Finite Group Databasebeta

← #125298

Subgroups — #125298

Order 512.

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

normal subgroups only

This group has 325 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#33232C211yesmaximal, characteristic
2#12658C211yesmaximal, characteristic
3#12594C211yesmaximal, characteristic
4#7117C214yesmaximal
5C2×#491—21no—
6C2×#491—21no—
7#2868C411yescharacteristic
8#2859C411yescharacteristic
9#2150—21no—
10C2×#347—21no—
11C4×M32—21no—
12#1960C412yes—
13#1960C2212yes—
14#1956—21no—
15#1956—21no—
16C4×#97C2211yescharacteristic
17#1375—21no—
18#1375C2212yes—
19#1375C2212yes—
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2×C412yes—
23C2×#125—21no—
24C2×M32—22no—
25C2×M32—22no—
26#491—44no—
27#491—42no—
28#491—42no—
29C2×#103—22no—
30C2×#103—22no—
31C2×#103—22no—
32C2×#103—22no—
33C2×#97C2×C411yescharacteristic
34C2×#97D411yescharacteristic
35C2×#97D411yescharacteristic
36C2×#97C2311yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#389—42no—
40#384—21no—
41#384—42no—
42#382C2×C412yes—
43C43C2×C411yescharacteristic
44C4⋊C16—24no—
45#347—42no—
46#346—22no—
47C4×C16—22no—
48C22×Q8—22no—
49C22×C8C22⋊C411yescharacteristic
50C22×C8C22⋊C411yescharacteristic
51C22×C8C2×D411yescharacteristic
52C22×C8C2×D411yescharacteristic
53C4⋊Q8—24no—
54C4⋊Q8—42no—
55C4⋊Q8—42no—
56C4⋊Q8—42no—
57C4⋊Q8—41no—
58C4⋊Q8—42no—
59C4⋊Q8—41no—
60#125—42no—
61#125—42no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C42—22no—
69C2×C42C22⋊C411yescharacteristic
70C2×C42C22⋊C411yescharacteristic
71C2×C42C22×C411yescharacteristic
72M32—42no—
73M32—42no—
74M32—42no—
75M32—42no—
76C2×C16—24no—
77C2×C16—24no—
78#103—44no—
79#103—44no—
80#103—42no—
81#103—42no—
82#103—44no—
83#97—22no—
84#97—22no—
85#97—41no—
86#97D812yes—
87#97SD1612yes—
88C4×C8—21no—
89C4×C8—21no—
90C4×C8—22no—
91#95—42no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×Q8—42no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4#10012yes—
103C22×C4C2×C22⋊C411yescharacteristic
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8#10212yes—
114C2×C8C2×D811yescharacteristic
115C2×C8C2×SD1611yescharacteristic
116C2×C8AGL(1,ℤ/8)12yes—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—84no—
120C4⋊C4—84no—
121C4⋊C4—84no—
122C4⋊C4—84no—
123C4⋊C4—82no—
124C4⋊C4—82no—
125C42—21no—
126C42—21no—
127C42—44no—
128C42#10212yes—
129C42#10411yescharacteristic
130C42#10412yes—
131C42#10411yescharacteristic
132C42#13112yes—
133C16—44no—
134C16—44no—
135C23C2×#10011yescharacteristic
136Q8—82no—
137Q8—82no—
138Q8—82no—
139Q8—82no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—84no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4#32912yes—
151C2×C4#32912yes—
152C2×C4#40811yescharacteristic
153C2×C4#41011yescharacteristic
154C2×C4C2×#10211yescharacteristic
155C2×C4#41811yescharacteristic
156C2×C4C2×#10411yescharacteristic
157C2×C4C2×#10411yescharacteristic
158C8—22no—
159C8—21no—
160C8—21no—
161C8—22no—
162C8—41no—
163C22—22no—
164C22#146911yescharacteristic
165C22C2×#32911yescharacteristic
166C22#154511yescharacteristic
167C4—22no—
168C4—42no—
169C4—42no—
170C4—162no—
171C4—162no—
172C4#135614yes—
173C4#148411yescharacteristic
174C4#149211yescharacteristic
175C4#150211yescharacteristic
176C4#150511yescharacteristic
177C2—21no—
178C2—21no—
179C2#767911yescharacteristic
180C2C2×#135612yes—
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
#332322
#126582
#125942
#71172