Finite Group Databasebeta

← #513102

Subgroups — #513102

Order 512.

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

normal subgroups only

This group has 363 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#33608C212yesmaximal
2#13716C214yesmaximal, complemented
3#13698C212yesmaximal, complemented
4#13695C212yesmaximal, complemented
5#13665C214yesmaximal, complemented
6#13554C211yesmaximal, characteristic
7C22×Q32C2212yes—
8C2×#504C2211yescharacteristic
9#3059—22no—
10C2×Q64C2212yes—
11C2×Q64C2212yes—
12C2×M64C2211yescharacteristic
13C22×C32C2211yescharacteristic
14#2219—22no—
15#2219—22no—
16#2216C2214yes—
17#2216C2214yes—
18#2158C2212yes—
19#2141—22no—
20#1424C2212yes—
21#1423C2212yes—
22#1416C2214yes—
23#1416C2214yes—
24C22⋊C32C2214yes—
25C22×Q16—22no—
26C2×#131D411yescharacteristic
27#536—42no—
28C2×Q32—22no—
29C2×Q32—42no—
30C2×Q32—42no—
31C2×Q32C2314yes—
32#504—41no—
33#504—41no—
34C2×M32D412yes—
35C2×M32D411yescharacteristic
36C22×C16C2312yes—
37C22×C16C2311yescharacteristic
38#470—42no—
39#470—42no—
40#467—24no—
41#467—24no—
42#427—22no—
43#419—42no—
44Q64—42no—
45Q64—42no—
46M64—41no—
47C2×C32—41no—
48C2×C32C2312yes—
49C2×C32C2312yes—
50#367—22no—
51#366C2314yes—
52#358—24no—
53#358—24no—
54C2×#41D811yescharacteristic
55C22×Q8—42no—
56C2×Q16—24no—
57C2×Q16—44no—
58C2×Q16—44no—
59C2×Q16—42no—
60#131—41no—
61#131—41no—
62C2×M16D812yes—
63C2×M16D811yescharacteristic
64C22×C8C2×D412yes—
65C22×C8C2×D411yescharacteristic
66C4⋊Q8—82no—
67#124—82no—
68#123—82no—
69#123—82no—
70C22⋊Q8—44no—
71C22⋊Q8—44no—
72#117—42no—
73Q32—24no—
74Q32—24no—
75Q32—44no—
76M32—42no—
77M32—41no—
78C2×C16—21no—
79C2×C16—21no—
80C2×C16—22no—
81C2×C16—22no—
82C2×C16C2×D411yescharacteristic
83C2×C16C2×D412yes—
84C2×C16C2411yescharacteristic
85#107—24no—
86#106—42no—
87#103—44no—
88#103—44no—
89C32—22no—
90C32—22no—
91#41—41no—
92#41—41no—
93C2×Q8—44no—
94C2×Q8—84no—
95C2×Q8—84no—
96C2×Q8—82no—
97C2×Q8D1611yescharacteristic
98C2×D4D1611yescharacteristic
99C2×D4D1612yes—
100C22×C4—82no—
101C22×C4C2×D812yes—
102C22×C4C2×D811yescharacteristic
103Q16—44no—
104Q16—44no—
105Q16—84no—
106M16—42no—
107M16—41no—
108C2×C8—21no—
109C2×C8—22no—
110C2×C8—21no—
111C2×C8—22no—
112C2×C8C2×D811yescharacteristic
113C2×C8C2×D812yes—
114C2×C8C22×D411yescharacteristic
115C4⋊C4—44no—
116C4⋊C4—84no—
117C4⋊C4—84no—
118C22⋊C4—82no—
119C22⋊C4—84no—
120C22⋊C4—84no—
121C42—82no—
122C16—21no—
123C16—22no—
124C16H(5,𝔽2)11yescharacteristic
125C16H(5,𝔽2)11yescharacteristic
126C23C2×D1611yescharacteristic
127C23C2×D1612yes—
128Q8—41no—
129Q8—84no—
130Q8—84no—
131Q8—164no—
132D4—41no—
133D4—42no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—84no—
139C2×C4—84no—
140C2×C4—164no—
141C2×C4C2×D1612yes—
142C2×C4C2×D1611yescharacteristic
143C2×C4C22×D811yescharacteristic
144C8—21no—
145C8—22no—
146C8#53411yescharacteristic
147C8#57511yescharacteristic
148C22—22no—
149C22—24no—
150C22—22no—
151C22—21no—
152C22C22×D1611yescharacteristic
153C4—21no—
154C4—22no—
155C4—164no—
156C4—164no—
157C4#318611yescharacteristic
158C4#341411yescharacteristic
159C2—42no—
160C2—41no—
161C2#3375211yescharacteristic
162C2#3398412yes—
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
prime radicalC2×#41D8
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13716C2semidirect
3. #13698C2semidirect
4. #13695C2semidirect
5. #13665C2semidirect

Maximal subgroups

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

subgroupindex
#336082
#137162
#136982
#136952
#136652
#135542