Finite Group Databasebeta

← #6499334

Subgroups — #6499334

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2#36640C211yesmaximal, characteristic
3#33258C212yesmaximal, complemented
4#33256C212yesmaximal, complemented
5#30307C214yesmaximal, complemented
6#29316C218yesmaximal, complemented
7#29300C218yesmaximal
8#21984C214yesmaximal
9C4×#431C211yesmaximal, characteristic
10C2×#431C2212yes—
11#2864—22no—
12#2864C2214yes—
13#2859C2212yes—
14#2859C2212yes—
15#2859C2212yes—
16#2859C2212yes—
17#2851—22no—
18#2851—22no—
19#2851—22no—
20#2851C2214yes—
21C4×#125—22no—
22C4×#125—22no—
23C4×#125—22no—
24C4×#125C2214yes—
25#2296—22no—
26#2296C2214yes—
27#2285C2212yes—
28#2285C2212yes—
29C2×C43C2211yescharacteristic
30#2085C22116yes—
31#2059C22116yes—
32#1949C22116yes—
33#1942C22116yes—
34#1941C22116yes—
35#1854C22116yes—
36#1845C2214yes—
37#1844C2214yes—
38#1834C2214yes—
39#1834C2214yes—
40C4×C4⋊C8C2214yes—
41C4×C22⋊C8C2214yes—
42#1744C2212yes—
43C42×C8C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125C2314yes—
49C2×#117—22no—
50C2×#117C2314yes—
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42C2312yes—
54C22×C42C2311yescharacteristic
55#431Q8116yes—
56C2×#107C23116yes—
57C2×#106C23116yes—
58C2×C4⋊C8C2314yes—
59C2×C22⋊C8C2314yes—
60C2×C4×C8C2314yes—
61C2×C4×C8C2314yes—
62C2×C4×C8C2312yes—
63C2×C4×C8C2312yes—
64#389—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—24no—
69#389C2318yes—
70#389C2318yes—
71#389C2318yes—
72#387—28no—
73#387—28no—
74#387—28no—
75#387—28no—
76#385—28no—
77#385—28no—
78#384C2314yes—
79#384C2314yes—
80#382—22no—
81#382—22no—
82#382—22no—
83#382—22no—
84#382C2314yes—
85#382C2314yes—
86#380—22no—
87#380—22no—
88#380—22no—
89#380C2314yes—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—28no—
94C4×C4⋊C4—28no—
95C4×C4⋊C4C2314yes—
96C43D412yes—
97C43D412yes—
98C43C2312yes—
99C43C2311yescharacteristic
100C43C2311yescharacteristic
101#340C23116yes—
102#340C23116yes—
103#336C2318yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D414yes—
107C23×C4C2411yescharacteristic
108C22×C8C2414yes—
109C22×C8C2414yes—
110#125—48no—
111#125—48no—
112#125—48no—
113#125—48no—
114#117—48no—
115#117—48no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4C2418yes—
125C2×C4⋊C4C2418yes—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C42—28no—
129C2×C42—24no—
130C2×C42—24no—
131C2×C42—44no—
132C2×C42—44no—
133C2×C42C2×D412yes—
134C2×C42C2×D414yes—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C2×C42C2×D414yes—
138C2×C42C2×D414yes—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D412yes—
142C2×C42C2×Q818yes—
143C2×C42C2412yes—
144C2×C42C2412yes—
145C2×C42C2411yescharacteristic
146C2×C42C2411yescharacteristic
147#107—416no—
148#106—416no—
149C4⋊C8C2×Q8116yes—
150C22⋊C8C2×Q8116yes—
151C4×C8—44no—
152C4×C8—44no—
153C4×C8C2×Q818yes—
154C4×C8C2×Q818yes—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—28no—
159#95—28no—
160#95—28no—
161#95—28no—
162#95—24no—
163C24C22×D411yescharacteristic
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—28no—
168C22×C4—28no—
169C22×C4—28no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4C4⋊Q814yes—
173C22×C4C4⋊Q814yes—
174C22×C4C22×D414yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D414yes—
179C22×C4C22×D411yescharacteristic
180C22×C4C22×D411yescharacteristic
181C22×C4C22×Q814yes—
182C22×C4C2511yescharacteristic
183C2×C8—28no—
184C2×C8—28no—
185C2×C8C22×Q818yes—
186C2×C8C22×Q818yes—
187C4⋊C4—216no—
188C4⋊C4—216no—
189C4⋊C4—416no—
190C4⋊C4—416no—
191C4⋊C4—416no—
192C4⋊C4—416no—
193C4⋊C4—48no—
194C4⋊C4—48no—
195C4⋊C4—48no—
196C4⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C42—28no—
200C42—24no—
201C42—24no—
202C42—24no—
203C42—24no—
204C42—24no—
205C42—44no—
206C42—44no—
207C42—48no—
208C42—48no—
209C42C4⋊Q814yes—
210C42C4⋊Q814yes—
211C42#13518yes—
212C42C22×Q814yes—
213C42C22×Q814yes—
214C23—22no—
215C23—22no—
216C23—22no—
217C23C2×C4⋊Q812yes—
218C23C23×D411yescharacteristic
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—28no—
225C2×C4—28no—
226C2×C4—28no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—416no—
231C2×C4—48no—
232C2×C4—48no—
233C2×C4—48no—
234C2×C4—48no—
235C2×C4—48no—
236C2×C4—48no—
237C2×C4C2×C4⋊Q812yes—
238C2×C4C2×C4⋊Q812yes—
239C2×C4C2×C4⋊Q812yes—
240C2×C4C2×C4⋊Q812yes—
241C2×C4C2×C4⋊Q812yes—
242C2×C4C2×#13518yes—
243C2×C4C2×#13514yes—
244C2×C4#57514yes—
245C2×C4C23×Q812yes—
246C2×C4#58514yes—
247C8—28no—
248C8—28no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—44no—
253C22C22×C4⋊Q811yescharacteristic
254C22#344912yes—
255C22C22×#13512yes—
256C22C2×#57512yes—
257C4—22no—
258C4—28no—
259C4—22no—
260C4—22no—
261C4—44no—
262C4—88no—
263C4—88no—
264C4#316014yes—
265C4#316614yes—
266C4#350414yes—
267C2—42no—
268C2#3533511yescharacteristic
269C2C2×#316011yescharacteristic
270C2C2×#316611yescharacteristic
271C2#6094414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33258C2semidirect
4. #33256C2semidirect
5. #30307C2semidirect
6. #29316C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#332582
#332562
#303072
#293162
#293002
#219842
C4×#4312