Finite Group Databasebeta

← #6510737

Subgroups — #6510737

Order 512.

1942 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3#29110C214yesmaximal, complemented
4#27966C214yesmaximal, complemented
5#27717C2116yesmaximal, complemented
6#21758C212yesmaximal
7C2×#1823C212yesmaximal, complemented
8#20526C211yesmaximal, characteristic, complemented
9C22×#123—22no—
10C22×#121C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2851C2214yes—
14#2646—22no—
15#2569—24no—
16#2501—24no—
17#2501C2218yes—
18#2442C2218yescomplemented
19C2×#399—22no—
20#2315—22no—
21C2×#390C2212yes—
22C2×#384C2212yes—
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#2100C22116yescomplemented
26#2073C2218yescomplemented
27#2025C2218yescomplemented
28#1992C2218yescomplemented
29#1936C22116yescomplemented
30#1893C2218yes—
31#1874C2218yes—
32#1836C2218yescomplemented
33#1823C2214yes—
34#1823C2218yes—
35#1823C2218yescomplemented
36#1786C2214yes—
37#1752C2218yescomplemented
38#1749C2214yes—
39C2×#340C2218yescomplemented
40C23×D4—21no—
41C23×D4C2311yescharacteristic
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2314yes—
45C2×#125C2314yes—
46C2×#123—28no—
47C2×#123—28no—
48C2×#123—416no—
49C2×#121—22no—
50C2×#121—28no—
51C2×#121C2314yes—
52C22×C4⋊C4—28no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2314yes—
59C22×C4⋊C4C2314yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—24no—
63C22×C22⋊C4D412yes—
64C22×C42C2312yes—
65C2×#102C23116yes—
66C2×C22⋊C8—28no—
67C2×C22⋊C8C2314yes—
68C2×C22⋊C8C2314yes—
69C2×#97C2318yes—
70#400—28no—
71#399—216no—
72#398—28no—
73#390—24no—
74#390C2318yes—
75#389C2318yes—
76#387—28no—
77#385—28no—
78#385—28no—
79#384—24no—
80#384C2318yes—
81#382C2314yes—
82#380—22no—
83#380—22no—
84#380—28no—
85#380C2314yes—
86#380C2314yes—
87C4×C22⋊C4D414yes—
88C4×C22⋊C4C2314yes—
89C2×#95—22no—
90C2×#95D412yes—
91#340C23116yes—
92#340C23116yes—
93#340C23116yes—
94#336C2318yes—
95C25—21no—
96C25C2×D411yescharacteristic
97C22×D4—24no—
98C22×D4—21no—
99C22×D4—24no—
100C22×D4—44no—
101C22×D4—44no—
102C22×D4—48no—
103C22×D4C2412yes—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—44no—
109C23×C4—44no—
110C23×C4—48no—
111C23×C4C2×D414yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4#4118yes—
115C23×C4C2412yes—
116C23×C4C2411yescharacteristic
117C22×C8—28no—
118C22×C8—48no—
119C22×C8C2418yes—
120#125—48no—
121#123—432no—
122#123—432no—
123#121—28no—
124#121—28no—
125#121—48no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—216no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—48no—
138C2×C4⋊C4—416no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—48no—
141C2×C4⋊C4C2414yes—
142C2×C4⋊C4C2412yes—
143C2×C4⋊C4C2418yes—
144C2×C22⋊C4—28no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—416no—
150C2×C22⋊C4—48no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4C2×D414yes—
153C2×C22⋊C4C2×D414yes—
154C2×C42—44no—
155C2×C42C2×D414yes—
156C2×C42C2414yes—
157#102—416no—
158C22⋊C8—28no—
159C22⋊C8—28no—
160#97—48no—
161#95—24no—
162#95—24no—
163#95—28no—
164#95—28no—
165#95C2×D418yes—
166C24—22no—
167C24—42no—
