Finite Group Databasebeta

← #149669

Subgroups — #149669

Order 512.

252 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 84 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#33432C211yesmaximal, characteristic
2#13556C212yesmaximal, complemented
3#7365C212yesmaximal, complemented
4#7336C212yesmaximal, complemented
5C2×#504C2211yescharacteristic
6C23×C16C411yescharacteristic
7#2929C411yescharacteristic
8C2×M64—22no—
9#2168—22no—
10D4×C16—22no—
11#2151C2211yescharacteristic
12#2113C411yescharacteristic
13#2112C2211yescharacteristic
14#2108C411yescharacteristic
15#1400—22no—
16#1400C2214yes—
17C2×#131C2×C411yescharacteristic
18C23×C8C2×C411yescharacteristic
19#525C811yescharacteristic
20#504—41no—
21#504—41no—
22C2×M32C2×C411yescharacteristic
23C2×M32D412yes—
24C2×M32C2311yescharacteristic
25C22×C16—21no—
26C22×C16—21no—
27C22×C16—41no—
28C22×C16—41no—
29C22×C16C2×C411yescharacteristic
30C22×C16C2×C411yescharacteristic
31C22×C16D412yes—
32#435—22no—
33C8×D4—22no—
34#424C811yescharacteristic
35#408C811yescharacteristic
36#407C811yescharacteristic
37#405C2×C411yescharacteristic
38M64—24no—
39M64—42no—
40C2×C32—42no—
41C4⋊C16—22no—
42C22⋊C16—22no—
43C22⋊C16—22no—
44#346—21no—
45C4×C16—21no—
46C2×#41C2×C811yescharacteristic
47C23×C4C2×C811yescharacteristic
48#131—41no—
49#131—41no—
50C2×M16C2×C811yescharacteristic
51C2×M16C2×C811yescharacteristic
52C2×M16M1612yes—
53C22×C8—21no—
54C22×C8—21no—
55C22×C8—41no—
56C22×C8—41no—
57C22×C8C22⋊C412yes—
58C22×C8C2×C811yescharacteristic
59C22×C8C22×C411yescharacteristic
60#123C1612yes—
61C22⋊Q8—21no—
62#121—21no—
63#120C1612yes—
64C4×D4—22no—
65#117C2×C811yescharacteristic
66M32—42no—
67M32—41no—
68M32—41no—
69C2×C16—22no—
70C2×C16—21no—
71C2×C16—22no—
72C2×C16—21no—
73C2×C16—21no—
74C2×C16—21no—
75C2×C16—42no—
76C2×C16—41no—
77C2×C16—41no—
78C2×C16—41no—
79C2×C16C22⋊C411yescharacteristic
80C2×C16C22⋊C411yescharacteristic
81C2×C16C2×D412yes—
82C4⋊C8—22no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85#97—21no—
86C4×C8—21no—
87C32—44no—
88C24C2×C1611yescharacteristic
89#41—41no—
90#41—41no—
91C2×Q8M3211yescharacteristic
92C2×D4—22no—
93C2×D4C2×C1612yes—
94C2×D4M3211yescharacteristic
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4C22⋊C812yes—
100C22×C4C2×C1611yescharacteristic
101C22×C4C22×C811yescharacteristic
102M16—41no—
103M16—42no—
104M16—41no—
105C2×C8—21no—
106C2×C8—22no—
107C2×C8—21no—
108C2×C8—22no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—42no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8C22⋊C812yes—
116C2×C8C2×C22⋊C411yescharacteristic
117C2×C8C2×M1611yescharacteristic
118C4⋊C4—22no—
119C4⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4C2×C1612yes—
123C42—21no—
124C16—22no—
125C16—21no—
126C16—41no—
127C16—41no—
128C16#9911yescharacteristic
129C16#9911yescharacteristic
130C23—21no—
131C23—41no—
132C23—42no—
133C23C22⋊C1612yes—
134C23C22×C1611yescharacteristic
135Q8—41no—
136D4—42no—
137D4—41no—
138D4—44no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4C22⋊C1612yes—
150C2×C4C2×C22⋊C811yescharacteristic
151C2×C4C2×M3211yescharacteristic
152C8—21no—
153C8—22no—
154C8—41no—
155C8—41no—
156C8C2×#9911yescharacteristic
157C8#41311yescharacteristic
158C22—22no—
159C22—21no—
160C22—22no—
161C22—42no—
162C22—41no—
163C22—41no—
164C22—42no—
165C22—42no—
166C22—41no—
167C22C2×C22⋊C1611yescharacteristic
168C4—21no—
169C4—22no—
170C4—42no—
171C4—41no—
172C4#145911yescharacteristic
173C4#211511yescharacteristic
174C2—21no—
175C2—42no—
176C2—42no—
177C2—41no—
178C2#1146511yescharacteristic
subgroupG/H
centerC16#99
commutatorC23C22×C16
Frattini, cosocleC2×M32C23
socleC2#11465
prime radical#525C8
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C16 ◁ C2×C16 ◁ C2×#504 ◁ G
LECSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13556C2semidirect
3. #7365C2semidirect
4. #7336C2semidirect

Maximal subgroups

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

subgroupindex
#334322
#135562
#73652
#73362