Finite Group Databasebeta

← #124901

Subgroups — #124901

Order 512.

301 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C4×#491C211yesmaximal, characteristic
2#12658C212yesmaximal
3#7130C212yesmaximal
4#7117C212yesmaximal
5C2×#491C411yescharacteristic
6C4×C4⋊Q8C411yescharacteristic
7C4×#125C411yescharacteristic
8#2150—22no—
9C2×#347—22no—
10#1992C2211yescharacteristic
11#1960C2211yescharacteristic
12#1935—22no—
13#1836C411yescharacteristic
14C4×#103—22no—
15C4×#97C2211yescharacteristic
16#1375C2214yes—
17C2×C4⋊Q8C2×C411yescharacteristic
18C2×#125C2×C411yescharacteristic
19C2×C4×Q8—21no—
20C2×M32—24no—
21#491—22no—
22#491—22no—
23C2×#103—22no—
24C2×#103—22no—
25C2×#97C2×C411yescharacteristic
26C2×#97D412yes—
27C2×#97C2311yescharacteristic
28C2×C4×C8D412yes—
29#391—21no—
30#389—21no—
31#389—41no—
32#387—41no—
33#384C2×C411yescharacteristic
34#382C2×C411yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4—41no—
37C4×C4⋊C4—41no—
38C43C2×C411yescharacteristic
39C4⋊C16—24no—
40C4⋊C16—24no—
41#347—44no—
42#340—22no—
43#340—22no—
44C22×Q8—21no—
45C22×C8C22⋊C412yes—
46C22×C8C2×D412yes—
47C4⋊Q8—21no—
48C4⋊Q8—41no—
49C4⋊Q8M1612yes—
50#125—21no—
51#125—41no—
52#125M1612yes—
53C4×Q8—42no—
54C4×Q8—42no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—41no—
63C2×C42—21no—
64C2×C42—21no—
65C2×C42—41no—
66C2×C42—41no—
67C2×C42C22⋊C411yescharacteristic
68C2×C42C22⋊C411yescharacteristic
69C2×C42C22×C411yescharacteristic
70M32—44no—
71M32—44no—
72C2×C16—28no—
73#103—28no—
74#103—42no—
75#103—42no—
76#97—22no—
77#97—22no—
78#97—21no—
79#97—21no—
80#97—41no—
81C4×C8—22no—
82C4×C8—22no—
83#95—41no—
84#95—41no—
85#95—42no—
86#95—41no—
87C2×Q8—22no—
88C2×Q8—41no—
89C2×Q8—42no—
90C2×Q8—42no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4#10011yescharacteristic
98C22×C4#10011yescharacteristic
99C22×C4C2×C22⋊C411yescharacteristic
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8AGL(1,ℤ/8)12yes—
106C2×C8#13712yes—
107C4⋊C4—22no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C4⋊C4—41no—
113C4⋊C4—42no—
114C4⋊C4—42no—
115C4⋊C4—82no—
116C4⋊C4—82no—
117C4⋊C4—82no—
118C42—21no—
119C42—22no—
120C42—42no—
121C42—82no—
122C42—81no—
123C42—81no—
124C42—81no—
125C42#10412yes—
126C42#10412yes—
127C42C2×M1611yescharacteristic
128C42C2×M1611yescharacteristic
129C16—48no—
130C23C2×#10011yescharacteristic
131Q8—44no—
132Q8—82no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4—82no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4#32911yescharacteristic
154C2×C4#32912yes—
155C2×C4#32911yescharacteristic
156C2×C4#40711yescharacteristic
157C2×C4#41011yescharacteristic
158C2×C4#41712yes—
159C2×C4C2×#10412yes—
160C8—22no—
161C8—22no—
162C8—42no—
163C22—21no—
164C22—21no—
165C22#146611yescharacteristic
166C22C2×#32911yescharacteristic
167C22#156011yescharacteristic
168C4—21no—
169C4—22no—
170C4—41no—
171C4—41no—
172C4—81no—
173C4—82no—
174C4—82no—
175C4—161no—
176C4—161no—
177C4#135612yes—
178C4#148912yes—
179C4#150912yes—
180C2—21no—
181C2—21no—
182C2#763511yescharacteristic
183C2C2×#135611yescharacteristic
184C2#1191111yescharacteristic
subgroupG/H
center, socleC22C2×#329
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23C2×#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C4×#4912
#126582
#71302
#71172