Finite Group Databasebeta

← #6633253

Subgroups — #6633253

Order 512.

1712 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37387C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic
3#32907C212yesmaximal, complemented
4#31662C214yesmaximal, complemented
5#31644C214yesmaximal, complemented
6#30760C214yesmaximal, complemented
7#30741C214yesmaximal, complemented
8#29714C212yesmaximal, complemented
9#29341C212yesmaximal, complemented
10#25732C212yesmaximal
11#23944C212yesmaximal, complemented
12#23942C212yesmaximal, complemented
13#21921C211yesmaximal, characteristic, complemented
14C2×#541C2212yes—
15C2×#535C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17C2×#497C2212yes—
18C2×#492C2212yes—
19C2×#462C2212yescomplemented
20C2×#461C2212yescomplemented
21C2×#452C2214yes—
22C2×#450C2214yes—
23#2857C2212yes—
24#2854—22no—
25#2726—22no—
26#2726—22no—
27#2722—24no—
28#2624—22no—
29#2621C2212yes—
30#2565C2214yescomplemented
31#2546C2214yescomplemented
32#2448C2214yes—
33#2445C2214yes—
34#2418—24no—
35#2316C2211yescharacteristic
36#2311C2211yescharacteristic
37#2309C2211yescharacteristic
38#2296C2211yescharacteristic
39C2×#390C2211yescharacteristic
40#2101C2214yescomplemented
41#2071C2214yes—
42#2055C2214yes—
43#2055C2214yes—
44#2043C2214yescomplemented
45#2026C2212yescomplemented
46#2025C2214yes—
47#2025C2212yescomplemented
48#2016C2212yescomplemented
49#2015C2214yes—
50#2014C2214yes—
51#2013C2214yes—
52#2013C2214yes—
53#2013C2212yescomplemented
54#1961C2212yes—
55#1958C2212yes—
56#1935C2214yescomplemented
57#1933C2214yescomplemented
58#1917C2214yescomplemented
59#1916C2214yescomplemented
60#1880C2214yes—
61#1880C2214yes—
62#1880C2214yescomplemented
63#1878C2214yescomplemented
64#1878C2214yescomplemented
65#1877C2214yescomplemented
66#1873C2214yes—
67#1872C2214yes—
68#1818C2212yes—
69#1803C2212yescomplemented
70#1802C2212yescomplemented
71#1797C2212yes—
72#1750C2212yes—
73C22×#41C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C22×Q16C2314yes—
76C22×SD16C2314yes—
77C22×SD16C2314yes—
78C22×SD16C2314yes—
79#541—48no—
80#535—42no—
81#535—42no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#127C2312yes—
86C2×#126—24no—
87C2×#125—24no—
88C2×#125C2312yes—
89C2×#124C2312yes—
90C2×#123—24no—
91C2×C22⋊Q8C2314yes—
92C2×#121—22no—
93C2×#120—22no—
94C4×#41—42no—
95C4×#41—42no—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4C2312yes—
100C2×#117—21no—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—22no—
106C22×C42—21no—
107C22×C42C2311yescharacteristic
108#497—44no—
109#492—44no—
110#462—44no—
111#461—44no—
112#452—48no—
113#450—48no—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8C2314yes—
125C2×#97C2312yes—
126C2×C4×C8C2312yes—
127#399—24no—
128#398—24no—
129#398—24no—
130#398—24no—
131#398—24no—
132#398—24no—
133#397—24no—
134#397—24no—
135#394—24no—
136#394—24no—
137#391C2314yes—
138#391C2314yes—
139#391C2312yes—
140#390—22no—
141#390—21no—
142#390—21no—
143#390C2314yes—
144#390C2314yes—
145#390C2312yes—
146#389—22no—
147#389—24no—
148#389—24no—
149#389—22no—
150#389—22no—
151#389D412yes—
152#389D412yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#387—24no—
157#387—24no—
158#386—22no—
159#386—24no—
160#385D414yes—
161#384C2312yes—
162#382C2312yes—
163#380—22no—
164#380—22no—
165#380—22no—
166#380—22no—
167#379—22no—
168C4×C4⋊C4C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#336C2314yes—
174#336C2314yes—
175#336C2312yes—
176#336C2312yes—
177#336C2314yes—
178C25—21no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—24no—
182C2×#41—42no—
183C22×Q8C2414yes—
184C22×Q8C2411yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—42no—
192C22×D4C2412yes—
193C22×D4C2411yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4C2×D412yes—
198C23×C4C2411yescharacteristic
199C2×Q16—28no—
200C2×Q16—44no—
201C2×Q16—44no—
202C2×SD16—28no—
203C2×SD16—44no—
204C2×SD16—44no—
205C2×SD16—44no—
206C2×SD16—44no—
207C2×SD16—44no—
208C2×SD16—44no—
209C2×SD16—44no—
210C2×SD16—44no—
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—44no—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4⋊Q8—42no—
218#127—24no—
219#127—42no—
