Finite Group Databasebeta

← #513160

Subgroups — #513160

Order 512.

315 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33741C211yesmaximal, characteristic, complemented
2#33740C211yesmaximal, characteristic
3#13708C212yesmaximal, complemented
4#13707C212yesmaximal, complemented
5#13699C212yesmaximal, complemented
6#13696C212yesmaximal, complemented
7#13687C212yesmaximal, complemented
8#13686C212yesmaximal, complemented
9#13628C211yesmaximal, characteristic, complemented
10#3177—21no—
11#3176—21no—
12C2×SD64C2212yes—
13C2×SD64C2212yes—
14C2×M64C2211yescharacteristic
15C22×C32C2211yescharacteristic
16#2253C2211yescharacteristic
17#2253C2211yescharacteristic
18#2247C2211yescharacteristic
19#2246C2211yescharacteristic
20#2245—21no—
21#2244—21no—
22#2219—21no—
23#2218—21no—
24#2216C2212yes—
25#2215C2212yes—
26#2177—21no—
27#2176—21no—
28C4×Q32C2212yes—
29C4×D16C2212yes—
30#2108C2211yescharacteristic
31#1424C2212yes—
32#1423C2212yes—
33C4⋊C32C2212yes—
34C4⋊C32C2212yes—
35#1416C2212yes—
36#1416C2212yes—
37#1415C2212yes—
38#1415C2212yes—
39#541—41no—
40#540—41no—
41C2×Q32—21no—
42C2×Q32C2312yes—
43C2×D16—21no—
44C2×D16C2312yes—
45C2×M32D411yescharacteristic
46C22×C16C2311yescharacteristic
47#499—21no—
48#499—21no—
49#494—21no—
50#493—21no—
51#491—41no—
52#490—41no—
53#470—41no—
54#469—41no—
55#467—22no—
56#466—22no—
57#442—41no—
58#441—41no—
59C4×Q16—22no—
60C4×D8—22no—
61#405D411yescharacteristic
62SD64—42no—
63SD64—42no—
64M64—41no—
65C2×C32—41no—
66C2×C32C2312yes—
67C2×C32C2312yes—
68#367—21no—
69#367—21no—
70#366C2312yes—
71#366C2312yes—
72#358—22no—
73#358—22no—
74#357—22no—
75#357—22no—
76#346D411yescharacteristic
77#346D411yescharacteristic
78C4×C16C2311yescharacteristic
79C4×C16C2311yescharacteristic
80C2×Q16—22no—
81C2×Q16—41no—
82C2×D8—22no—
83C2×D8—41no—
84C2×M16D811yescharacteristic
85C22×C8C2×D411yescharacteristic
86C4⋊Q8—41no—
87#127—41no—
88#126—82no—
89#125—41no—
90#125—41no—
91#125—82no—
92#123—82no—
93C22⋊Q8—42no—
94#121—42no—
95#121—82no—
96C4×Q8—42no—
97C4×D4—42no—
98#117D811yescharacteristic
99Q32—22no—
100Q32—22no—
101Q32—42no—
102D16—22no—
103D16—22no—
104D16—42no—
105M32—41no—
106C2×C16—21no—
107C2×C16—21no—
108C2×C16C2×D411yescharacteristic
109C2×C16C2×D411yescharacteristic
110C2×C16C2×D411yescharacteristic
111C2×C16C2411yescharacteristic
112#107—22no—
113#107—22no—
114#106—41no—
115#106—41no—
116#103—42no—
117#103—42no—
118#102—42no—
119#102—42no—
120#97D811yescharacteristic
121#97D811yescharacteristic
122C4×C8C2×D411yescharacteristic
123C4×C8C2×D411yescharacteristic
124C32—22no—
125C32—22no—
126C2×Q8—42no—
127C2×D4—42no—
128C2×D4—82no—
129C2×D4—82no—
130C22×C4—82no—
131C22×C4C2×D811yescharacteristic
132Q16—42no—
133Q16—42no—
134Q16—82no—
135D8—42no—
136D8—42no—
137D8—82no—
138M16—41no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8C2×D811yescharacteristic
142C2×C8C2×D811yescharacteristic
143C2×C8C2×D811yescharacteristic
144C2×C8C22×D411yescharacteristic
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—82no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C4⋊C4D1612yes—
153C22⋊C4—82no—
154C22⋊C4—82no—
155C22⋊C4—82no—
156C22⋊C4D1612yes—
157C42—82no—
158C42C2×D811yescharacteristic
159C42C2×D811yescharacteristic
160C16—21no—
161C16—21no—
162C16—21no—
163C16H(5,𝔽2)11yescharacteristic
164C162−1+411yescharacteristic
165C23—82no—
166C23C2×D1611yescharacteristic
167Q8—82no—
168Q8—82no—
169D4—82no—
170D4—82no—
171D4—322no—
172D4—322no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—322no—
179C2×C4C2×D1612yes—
180C2×C4C2×D1612yes—
181C2×C4C2×D1611yescharacteristic
182C2×C4C22×D811yescharacteristic
183C8—21no—
184C8—21no—
185C8—21no—
186C8#53711yescharacteristic
187C8#57511yescharacteristic
188C22—21no—
189C22—22no—
190C22—162no—
191C22—162no—
192C22—162no—
193C22C22×D1611yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—162no—
198C4—162no—
199C4—162no—
200C4#324511yescharacteristic
201C4#341411yescharacteristic
202C2—41no—
203C2—322no—
204C2#3381111yescharacteristic
205C2#3398312yes—
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
prime radical#33740C2
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33741C2semidirect
3. #13708C2semidirect
4. #13707C2semidirect
5. #13699C2semidirect
6. #13696C2semidirect
7. #13687C2semidirect
8. #13686C2semidirect
9. #13628C2semidirect

Maximal subgroups

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

subgroupindex
#337412
#337402
#137082
#137072
#136992
#136962
#136872
#136862
#136282