Finite Group Databasebeta

← #124206

Subgroups — #124206

Order 512.

305 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 90 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#12594C212yesmaximal
3#7130C212yesmaximal
4#7117C212yesmaximal
5C2×#491C411yescharacteristic
6C4×C4⋊Q8C411yescharacteristic
7C4×#125C411yescharacteristic
8C4×M32—22no—
9#1992C2211yescharacteristic
10#1960C2211yescharacteristic
11#1935—22no—
12#1836C411yescharacteristic
13C4×#103—22no—
14C4×#97C2211yescharacteristic
15#1375—22no—
16#1375C2214yes—
17C2×C4⋊Q8C2×C411yescharacteristic
18C2×#125C2×C411yescharacteristic
19C2×C4×Q8—21no—
20C2×M32—24no—
21#491—22no—
22#491C814yes—
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
39#346—24no—
40C4×C16—24no—
41#340—22no—
42#340—22no—
43C22×Q8—21no—
44C22×C8C22⋊C412yes—
45C22×C8C2×D412yes—
46C4⋊Q8—21no—
47C4⋊Q8—41no—
48C4⋊Q8C2×C812yes—
49#125—21no—
50#125—41no—
51#125C2×C812yes—
52C4×Q8—42no—
53C4×Q8—42no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42—41no—
65C2×C42—41no—
66C2×C42C22⋊C411yescharacteristic
67C2×C42C22⋊C411yescharacteristic
68C2×C42C22×C411yescharacteristic
69M32—44no—
70M32—44no—
71C2×C16—28no—
72#103—28no—
73#103—44no—
74#97—24no—
75#97—41no—
76#97C2×C812yes—
77#97M1612yes—
78C4×C8—22no—
79C4×C8—22no—
80#95—41no—
81#95—41no—
82#95—42no—
83#95—41no—
84C2×Q8—22no—
85C2×Q8—41no—
86C2×Q8—42no—
87C2×Q8—42no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4#10011yescharacteristic
95C22×C4#10011yescharacteristic
96C22×C4C2×C22⋊C411yescharacteristic
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8C22⋊C814yes—
102C2×C8#13514yes—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—41no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—82no—
112C4⋊C4—82no—
113C4⋊C4—82no—
114C42—21no—
115C42—22no—
116C42—42no—
117C42—82no—
118C42—81no—
119C42—81no—
120C42—81no—
121C42#10412yes—
122C42#10412yes—
123C42C22×C811yescharacteristic
124C42C2×M1611yescharacteristic
125C16—48no—
126C23C2×#10011yescharacteristic
127Q8—44no—
128Q8—82no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—82no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4#32911yescharacteristic
150C2×C4#32912yes—
151C2×C4#32911yescharacteristic
152C2×C4C2×C22⋊C811yescharacteristic
153C2×C4#41011yescharacteristic
154C2×C4#41612yes—
155C2×C4C2×#10412yes—
156C8—24no—
157C8—24no—
158C22—21no—
159C22—21no—
160C22#146011yescharacteristic
161C22C2×#32911yescharacteristic
162C22#154311yescharacteristic
163C4—21no—
164C4—22no—
165C4—41no—
166C4—41no—
167C4—81no—
168C4—82no—
169C4—82no—
170C4—161no—
171C4—161no—
172C4#135612yes—
173C4#147612yes—
174C4#150012yes—
175C2—21no—
176C2—21no—
177C2#756711yescharacteristic
178C2C2×#135611yescharacteristic
179C2#1191111yescharacteristic
subgroupG/H
centerC2×C4#329
commutatorC42C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#329
prime radicalC23C2×#100
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
#125942
#71302
#71172