Finite Group Databasebeta

← #200757

Subgroups — #200757

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60968C211yesmaximal, characteristic
2#60237C211yesmaximal, characteristic, complemented
3#17671C211yesmaximal, characteristic, complemented
4C2×#1514C211yesmaximal, characteristic, complemented
5C2×#1514C211yesmaximal, characteristic, complemented
6#9417C218yesmaximal, complemented
7#7995C211yesmaximal, characteristic, complemented
8#7975C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3253—22no—
12#3227C412yescomplemented
13#3227C412yescomplemented
14#3205C412yescomplemented
15#3205C412yescomplemented
16#3192—22no—
17#3184C2214yes—
18#3109—22no—
19#3109—22no—
20#2981—22no—
21#2980C2214yescomplemented
22#2974—22no—
23#2973—22no—
24#2935—22no—
25#2932—22no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#432C2211yescharacteristic
30#2880—22no—
31C2×#383C2211yescharacteristic
32#1536C2214yescomplemented
33#1530—22no—
34#1527—22no—
35#1514—22no—
36#1514—22no—
37#1514C2214yescomplemented
38#1514C2214yescomplemented
39C2×#327—22no—
40C2×#327—22no—
41#1291C2214yescomplemented
42#1288C2214yescomplemented
43C22×#41—21no—
44C22×#41—21no—
45C2×#131—24no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532C2×C412yes—
57#532C2×C412yes—
58#532D412yes—
59#532D412yes—
60#532C2312yes—
61#532C2312yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#127C2×C411yescharacteristic
65C2×#127C2×C411yescharacteristic
66C2×#124—21no—
67C2×#124—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121D412yes—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4D412yes—
76C22×C42C2311yescharacteristic
77#498C2×C414yescomplemented
78#493C2×C414yescomplemented
79#486C2×C414yescomplemented
80#486C2×C414yescomplemented
81#481—24no—
82#481—24no—
83#476—24no—
84#476—24no—
85#469—28no—
86#459—24no—
87#459—24no—
88#447—24no—
89#447—24no—
90#442—28no—
91#436—24no—
92#435—28no—
93#435—28no—
94#435—28no—
95C8×D4—24no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432D412yes—
104#432D412yes—
105#432C2312yes—
106#432C2312yes—
107#432C2312yes—
108#431—21no—
109#431—21no—
110#431C2×C412yes—
111#431C2×C412yes—
112#423—21no—
113#423—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116#408—24no—
117#408—28no—
118#407—24no—
119#407—21no—
120#407—21no—
121C2×C22⋊C8—24no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8D412yes—
124C4×M16D412yes—
125C4×M16C2312yes—
126C2×#97—21no—
127C2×#97—21no—
128#383—21no—
129#383—21no—
130#383—21no—
131#383C2312yes—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135#329—28no—
136#327—28no—
137#327—28no—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—41no—
145C2×#41—41no—
146C2×#41—84no—
147C2×#41—84no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—41no—
157C22×D4C22⋊C412yes—
158C23×C4—22no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C2×D8—28no—
165#131—88no—
166C2×M16—22no—
167C2×M16—28no—
168C2×M16—22no—
169C22×C8—22no—
170C22×C8—28no—
171C22×C8—41no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8C22⋊C412yes—
178C4⋊Q8C22×C412yes—
179C4⋊Q8C22×C412yes—
180#127—21no—
181#127—21no—
182#127—21no—
183#127—21no—
184#127—21no—
185#127C22⋊C412yes—
186#127C22×C412yes—
187#127C22×C412yes—
188#124—22no—
189#124—22no—
190#124—22no—
191#124—22no—
192#124—22no—
193#124—22no—
194#124—41no—
195#124—41no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—22no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—41no—
207#121—41no—
208#121—41no—
209#121—41no—
210#121C22⋊C414yes—
211#121C2×D414yes—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4C22⋊C414yes—
227C4×D4C2×D414yes—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C22⋊C412yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42C2411yescharacteristic
248#106—28no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—41no—
