Finite Group Databasebeta

← #198817

Subgroups — #198817

Order 512.

1203 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60803C211yesmaximal, characteristic
2#60237C211yesmaximal, characteristic, complemented
3C2×#1505C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic, complemented
5#17495C211yesmaximal, characteristic, complemented
6#9281C218yesmaximal, complemented
7#7967C211yesmaximal, characteristic, complemented
8#7855C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3214—22no—
12#3213—22no—
13#3207C2214yes—
14#3206C412yescomplemented
15#3205C412yescomplemented
16#3151C412yescomplemented
17#3141C412yescomplemented
18#3036—22no—
19#3006—22no—
20#2981C2214yescomplemented
21#2980—22no—
22#2974—22no—
23#2973—22no—
24#2935—22no—
25#2932—22no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#432C2211yescharacteristic
30#2880—22no—
31C2×#376C2211yescharacteristic
32#1505—22no—
33#1505—22no—
34#1505C2214yescomplemented
35#1505C2214yescomplemented
36C2×#327—22no—
37C2×#327—22no—
38#1496C2214yescomplemented
39#1490—22no—
40#1480—22no—
41#1287C2214yescomplemented
42#1274C2214yescomplemented
43C22×#41—21no—
44C22×#41—21no—
45C2×AGL(1,ℤ/8)—24no—
46C2×#131—24no—
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#532—21no—
57#532C2×C412yes—
58#532C2×C412yes—
59#532D412yes—
60#532D412yes—
61#532C2312yes—
62#532C2312yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#127C2×C411yescharacteristic
66C2×#127C2×C411yescharacteristic
67C2×#124—21no—
68C2×#124—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121D412yes—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4D412yes—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#496—24no—
80#493C2×C414yescomplemented
81#492C2×C414yescomplemented
82#489—24no—
83#487C2×C414yescomplemented
84#486C2×C414yescomplemented
85#463—28no—
86#461—24no—
87#459—24no—
88#449—28no—
89#448—24no—
90#447—24no—
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#432—21no—
104#432—21no—
105#432D412yes—
106#432C2312yes—
107#432C2312yes—
108#431C2×C412yes—
109#431D412yes—
110#431C2312yes—
111#431C2312yes—
112#423—21no—
113#423—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116#417—24no—
117#408—24no—
118#408—28no—
119#407—24no—
120#407—21no—
121#407—21no—
122C2×C22⋊C8—24no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8D412yes—
125C4×M16C2×C412yes—
126C4×M16D412yes—
127C2×#97—21no—
128C2×#97—21no—
129#376—21no—
130#376—21no—
131#376—21no—
132#376C2312yes—
133C2×#95—21no—
134C2×#95—21no—
135#327—28no—
136#327—28no—
137D4⋊C8—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
164AGL(1,ℤ/8)—88no—
165C2×SD16—28no—
166C2×D8—28no—
167#131—88no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—28no—
171C22×C8—22no—
172C22×C8—28no—
173C22×C8—41no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22⋊C412yes—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C412yes—
182#127—21no—
183#127—21no—
184#127—21no—
185#127—21no—
186#127—21no—
187#127C22⋊C412yes—
188#127C22×C412yes—
189#127C22×C412yes—
190#124—22no—
191#124—22no—
192#124—22no—
193#124—22no—
194#124—22no—
195#124—22no—
196#124—41no—
197#124—41no—
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—22no—
207#121—22no—
208#121—41no—
209#121—41no—
210#121—41no—
211#121—41no—
212#121C22⋊C414yes—
213#121C2×D414yes—
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—22no—
223C4×D4—22no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4C22⋊C414yes—
229C4×D4C2×D414yes—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C22⋊C412yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42—41no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42C2411yescharacteristic
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—41no—
261#103—28no—
262#102—28no—
263C22⋊C8—28no—
264C22⋊C8—28no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—41no—
272C22⋊C8C2×D414yes—
273#97—22no—
274#97—22no—
275#97—22no—
276#97—22no—
277#97—22no—
278#97—41no—
279C4×C8C22×C414yes—
280C4×C8C2×D414yes—
281#95—22no—
282#95—22no—
283#95—41no—
284#95—41no—
285C24—22no—
286C24—41no—
287C24#12111yescharacteristic
288#41—48no—
289#41—48no—
290#41—84no—
291#41—84no—
292#41—84no—
293#41—84no—
294#41—84no—
295#41—84no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—41no—
300C2×Q8—82no—
301C2×Q8—82no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4—84no—
324C2×D4—84no—
325C2×D4C2×C22⋊C414yes—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12311yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359SD16—88no—
360D8—88no—
361M16—44no—
362M16—44no—
363M16—88no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—44no—
368C2×C8—42no—
369C2×C8—88no—
370C2×C8—82no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4C2×C22⋊C414yes—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C42—24no—
388C42—42no—
389C42—44no—
390C42C2×C22⋊C411yescharacteristic
391C42C2×C22⋊C411yescharacteristic
392C42C4×D411yescharacteristic
393C42C4×D411yescharacteristic
394C42C4×D411yescharacteristic
395C42C4×D411yescharacteristic
396C42#12011yescharacteristic
397C42#12011yescharacteristic
398C42#12111yescharacteristic
399C42#12111yescharacteristic
400C42#12311yescharacteristic
401C42#12311yescharacteristic
402C42C23×C411yescharacteristic
403C42C22×D411yescharacteristic
404C42C22×D411yescharacteristic
405C42C2×#4111yescharacteristic
406C23—22no—
407C23—22no—
408C23—24no—
409C23—41no—
410C23—44no—
411C23—42no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23#38611yescharacteristic
418C23#38611yescharacteristic
419C23C2×#12111yescharacteristic
420Q8—44no—
421Q8—44no—
422Q8—82no—
423Q8—82no—
424D4—44no—
425D4—44no—
426D4—44no—
427D4—48no—
428D4—44no—
429D4—44no—
430D4—82no—
431D4—82no—
432D4—82no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437D4—84no—
438D4—84no—
439D4—84no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—88no—
470C2×C4#38611yescharacteristic
471C2×C4#38611yescharacteristic
472C2×C4#38611yescharacteristic
473C2×C4#38611yescharacteristic
474C2×C4#38611yescharacteristic
475C2×C4#38611yescharacteristic
476C2×C4C22×C22⋊C411yescharacteristic
477C2×C4C2×C4×D411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×#12011yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×#12311yescharacteristic
482C8—44no—
483C8—44no—
484C8—44no—
485C8—82no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—164no—
501C22—164no—
502C22#190412yes—
503C22#191012yes—
504C22C2×#38611yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—44no—
517C4—84no—
518C4—82no—
519C4#295612yes—
520C2—22no—
521C2—22no—
522C2—41no—
523C2—84no—
524C2—162no—
525C2#2091811yescharacteristic
526C2C2×#190411yescharacteristic
527C2C2×#191011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#60803C2
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. C2×#1505C2semidirect
4. C2×#1505C2semidirect
5. #17495C2semidirect
6. #9281C2semidirect
7. #7967C2semidirect
8. #7855C2semidirect
14. #3206C4semidirect
15. #3205C4semidirect
16. #3151C4semidirect
17. #3141C4semidirect
20. #2981C22semidirect
34. #1505C22semidirect
35. #1505C22semidirect
38. #1496C22semidirect
41. #1287C22semidirect
42. #1274C22semidirect
80. #493C2×C4semidirect
81. #492C2×C4semidirect
83. #487C2×C4semidirect
84. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#608032
#602372
C2×#15052
#174952
#92812
#79672
#78552