Finite Group Databasebeta

← #149867

Subgroups — #149867

Order 512.

225 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 56 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#21737C211yesmaximal, characteristic
2#11539C212yesmaximal
3#7374C212yesmaximal
4#7360C212yesmaximal
5#3162C411yescharacteristic
6#2960C411yescharacteristic
7#2960C2211yescharacteristic
8#2166—22no—
9#1995C411yescharacteristic
10#1995C2211yescharacteristic
11#1945—22no—
12#1849C2211yescharacteristic
13#1773—22no—
14#1726C411yescharacteristic
15#1328—22no—
16#1327C2214yes—
17#533C2×C411yescharacteristic
18#533C2×C411yescharacteristic
19#501—22no—
20#501—22no—
21#446—41no—
22#446—41no—
23C8×Q8—41no—
24C8×Q8—41no—
25C2×#108—22no—
26#427—22no—
27#424—21no—
28#424D412yes—
29#423C2×C411yescharacteristic
30#423C2311yescharacteristic
31#405—21no—
32#405D412yes—
33C2×#97—21no—
34C2×#97C2×C411yescharacteristic
35C2×C4×C8—21no—
36C2×C4×C8C2×C411yescharacteristic
37#376C2×C411yescharacteristic
38#364—42no—
39#349—24no—
40#347—42no—
41#339—22no—
42#339—22no—
43C2×M16—21no—
44C2×M16C2×D412yes—
45C22×C8—21no—
46C22×C8C22⋊C411yescharacteristic
47C22×C8C22⋊C411yescharacteristic
48C4⋊Q8—21no—
49C4⋊Q8—21no—
50#125—21no—
51#125M1612yes—
52C22⋊Q8—41no—
53C22⋊Q8—41no—
54C4×Q8—41no—
55C4×Q8—41no—
56#117—21no—
57#117C22⋊C412yes—
58C2×C42—21no—
59C2×C42C22×C411yescharacteristic
60M32—44no—
61#108—44no—
62#107—24no—
63#106—24no—
64C4⋊C8—41no—
65C4⋊C8—42no—
66C4⋊C8—41no—
67C4⋊C8—42no—
68#97—24no—
69#97—22no—
70#97—21no—
71#97—41no—
72#97M1612yes—
73C4×C8—22no—
74C4×C8—21no—
75C4×C8—21no—
76C4×C8—42no—
77C4×C8—41no—
78#95—41no—
79C2×Q8—41no—
80C2×Q8—41no—
81C22×C4—21no—
82C22×C4—41no—
83C22×C4C2×C22⋊C411yescharacteristic
84M16—42no—
85M16—42no—
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—21no—
91C2×C8—22no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8#9911yescharacteristic
95C2×C8#9911yescharacteristic
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—42no—
99C4⋊C4—42no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C22⋊C4—22no—
103C22⋊C4—41no—
104C42—21no—
105C42—22no—
106C42—22no—
107C42—41no—
108C42—41no—
109C42C2×M1611yescharacteristic
110C42C2×M1611yescharacteristic
111C16—84no—
112C23#41311yescharacteristic
113Q8—82no—
114Q8—82no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—81no—
123C2×C4—82no—
124C2×C4#40711yescharacteristic
125C2×C4C2×#9911yescharacteristic
126C8—24no—
127C8—21no—
128C8—42no—
129C8—41no—
130C8—81no—
131C8#35612yes—
132C22—21no—
133C22—41no—
134C22#146511yescharacteristic
135C4—21no—
136C4—41no—
137C4—41no—
138C4—82no—
139C4—82no—
140C4—82no—
141C4#212611yescharacteristic
142C4C2×#35611yescharacteristic
143C2—21no—
144C2—41no—
145C2#1153211yescharacteristic
subgroupG/H
centerC4#2126
commutatorC2×C42C22×C4
Frattini, cosocle#423C23
socleC2#11532
prime radicalC23#413
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2960 ◁ G
LECSG ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#217372
#115392
#73742
#73602