257#102—28no—
258#102—28no—
259C22⋊C8—28no—
260C22⋊C8—28no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268C22⋊C8C2×D414yes—
269#97—22no—
270#97—22no—
271#97—22no—
272#97—22no—
273#97—41no—
274#97C2×D414yes—
275C4×C8—22no—
276C4×C8C22×C414yes—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—41no—
281#95—41no—
282#95—41no—
283C24—22no—
284C24—41no—
285C24#12111yescharacteristic
286#41—48no—
287#41—84no—
288#41—84no—
289#41—84no—
290#41—84no—
291#41—84no—
292#41—84no—
293#41—84no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—82no—
299C2×Q8—82no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—82no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—84no—
322C2×D4—84no—
323C2×D4C2×C22⋊C414yes—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—81no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12011yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12311yescharacteristic
353C22×C4#12311yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4111yescharacteristic
357D8—88no—
358M16—44no—
359M16—82no—
360M16—88no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—44no—
365C2×C8—42no—
366C2×C8—88no—
367C2×C8—82no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4C2×C22⋊C414yes—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C42C2×C22⋊C411yescharacteristic
385C42C2×C22⋊C411yescharacteristic
386C42C4×D411yescharacteristic
387C42C4×D411yescharacteristic
388C42C4×D411yescharacteristic
389C42C4×D411yescharacteristic
390C42#12011yescharacteristic
391C42#12011yescharacteristic
392C42#12111yescharacteristic
393C42#12111yescharacteristic
394C42#12311yescharacteristic
395C42#12311yescharacteristic
396C42C23×C411yescharacteristic
397C42C22×D411yescharacteristic
398C42C22×D411yescharacteristic
399C42C2×#4111yescharacteristic
400C23—22no—
401C23—22no—
402C23—24no—
403C23—41no—
404C23—44no—
405C23—42no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23#38611yescharacteristic
412C23#38611yescharacteristic
413C23C2×#12111yescharacteristic
414Q8—44no—
415Q8—82no—
416Q8—82no—
417Q8—82no—
418D4—44no—
419D4—44no—
420D4—44no—
421D4—44no—
422D4—48no—
423D4—82no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—82no—
428D4—82no—
429D4—82no—
430D4—82no—
431D4—84no—
432D4—84no—
433D4—84no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—164no—
464C2×C4#38611yescharacteristic
465C2×C4#38611yescharacteristic
466C2×C4#38611yescharacteristic
467C2×C4#38611yescharacteristic
468C2×C4#38611yescharacteristic
469C2×C4#38611yescharacteristic
470C2×C4C22×C22⋊C411yescharacteristic
471C2×C4C2×C4×D411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×#12011yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×#12311yescharacteristic
476C8—44no—
477C8—44no—
478C8—44no—
479C8—82no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—164no—
495C22—164no—
496C22#190312yes—
497C22#190412yes—
498C22C2×#38611yescharacteristic
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—44no—
512C4—82no—
513C4—162no—
514C2—22no—
515C2—22no—
516C2—41no—
517C2—84no—
518C2—162no—
519C2#2094711yescharacteristic
520C2C2×#190311yescharacteristic
521C2C2×#190411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#60968C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60237C2semidirect
3. #17671C2semidirect
4. C2×#1514C2semidirect
5. C2×#1514C2semidirect
6. #9417C2semidirect
7. #7995C2semidirect
8. #7975C2semidirect
12. #3227C4semidirect
13. #3227C4semidirect
14. #3205C4semidirect
15. #3205C4semidirect
21. #2980C22semidirect
32. #1536C22semidirect
37. #1514C22semidirect
38. #1514C22semidirect
41. #1291C22semidirect
42. #1288C22semidirect
77. #498C2×C4semidirect
78. #493C2×C4semidirect
79. #486C2×C4semidirect
80. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#609682
#602372
#176712
C2×#15142
#94172
#79952
#79752