Finite Group Databasebeta

← #6633293

Subgroups — #6633293

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37298C211yesmaximal, characteristic, complemented
2#35679C211yesmaximal, characteristic, complemented
3#32912C212yesmaximal, complemented
4#31651C218yesmaximal, complemented
5#30752C218yesmaximal, complemented
6#29721C212yesmaximal, complemented
7#29337C212yesmaximal, complemented
8#25731C212yesmaximal, complemented
9#23940C212yesmaximal, complemented
10#23939C212yesmaximal, complemented
11#21917C211yesmaximal, characteristic, complemented
12C2×#540C2212yes—
13C22×#124C2211yescharacteristic
14C2×#519C2211yescharacteristic
15C2×#496C2214yes—
16C2×#460C2214yescomplemented
17C2×#448C2214yes—
18C2×#447C2214yes—
19#2856C2212yes—
20#2682—24no—
21#2675—24no—
22#2619C2212yes—
23#2566C2214yescomplemented
24#2541C2214yescomplemented
25#2482—24no—
26#2448—24no—
27#2448C2218yes—
28#2310C2211yescharacteristic
29#2309C2211yescharacteristic
30#2309C2211yescharacteristic
31#2296C2211yescharacteristic
32#2296C2211yescharacteristic
33#2099C2214yescomplemented
34#2073C2214yes—
35#2054C2218yes—
36#2041C2214yescomplemented
37#2026C2214yescomplemented
38#2025C2214yes—
39#2014C2214yescomplemented
40#2012C2218yes—
41#2011C2214yes—
42#2011C2214yes—
43#1959C2214yes—
44#1936C2218yescomplemented
45#1917C2218yescomplemented
46#1880C2218yescomplemented
47#1879C2218yes—
48#1877C2218yescomplemented
49#1875C2214yes—
50#1875C2214yes—
51#1818C2212yes—
52#1805C2214yescomplemented
53#1798C2212yes—
54#1749C2212yes—
55C23×Q8C2311yescharacteristic
56C23×D4C2311yescharacteristic
57C22×SD16C2318yes—
58C22×D8C2318yes—
59#540—48no—
60C2×#127C2312yes—
61C2×#127C2312yes—
62C2×#127C2312yes—
63C2×#125C2312yes—
64C2×#124—22no—
65C2×#124—44no—
66C2×#124C2314yes—
67C2×#123—24no—
68C2×#121—24no—
69C2×#121—28no—
70C2×#121—24no—
71C2×#121C2314yes—
72#519—44no—
73C2×C4×Q8—22no—
74C2×C4×Q8C2314yes—
75C2×C4×D4C2314yes—
76C2×#117—22no—
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C22⋊C4—22no—
82C22×C42C2311yescharacteristic
83#496—48no—
84#460—48no—
85#448—48no—
86#447—48no—
87C2×C4⋊C8C2314yes—
88C2×#103C2318yes—
89C2×#102C2318yes—
90C2×#102C2318yes—
91C2×#102C2318yes—
92C2×C22⋊C8C2314yes—
93C2×#97C2314yes—
94#399—28no—
95#397—28no—
96#396—28no—
97#394—24no—
98#392—24no—
99#392—28no—
100#391—22no—
101#391C2314yes—
102#390—22no—
103#390—22no—
104#390—24no—
105#390C2314yes—
106#390C2314yes—
107#390C2314yes—
108#390C2314yes—
109#389D414yes—
110#389C2314yes—
111#388—28no—
112#388—28no—
113#385—22no—
114#385—28no—
115#385D414yes—
116#384C2312yes—
117#382C2312yes—
118#381—22no—
119#379—22no—
120#379—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2314yes—
123#340C2318yes—
124#340C2318yes—
125#336C2314yes—
126#336C2314yes—
127#336C2314yes—
128#336C2314yes—
129C25—21no—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—24no—
133C22×Q8—42no—
134C22×Q8C2412yes—
135C22×D4—24no—
136C22×D4—21no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4C2414yes—
144C22×D4C2412yes—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
149C2×SD16—216no—
150C2×SD16—48no—
151C2×SD16—48no—
152C2×D8—48no—
153C2×D8—48no—
154C2×D8—48no—
155C22×C8C2418yes—
156#127—44no—
157#127—44no—
158#127—44no—
159#125—44no—
160#124—28no—
161#124—44no—
162#124—48no—
163#123—416no—
164#121—28no—
165#121—28no—
166#121—416no—
167#121—88no—
168#121—816no—
169C4×Q8—28no—
170C4×Q8—44no—
171C4×Q8—48no—
172C4×D4—44no—
173C4×D4—44no—
174#117—24no—
175#117—42no—
176#117—42no—
177#117—42no—
178#117—48no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D418yes—
186C2×C4⋊C4C2412yes—
