Q8—21no—
92C2×C4×D4—22no—
93C2×C4×D4—21no—
94C2×C4×D4—24no—
95C2×C4×D4—22no—
96C2×C4×D4—44no—
97C2×C4×D4C2314yes—
98C2×#117C2312yes—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—21no—
107C22×C42C2311yescharacteristic
108#476—48no—
109#428—44no—
110C2×#107C2314yes—
111C2×#106—44no—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2314yes—
116C2×C4⋊C8C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121#408—28no—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2314yes—
124#399—24no—
125#398—24no—
126#398—24no—
127#396—24no—
128#396—24no—
129#395—24no—
130#391—22no—
131#391—22no—
132#391D412yes—
133#390—22no—
134#390—22no—
135#390—24no—
136#390D412yes—
137#390C2312yes—
138#390C2312yes—
139#389—22no—
140#389—24no—
141#389C2312yes—
142#389C2312yes—
143#389C2314yes—
144#389C2314yes—
145#388—24no—
146#388—24no—
147#387—24no—
148#386—22no—
149#386—22no—
150#386—24no—
151#386—24no—
152#386D412yes—
153#386D412yes—
154#385—21no—
155#385—21no—
156#385—24no—
157#385—24no—
158#384C2312yes—
159#384C2312yes—
160#380—22no—
161#380C2314yes—
162#380C2314yes—
163#379—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4C2314yes—
167C4×C22⋊C4—22no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#336C2312yes—
179#336C2312yes—
180#336C2312yes—
181#336C2312yes—
182C25—21no—
183C2×#41—42no—
184C2×#41—42no—
185C2×#41C2×Q814yes—
186C2×#41#4114yes—
187C22×Q8—22no—
188C22×Q8C2×D411yescharacteristic
189C22×D4—24no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4C2×D412yes—
195C22×D4C2×D411yescharacteristic
196C22×D4C2×Q814yes—
197C22×D4C2412yes—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—44no—
203C23×C4—84no—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C2×M16—24no—
209C2×M16—42no—
210C2×M16—42no—
211C22×C8—24no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8C2414yes—
215C22×C8C2414yes—
216C4⋊Q8—44no—
217#127—28no—
218#125—44no—
219#124—28no—
220C22⋊Q8—88no—
221C22⋊Q8—84no—
222#121—28no—
223#121C2×Q8116yes—
224#120—216no—
225C4×Q8—44no—
226C4×D4—28no—
227C4×D4—44no—
228C4×D4—48no—
229C4×D4—44no—
230C4×D4—48no—
231C4×D4—88no—
232#117—22no—
233#117—24no—
234#117—42no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117—42no—
239#117C2×Q814yes—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2411yescharacteristic
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2411yescharacteristic
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D414yes—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—42no—
284C2×C42C2×D412yes—
285C2×C42C2412yes—
286C2×C42C2412yes—
287#107—44no—
288#106—28no—
289#106—28no—
290#106—44no—
291C4⋊C8—44no—
292C4⋊C8—44no—
293#102—44no—
294#102—44no—
295#102—44no—
296#102—44no—
297C22⋊C8—28no—
298C22⋊C8—28no—
299#95—22no—
300#95—24no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95C2×D414yes—
307#95C2×D414yes—
308C24—22no—
309C24—24no—
310C24—42no—
311C24—42no—
312C24—82no—
313C24—82no—
314C24C22×D411yescharacteristic
315C24C22×D412yes—
316#41—48no—
317#41—48no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—44no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×Q8C22⋊Q814yes—
324C2×D4—216no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—48no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—48no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4C22⋊Q818yes—
340C2×D4C22⋊Q814yes—
341C2×D4C22×Q814yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—48no—
362C22×C4—84no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—816no—
368C22×C4C22×D414yes—
369C22×C4C22×D412yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×Q812yes—
374C22×C4C22×Q814yes—
375C22×C4C2×#4112yes—
376C22×C4C2×#4114yes—
377C22×C4C2511yescharacteristic
378M16—48no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—48no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4—82no—
405C4⋊C4—84no—
406C4⋊C4—82no—
407C4⋊C4C22×Q814yes—
408C4⋊C4H(5,𝔽2)14yes—
409C22⋊C4—24no—
410C22⋊C4—216no—
411C22⋊C4—28no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—44no—
416C22⋊C4—42no—
417C22⋊C4—84no—
418C22⋊C4—84no—
419C22⋊C4—82no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—42no—
426C42—44no—
427C42—42no—
428C42—82no—
429C42—84no—
430C23—24no—
431C23—22no—
432C23—21no—
433C23—22no—
434C23—21no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—48no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—88no—
445C23—84no—
446C23—88no—
447C23—84no—
448C23C2×C22⋊Q812yes—
449C23C2×C22⋊Q814yes—
450C23C23×D411yescharacteristic
451Q8—44no—
452Q8—164no—
453D4—28no—
454D4—44no—
455D4—48no—
456D4—48no—
457D4—44no—
458D4—816no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—816no—
494C2×C4—84no—
495C2×C4—82no—
496C2×C4—164no—
497C2×C4—164no—
498C2×C4C2×C22⋊Q812yes—
499C2×C4C2×C22⋊Q814yes—
500C2×C4#57714yes—
501C2×C4C23×Q811yescharacteristic
502C2×C4C22×#4111yescharacteristic
503C2×C4C2×H(5,𝔽2)12yes—
504C2×C4#58512yes—
505C8—44no—
506C8—44no—
507C22—28no—
508C22—24no—
509C22—22no—
510C22—24no—
511C22—44no—
512C22—42no—
513C22—41no—
514C22—42no—
515C22—41no—
516C22—42no—
517C22—48no—
518C22—44no—
519C22—816no—
520C22—84no—
521C22—82no—
522C22—82no—
523C22C22×C22⋊Q811yescharacteristic
524C22#348411yescharacteristic
525C22#348511yescharacteristic
526C22C2×#57712yes—
527C22#358512yes—
528C4—22no—
529C4—24no—
530C4—24no—
531C4—44no—
532C4—44no—
533C4—44no—
534C4—42no—
535C4—84no—
536C4—81no—
537C4—81no—
538C4—84no—
539C4#345912yes—
540C4#350112yes—
541C2—44no—
542C2—42no—
543C2—84no—
544C2#3498511yescharacteristic
545C2#6049512yes—
546C2#6163111yescharacteristic
547C2#6165111yescharacteristic
548C2#6180812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37307C2semidirect
3. #33157C2semidirect
4. #32956C2semidirect
5. #29971C2semidirect
6. #29962C2semidirect
7. #26436C2semidirect
8. #26423C2semidirect
9. #26359C2semidirect
10. #26338C2semidirect
11. #21937C2semidirect
13. #20848C2semidirect
18. C2×#428C22semidirect
19. C22×#106C22semidirect
27. #2441C22semidirect
28. #2440C22semidirect
40. #2099C22semidirect
41. #2099C22semidirect
44. #2088C22semidirect
45. #2087C22semidirect
48. #2058C22semidirect
49. #2057C22semidirect
52. #1989C22semidirect
53. #1948C22semidirect
54. #1920C22semidirect
55. #1919C22semidirect
63. #1854C22semidirect
65. #1853C22semidirect
66. #1851C22semidirect
67. #1850C22semidirect
73. #1736C22semidirect
74. #1735C22semidirect
81. #535Q8semidirect

Maximal subgroups

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

subgroupindex
#373072
#342652
#331572
#329562
#299712
#299622
#264362
#264232
#263592
#263382
#219372
#217702
#208482