Finite Group Databasebeta

← #6633263

Subgroups — #6633263

Order 512.

1759 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37387C211yesmaximal, characteristic, complemented
2#35152C211yesmaximal, characteristic
3#32905C212yesmaximal, complemented
4#31662C214yesmaximal, complemented
5#31642C214yesmaximal, complemented
6#30757C214yesmaximal, complemented
7#30741C214yesmaximal, complemented
8#29717C212yesmaximal, complemented
9#29344C212yesmaximal, complemented
10#25732C212yesmaximal
11#23944C212yesmaximal, complemented
12#23942C212yesmaximal, complemented
13#21919C211yesmaximal, characteristic, complemented
14C2×#541C2212yes—
15C2×#535C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#497C2212yes—
18C2×#492C2212yes—
19C2×#461C2214yes—
20C2×#452C2214yes—
21C2×#450C2214yes—
22#2857C2212yes—
23#2849—22no—
24#2726—22no—
25#2726—22no—
26#2722—22no—
27#2722—22no—
28#2622—22no—
29#2621C2212yes—
30#2565C2214yescomplemented
31#2546C2214yescomplemented
32#2447C2214yes—
33#2445C2214yes—
34#2422—22no—
35#2420—22no—
36#2317C2211yescharacteristic
37#2311C2211yescharacteristic
38#2309C2211yescharacteristic
39#2297C2211yescharacteristic
40#2296C2211yescharacteristic
41#2101C2214yescomplemented
42#2072C2212yescomplemented
43#2071C2212yescomplemented
44#2056C2214yescomplemented
45#2055C2214yescomplemented
46#2043C2214yescomplemented
47#2026C2212yescomplemented
48#2025C2214yes—
49#2025C2212yescomplemented
50#2016C2212yescomplemented
51#2016C2212yescomplemented
52#2015C2214yes—
53#2014C2214yes—
54#2013C2214yes—
55#2013C2212yescomplemented
56#2013C2212yescomplemented
57#1961C2212yes—
58#1958C2212yes—
59#1935C2214yescomplemented
60#1933C2214yescomplemented
61#1917C2214yescomplemented
62#1916C2214yescomplemented
63#1880C2214yes—
64#1880C2214yes—
65#1880C2214yescomplemented
66#1878C2214yescomplemented
67#1877C2214yes—
68#1877C2214yes—
69#1873C2214yes—
70#1872C2214yes—
71#1818C2212yes—
72#1802C2214yes—
73#1797C2212yes—
74#1750C2212yes—
75C22×#41C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4—21no—
78C22×Q16C2314yes—
79C22×SD16C2314yes—
80C22×SD16C2314yes—
81C22×SD16C2314yes—
82#541—48no—
83#535—44no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×#127C2312yes—
88C2×#126—24no—
89C2×#125—24no—
90C2×#125C2312yes—
91C2×#124C2312yes—
92C2×#123—24no—
93C2×C22⋊Q8C2314yes—
94C2×#121—21no—
95C2×#121—21no—
96C2×#120—22no—
97C2×C4×Q8C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4—44no—
100C2×C4×D4C2314yes—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C42C2311yescharacteristic
107#497—44no—
108#492—44no—
109#461—48no—
110#452—48no—
111#450—48no—
112C2×C4⋊C8C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×C22⋊C8C2314yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124#398—24no—
125#398—24no—
126#398—22no—
127#398—22no—
128#398—24no—
129#398—24no—
130#397—24no—
131#397—22no—
132#397—22no—
133#397—24no—
134#394—24no—
135#394—24no—
136#391C2314yes—
137#391C2314yes—
138#391C2312yes—
139#391C2312yes—
140#390—22no—
141#390C2314yes—
142#390C2312yes—
143#390C2312yes—
144#389—24no—
145#389—24no—
146#389—22no—
147#389—22no—
148#389D412yes—
149#389D412yes—
150#389C2314yes—
151#387—24no—
152#387—24no—
153#386—21no—
154#386—21no—
155#386—22no—
156#386—24no—
157#385—24no—
158#385—22no—
159#385D412yes—
160#385D412yes—
161#384—21no—
162#384C2312yes—
163#382—21no—
164#382C2312yes—
165#381—24no—
166#380—22no—
167#380—22no—
168#380—22no—
169#379—22no—
170C4×C4⋊C4C2314yes—
171C4×C22⋊C4—22no—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#336C2312yes—
177#336C2312yes—
178#336C2314yes—
179#336C2312yes—
180#336C2312yes—
181#336C2314yes—
182C25—21no—
183C2×#41—24no—
184C2×#41—42no—
185C2×#41—42no—
186C22×Q8C2414yes—
187C22×Q8C2411yescharacteristic
188C22×D4—24no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—42no—
195C22×D4C2411yescharacteristic
196C22×D4C2412yes—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—24no—
200C23×C4—44no—
201C23×C4—84no—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2411yescharacteristic
205C2×Q16—44no—
206C2×Q16—44no—
207C2×Q16—44no—
208C2×SD16—28no—
209C2×SD16—28no—
210C2×SD16—44no—
211C2×SD16—44no—
212C2×SD16—44no—
213C2×SD16—44no—
214C2×SD16—44no—
215C2×SD16—44no—
216C2×SD16—44no—