187C2×C4⋊C4C2412yes—
188C2×C4⋊C4C2414yes—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—48no—
199C2×C22⋊C4C2×D412yes—
200C2×C22⋊C4C2×D412yes—
201C2×C42—24no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—42no—
205C2×C42C2×D412yes—
206C2×C42C2412yes—
207C2×C42C2412yes—
208C2×C42C2412yes—
209C4⋊C8—44no—
210#103—48no—
211#102—48no—
212#102—48no—
213#102—48no—
214C22⋊C8—28no—
215#97—44no—
216#95—24no—
217#95—28no—
218#95—24no—
219#95C2×D418yes—
220C24—22no—
221C24—24no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—84no—
226C24C22×D411yescharacteristic
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—416no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8H(5,𝔽2)14yes—
235C2×D4—24no—
236C2×D4—28no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—28no—
241C2×D4—24no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—48no—
245C2×D4—48no—
246C2×D4—48no—
247C2×D4—42no—
248C2×D4—88no—
249C2×D4—88no—
250C2×D4—88no—
251C2×D4—84no—
252C2×D4—84no—
253C2×D4—84no—
254C2×D4—88no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—88no—
268C22×C4C22×D414yes—
269C22×C4C22×D414yes—
270C22×C4C22×D414yes—
271C22×C4C22×D411yescharacteristic
272C22×C4H(5,𝔽2)14yes—
273C22×C4C2511yescharacteristic
274SD16—816no—
275D8—816no—
276C2×C8—28no—
277C2×C8—28no—
278C2×C8—28no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—42no—
285C4⋊C4—48no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C22⋊C4—24no—
291C22⋊C4—42no—
292C22⋊C4—416no—
293C22⋊C4—48no—
294C22⋊C4—416no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—48no—
298C22⋊C4—88no—
299C22⋊C4—88no—
300C22⋊C4—88no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—42no—
305C42—48no—
306C42—42no—
307C42—42no—
308C23—24no—
309C23—21no—
310C23—21no—
311C23—48no—
312C23—48no—
313C23—48no—
314C23—44no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—42no—
319C23—44no—
320C23—84no—
321C23—88no—
322C23—164no—
323C23—164no—
324C23#53512yes—
325C23C23×D411yescharacteristic
326Q8—28no—
327Q8—48no—
328Q8—44no—
329D4—48no—
330D4—48no—
331D4—48no—
332D4—84no—
333D4—168no—
334D4—168no—
335D4—168no—
336D4—168no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—84no—
358C2×C4—168no—
359C2×C4#53512yes—
360C2×C4C2×H(5,𝔽2)12yes—
361C2×C4C2×H(5,𝔽2)12yes—
362C2×C4#58512yes—
363C8—48no—
364C22—22no—
365C22—24no—
366C22—42no—
367C22—42no—
368C22—82no—
369C22—84no—
370C22—84no—
371C22—84no—
372C22—84no—
373C22—88no—
374C22—84no—
375C22—88no—
376C22—84no—
377C22—168no—
378C22C2×#53511yescharacteristic
379C22#348411yescharacteristic
380C22#348711yescharacteristic
381C22#358514yes—
382C4—22no—
383C4—44no—
384C4—44no—
385C4—44no—
386C4—44no—
387C4—84no—
388C4—82no—
389C4—82no—
390C4#353414yes—
391C2—42no—
392C2—164no—
393C2—162no—
394C2#3671011yescharacteristic
395C2#6082712yes—
396C2#6165112yes—
397C2#6194212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #37298C2semidirect
2. #35679C2semidirect
3. #32912C2semidirect
4. #31651C2semidirect
5. #30752C2semidirect
6. #29721C2semidirect
7. #29337C2semidirect
8. #25731C2semidirect
9. #23940C2semidirect
10. #23939C2semidirect
11. #21917C2semidirect
16. C2×#460C22semidirect
23. #2566C22semidirect
24. #2541C22semidirect
33. #2099C22semidirect
36. #2041C22semidirect
37. #2026C22semidirect
39. #2014C22semidirect
44. #1936C22semidirect
45. #1917C22semidirect
46. #1880C22semidirect
48. #1877C22semidirect
52. #1805C22semidirect

Maximal subgroups

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

subgroupindex
#372982
#356792
#329122
#316512
#307522
#297212
#293372
#257312
#239402
#239392
#219172