Finite Group Databasebeta

← #124920

Subgroups — #124920

Order 512.

289 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 72 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#21715C211yesmaximal, characteristic
2#12596C211yesmaximal, characteristic
3#12596C211yesmaximal, characteristic
4#7124C214yesmaximal
5C2×C8×Q8—21no—
6#2399C411yescharacteristic
7C2×#347—21no—
8C2×#347—21no—
9#1985—21no—
10#1940—21no—
11#1940C2212yes—
12#1843C411yescharacteristic
13C8×C4⋊C4—21no—
14C8×C4⋊C4C412yes—
15#1726C2211yescharacteristic
16#1377C2212yes—
17#1377C2212yes—
18#1376—21no—
19#1376—21no—
20C2×C4×Q8—21no—
21C2×M32—22no—
22C2×M32—22no—
23C8×Q8—44no—
24C2×C4⋊C8—22no—
25C2×C4⋊C8—21no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8C2×C412yes—
28C2×#97D411yescharacteristic
29C2×#97D411yescharacteristic
30C2×#97C2311yescharacteristic
31C2×C4×C8—22no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8D411yescharacteristic
34C2×C4×C8D411yescharacteristic
35#398—22no—
36#395—22no—
37C4×C4⋊C4—21no—
38C4×C4⋊C4C2×C412yes—
39#376C2×C411yescharacteristic
40#347—24no—
41#347—24no—
42#336—22no—
43#336—22no—
44C22×Q8—21no—
45C22×C8—22no—
46C22×C8C22⋊C411yescharacteristic
47C22×C8C22⋊C411yescharacteristic
48C22×C8C2×D411yescharacteristic
49C22×C8C2×D411yescharacteristic
50C4×Q8—42no—
51C4×Q8—42no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C42—22no—
57C2×C42C22⋊C411yescharacteristic
58C2×C42C22⋊C411yescharacteristic
59C2×C42C22×C411yescharacteristic
60M32—24no—
61M32—24no—
62M32—42no—
63M32—42no—
64C2×C16—42no—
65C2×C16—42no—
66C4⋊C8—42no—
67C4⋊C8—42no—
68C4⋊C8—44no—
69C4⋊C8—42no—
70#97—22no—
71#97—22no—
72#97—41no—
73C4×C8—22no—
74C4×C8—22no—
75C4×C8—22no—
76C4×C8—44no—
77#95—21no—
78#95—21no—
79#95—22no—
80#95M1612yes—
81#95M1612yes—
82C2×Q8—42no—
83C2×Q8—44no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4C2×C22⋊C411yescharacteristic
89C22×C4C2×M1612yes—
90C2×C8—21no—
91C2×C8—21no—
92C2×C8—21no—
93C2×C8—21no—
94C2×C8—21no—
95C2×C8—21no—
96C2×C8—21no—
97C2×C8—21no—
98C2×C8—21no—
99C2×C8—42no—
100C2×C8—42no—
101C2×C8—42no—
102C2×C8#10112yes—
103C2×C8AGL(1,ℤ/8)12yes—
104C2×C8#13712yes—
105C4⋊C4—44no—
106C4⋊C4—44no—
107C4⋊C4—44no—
108C4⋊C4—42no—
109C42—21no—
110C42—21no—
111C42—21no—
112C42—21no—
113C42—42no—
114C42—42no—
115C42#10412yes—
116C42#10412yes—
117C16—44no—
118C16—44no—
119C23#40711yescharacteristic
120Q8—48no—
121C2×C4—24no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4#41011yescharacteristic
134C2×C4C2×#10111yescharacteristic
135C2×C4#41811yescharacteristic
136C2×C4#41911yescharacteristic
137C2×C4C2×#10411yescharacteristic
138C2×C4C2×#10411yescharacteristic
139C8—22no—
140C8—22no—
141C8—22no—
142C8—41no—
143C8—82no—
144C22—22no—
145C22#146711yescharacteristic
146C22#149011yescharacteristic
147C22#149111yescharacteristic
148C4—42no—
149C4—44no—
150C4—42no—
151C4—42no—
152C4—82no—
153C4#150511yescharacteristic
154C4#150611yescharacteristic
155C4#152611yescharacteristic
156C4#155611yescharacteristic
157C2—21no—
158C2—21no—
159C2#763811yescharacteristic
160C2#1153912yes—
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
socleC22#1467
prime radicalC23#407
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1726 ◁ 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
#217152
#125962
#71242