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—44no—
220C4⋊Q8—44no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223#127—24no—
224#127—42no—
225#126—416no—
226#125—44no—
227#125—416no—
228#124—24no—
229#124—42no—
230#123—88no—
231C22⋊Q8—28no—
232C22⋊Q8—28no—
233#121—44no—
234#121—44no—
235#120—48no—
236C4×Q8—44no—
237C4×Q8—44no—
238C4×D4—28no—
239C4×D4—44no—
240C4×D4—48no—
241#117—24no—
242#117—42no—
243#117—42no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D412yes—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—42no—
281C2×C42C2×D412yes—
282C2×C42C2412yes—
283C2×C42C2412yes—
284C2×C42C2412yes—
285C4⋊C8—44no—
286#103—44no—
287#103—44no—
288#103—44no—
289#103—44no—
290#103—44no—
291#102—44no—
292#102—44no—
293#102—44no—
294C22⋊C8—28no—
295#97—42no—
296C4×C8—42no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95C2×D414yes—
308#95C2×D414yes—
309C24—24no—
310C24—22no—
311C24—21no—
312C24—42no—
313C24—42no—
314C24—82no—
315C24—82no—
316C24C22×D411yescharacteristic
317#41—88no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—416no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4H(5,𝔽2)14yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—48no—
362C22×C4—84no—
363C22×C4—82no—
364C22×C4—816no—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D414yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4H(5,𝔽2)14yes—
372C22×C4C2511yescharacteristic
373Q16—88no—
374SD16—88no—
375SD16—88no—
376SD16—88no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C4⋊C4—24no—
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—44no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C4⋊C4—84no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C22⋊C4—24no—
405C22⋊C4—42no—
406C22⋊C4—48no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—42no—
410C22⋊C4—48no—
411C22⋊C4—44no—
412C22⋊C4—84no—
413C22⋊C4—82no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C42—22no—
417C42—22no—
418C42—24no—
419C42—24no—
420C42—42no—
421C42—42no—
422C42—48no—
423C23—22no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—24no—
428C23—48no—
429C23—44no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—88no—
439C23—84no—
440C23—84no—
441C23—88no—
442C23#53512yes—
443C23C23×D411yescharacteristic
444Q8—44no—
445Q8—44no—
446Q8—44no—
447Q8—44no—
448Q8—44no—
449D4—28no—
450D4—48no—
451D4—44no—
452D4—44no—
453D4—84no—
454D4—164no—
455D4—164no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—816no—
487C2×C4—84no—
488C2×C4—82no—
489C2×C4—84no—
490C2×C4#53512yes—
491C2×C4#57714yes—
492C2×C4C2×H(5,𝔽2)12yes—
493C2×C4C2×H(5,𝔽2)12yes—
494C2×C4#58512yes—
495C8—44no—
496C8—44no—
497C22—22no—
498C22—24no—
499C22—41no—
500C22—41no—
501C22—42no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—816no—
506C22—84no—
507C22—84no—
508C22—81no—
509C22—82no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22C2×#53511yescharacteristic
514C22#348411yescharacteristic
515C22#348711yescharacteristic
516C22C2×#57712yes—
517C22#358612yes—
518C4—22no—
519C4—24no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—81no—
525C4—84no—
526C4—84no—
527C4—82no—
528C4—82no—
529C4#353414yes—
530C2—42no—
531C2—84no—
532C2—161no—
533C2#3671011yescharacteristic
534C2#6083612yes—
535C2#6163111yescharacteristic
536C2#6164911yescharacteristic
537C2#6191112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37387C2semidirect
3. #32905C2semidirect
4. #31662C2semidirect
5. #31642C2semidirect
6. #30757C2semidirect
7. #30741C2semidirect
8. #29717C2semidirect
9. #29344C2semidirect
11. #23944C2semidirect
12. #23942C2semidirect
13. #21919C2semidirect
30. #2565C22semidirect
31. #2546C22semidirect
41. #2101C22semidirect
42. #2072C22semidirect
43. #2071C22semidirect
44. #2056C22semidirect
45. #2055C22semidirect
46. #2043C22semidirect
47. #2026C22semidirect
49. #2025C22semidirect
50. #2016C22semidirect
51. #2016C22semidirect
55. #2013C22semidirect
56. #2013C22semidirect
59. #1935C22semidirect
60. #1933C22semidirect
61. #1917C22semidirect
62. #1916C22semidirect
65. #1880C22semidirect
66. #1878C22semidirect

Maximal subgroups

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

subgroupindex
#373872
#351522
#329052
#316622
#316422
#307572
#307412
#297172
#293442
#257322
#239442
#239422
#219192