168C24—42no—
169C24—44no—
170C24—44no—
171C24—42no—
172C24—44no—
173C24#12414yes—
174C24AGL(1,ℤ/8)18yes—
175C24C22×D411yescharacteristic
176C24C22×D412yes—
177C2×D4—24no—
178C2×D4—28no—
179C2×D4—44no—
180C2×D4—48no—
181C2×D4—48no—
182C2×D4—48no—
183C2×D4—42no—
184C2×D4—88no—
185C2×D4—816no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—28no—
189C22×C4—28no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—416no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—48no—
201C22×C4—44no—
202C22×C4—48no—
203C22×C4—48no—
204C22×C4—48no—
205C22×C4—84no—
206C22×C4—84no—
207C22×C4—816no—
208C22×C4#12414yes—
209C22×C4#12418yes—
210C22×C4C22×D418yes—
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D412yes—
213C22×C4C2×#4114yes—
214C22×C4C2×#4114yes—
215C22×C4C2×#4114yes—
216C22×C4C2511yescharacteristic
217C2×C8—28no—
218C2×C8—216no—
219C2×C8—88no—
220C4⋊C4—28no—
221C4⋊C4—24no—
222C4⋊C4—216no—
223C4⋊C4—432no—
224C4⋊C4—48no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—88no—
228C4⋊C4—84no—
229C4⋊C4—88no—
230C22⋊C4—28no—
231C22⋊C4—28no—
232C22⋊C4—432no—
233C22⋊C4—48no—
234C22⋊C4—48no—
235C22⋊C4—48no—
236C42—28no—
237C42—44no—
238C23—22no—
239C23—24no—
240C23—22no—
241C23—24no—
242C23—44no—
243C23—44no—
244C23—416no—
245C23—44no—
246C23—42no—
247C23—44no—
248C23—44no—
249C23—84no—
250C23—816no—
251C23—84no—
252C23—84no—
253C23#490116yes—
254C23C2×#12412yes—
255C23C2×#12414yes—
256C23C2×AGL(1,ℤ/8)14yes—
257C23C2×AGL(1,ℤ/8)14yes—
258C23C2×AGL(1,ℤ/8)14yes—
259C23C23×D411yescharacteristic
260D4—48no—
261D4—84no—
262D4—816no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—28no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—28no—
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—44no—
272C2×C4—48no—
273C2×C4—416no—
274C2×C4—48no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—48no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—88no—
282C2×C4—816no—
283C2×C4—84no—
284C2×C4—168no—
285C2×C4C2×#12414yes—
286C2×C4C2×#12412yes—
287C2×C4C22×#4112yes—
288C2×C4#58514yes—
289C8—48no—
290C22—28no—
291C22—28no—
292C22—28no—
293C22—22no—
294C22—24no—
295C22—22no—
296C22—24no—
297C22—416no—
298C22—88no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—168no—
303C22C2×#49014yes—
304C22C2×#49018yes—
305C22C22×#12411yescharacteristic
306C22#348512yes—
307C22C22×AGL(1,ℤ/8)12yes—
308C22#358612yes—
309C4—24no—
310C4—22no—
311C4—48no—
312C4—88no—
313C4—84no—
314C4—82no—
315C4#349114yes—
316C2—24no—
317C2—28no—
318C2—162no—
319C2#3545511yescharacteristic
320C2C22×#49011yescharacteristic
321C2#6062011yescharacteristic
322C2#6170614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35020C2semidirect
3. #29110C2semidirect
4. #27966C2semidirect
5. #27717C2semidirect
7. C2×#1823C2semidirect
8. #20526C2semidirect
18. #2442C22semidirect
25. #2100C22semidirect
26. #2073C22semidirect
27. #2025C22semidirect
28. #1992C22semidirect
29. #1936C22semidirect
32. #1836C22semidirect
35. #1823C22semidirect
37. #1752C22semidirect
39. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#25012
#291102
#279662
#277172
#217582
C2×#18232
#205262