Finite Group Databasebeta

← #149663

Subgroups — #149663

Order 512.

288 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×D4×C16C211yesmaximal, characteristic
2C2×C22⋊C32C212yesmaximal, complemented
3#7362C212yesmaximal, complemented
4#7334C212yesmaximal, complemented
5C23×C16C411yescharacteristic
6C23×C16C2211yescharacteristic
7C2×C8×D4C411yescharacteristic
8C22×C32—22no—
9D4×C16—22no—
10D4×C16—22no—
11C2×C4⋊C16C2211yescharacteristic
12C2×C22⋊C16C411yescharacteristic
13C2×C22⋊C16C2211yescharacteristic
14C2×C4×C16C411yescharacteristic
15C22⋊C32—22no—
16C22⋊C32C2214yes—
17C23×C8C2×C411yescharacteristic
18C23×C8C2×C411yescharacteristic
19C2×C4×D4C811yescharacteristic
20C22×C16—21no—
21C22×C16—21no—
22C22×C16—42no—
23C22×C16—42no—
24C22×C16C2×C411yescharacteristic
25C22×C16C2×C411yescharacteristic
26C22×C16C2×C411yescharacteristic
27C22×C16D412yes—
28C22×C16D412yes—
29C22×C16C2311yescharacteristic
30C8×D4—22no—
31C8×D4—22no—
32C2×C4⋊C8C811yescharacteristic
33C2×C22⋊C8C811yescharacteristic
34C2×C22⋊C8C811yescharacteristic
35C2×C4×C8C2×C411yescharacteristic
36C2×C32—24no—
37C2×C32—44no—
38C4⋊C16—22no—
39C22⋊C16—22no—
40C22⋊C16—22no—
41C4×C16—21no—
42C4×C16—21no—
43C22×D4C1611yescharacteristic
44C23×C4C2×C811yescharacteristic
45C23×C4C2×C811yescharacteristic
46C22×C8—21no—
47C22×C8—21no—
48C22×C8—41no—
49C22×C8—41no—
50C22×C8—41no—
51C22×C8—41no—
52C22×C8C22⋊C412yes—
53C22×C8C2×C811yescharacteristic
54C22×C8C2×C811yescharacteristic
55C22×C8C2×C811yescharacteristic
56C22×C8M1612yes—
57C22×C8C22×C411yescharacteristic
58C4×D4—22no—
59C4×D4—22no—
60C2×C4⋊C4C1611yescharacteristic
61C2×C22⋊C4C1611yescharacteristic
62C2×C22⋊C4C1611yescharacteristic
63C2×C42C2×C811yescharacteristic
64C2×C16—24no—
65C2×C16—22no—
66C2×C16—21no—
67C2×C16—21no—
68C2×C16—42no—
69C2×C16—41no—
70C2×C16—42no—
71C2×C16—42no—
72C2×C16—41no—
73C2×C16—41no—
74C2×C16C22⋊C411yescharacteristic
75C2×C16C22⋊C411yescharacteristic
76C2×C16C2×D412yes—
77C4⋊C8—22no—
78C22⋊C8—22no—
79C22⋊C8—22no—
80C4×C8—21no—
81C4×C8—21no—
82C32—44no—
83C24C2×C1611yescharacteristic
84C24C2×C1611yescharacteristic
85C2×D4—22no—
86C2×D4—22no—
87C2×D4C3214yescomplemented
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4C22⋊C812yes—
95C22×C4C2×C1611yescharacteristic
96C22×C4C2×C1611yescharacteristic
97C22×C4C2×C1611yescharacteristic
98C22×C4C2×C1611yescharacteristic
99C22×C4M3212yes—
100C22×C4C22×C811yescharacteristic
101C2×C8—21no—
102C2×C8—22no—
103C2×C8—21no—
104C2×C8—22no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—41no—
108C2×C8—42no—
109C2×C8—42no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8C22⋊C812yes—
115C2×C8C2×C22⋊C411yescharacteristic
116C2×C8C2×M1611yescharacteristic
117C4⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42—21no—
122C16—22no—
123C16—21no—
124C16—41no—
125C16—41no—
126C16#9912yes—
127C23—21no—
128C23—21no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23C22⋊C1612yes—
134C23C2×C3212yes—
135C23C2×C3212yescomplemented
136C23M6412yes—
137C23C22×C1611yescharacteristic
138D4—28no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4C22⋊C1612yes—
155C2×C4C2×C3212yescomplemented
156C2×C4C2×C22⋊C811yescharacteristic
157C2×C4C2×M3211yescharacteristic
158C8—21no—
159C8—22no—
160C8—41no—
161C8—41no—
162C8C2×#9911yescharacteristic
163C8#41311yescharacteristic
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—22no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—41no—
174C22—41no—
175C22C22⋊C3214yes—
176C22C2×C22⋊C1611yescharacteristic
177C22C22×C3211yescharacteristic
178C22C2×M6411yescharacteristic
179C4—21no—
180C4—22no—
181C4—22no—
182C4—41no—
183C4—41no—
184C4#145911yescharacteristic
185C4#211511yescharacteristic
186C2—21no—
187C2—21no—
188C2—24no—
189C2—42no—
190C2#1146511yescharacteristic
191C2C2×C22⋊C3211yescharacteristic
192C2#1355611yescharacteristic
subgroupG/H
centerC2×C16C22⋊C4
commutatorC22C22×C32
Frattini, cosocleC22×C16C23
socleC22C2×C22⋊C16
prime radicalC22×D4C16
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C16 ◁ C23×C16 ◁ G
LECSG ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C32C2semidirect
3. #7362C2semidirect
4. #7334C2semidirect
87. C2×D4C32semidirect
135. C23C2×C32semidirect
155. C2×C4C2×C32semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C162
C2×C22⋊C322
#73622
#73342