220#126—416no—
221#125—44no—
222#125—416no—
223#124—24no—
224#124—42no—
225#123—88no—
226C22⋊Q8—28no—
227C22⋊Q8—28no—
228#121—48no—
229#120—48no—
230C4×Q8—24no—
231C4×Q8—42no—
232C4×Q8—44no—
233C4×Q8—44no—
234C4×D4—24no—
235C4×D4—42no—
236C4×D4—48no—
237#117—24no—
238#117—41no—
239#117—41no—
240#117—41no—
241#117—41no—
242#117—44no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
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—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—84no—
273C2×C22⋊C4C2×D412yes—
274C2×C22⋊C4C2×D412yes—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—44no—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2412yes—
284C2×C42C2412yes—
285C2×C42C2412yes—
286C4⋊C8—42no—
287C4⋊C8—42no—
288#103—44no—
289#103—44no—
290#103—44no—
291#103—44no—
292#103—44no—
293#102—44no—
294#102—44no—
295#102—44no—
296C22⋊C8—28no—
297#97—42no—
298C4×C8—42no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95C2×D414yes—
307#95C2×D414yes—
308C24—21no—
309C24—22no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—82no—
314C24—82no—
315C24C22×D411yescharacteristic
316#41—48no—
317#41—48no—
318C2×Q8—21no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8H(5,𝔽2)12yes—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—48no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4H(5,𝔽2)12yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4C22×D414yes—
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—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—44no—
400C4⋊C4—84no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—84no—
404C22⋊C4—24no—
405C22⋊C4—42no—
406C22⋊C4—48no—
407C22⋊C4—44no—
408C22⋊C4—42no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C22⋊C4—84no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—24no—
420C42—41no—
421C42—41no—
422C42—44no—
423C42—42no—
424C42—48no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—22no—
429C23—22no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—84no—
440C23—88no—
441C23—164no—
442C23—162no—
443C23—162no—
444C23#53512yes—
445C23C23×D411yescharacteristic
446Q8—24no—
447Q8—42no—
448Q8—44no—
449Q8—44no—
450Q8—44no—
451Q8—44no—
452D4—24no—
453D4—48no—
454D4—42no—
455D4—44no—
456D4—44no—
457D4—84no—
458D4—168no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—48no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—84no—
490C2×C4—88no—
491C2×C4—162no—
492C2×C4#53512yes—
493C2×C4#57714yes—
494C2×C4C2×H(5,𝔽2)12yes—
495C2×C4C2×H(5,𝔽2)11yescharacteristic
496C2×C4C2×H(5,𝔽2)11yescharacteristic
497C2×C4#58512yes—
498C8—44no—
499C8—44no—
500C22—22no—
501C22—24no—
502C22—42no—
503C22—41no—
504C22—41no—
505C22—42no—
506C22—44no—
507C22—82no—
508C22—81no—
509C22—81no—
510C22—82no—
511C22—84no—
512C22—84no—
513C22—82no—
514C22—82no—
515C22—168no—
516C22C2×#53511yescharacteristic
517C22#348411yescharacteristic
518C22#348711yescharacteristic
519C22C2×#57712yes—
520C22#358612yes—
521C4—22no—
522C4—24no—
523C4—42no—
524C4—44no—
525C4—42no—
526C4—44no—
527C4—84no—
528C4—84no—
529C4—82no—
530C4—82no—
531C4#353412yes—
532C4#353412yes—
533C2—42no—
534C2—82no—
535C2—162no—
536C2#3671011yescharacteristic
537C2#6083612yes—
538C2#6161812yes—
539C2#6193311yescharacteristic
540C2#6194311yescharacteristic
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. #32907C2semidirect
4. #31662C2semidirect
5. #31644C2semidirect
6. #30760C2semidirect
7. #30741C2semidirect
8. #29714C2semidirect
9. #29341C2semidirect
11. #23944C2semidirect
12. #23942C2semidirect
13. #21921C2semidirect
19. C2×#462C22semidirect
20. C2×#461C22semidirect
30. #2565C22semidirect
31. #2546C22semidirect
40. #2101C22semidirect
44. #2043C22semidirect
45. #2026C22semidirect
47. #2025C22semidirect
48. #2016C22semidirect
53. #2013C22semidirect
56. #1935C22semidirect
57. #1933C22semidirect
58. #1917C22semidirect
59. #1916C22semidirect
62. #1880C22semidirect
63. #1878C22semidirect
64. #1878C22semidirect
65. #1877C22semidirect
69. #1803C22semidirect
70. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#373872
#352542
#329072
#316622
#316442
#307602
#307412
#297142
#293412
#257322
#239442
#239422
#219212