Finite Group Databasebeta

← #149869

Subgroups — #149869

Order 512.

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

normal subgroups only

This group has 229 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#21734C211yesmaximal, characteristic
2#11539C212yesmaximal
3#7373C212yesmaximal
4#7361C212yesmaximal
5#3160C411yescharacteristic
6#2960C411yescharacteristic
7#2960C2211yescharacteristic
8#2166—22no—
9#1994C411yescharacteristic
10#1994C2211yescharacteristic
11#1946—22no—
12#1848C2211yescharacteristic
13#1773—22no—
14C4×#97C411yescharacteristic
15#1328—22no—
16#1327C2214yes—
17#533C2×C411yescharacteristic
18#533C2×C411yescharacteristic
19#500—21no—
20#499—22no—
21#498—21no—
22#446—41no—
23#446—41no—
24C8×Q8—41no—
25C8×Q8—41no—
26C2×#108—22no—
27#427—22no—
28#424—21no—
29#424D412yes—
30#423C2×C411yescharacteristic
31#423C2311yescharacteristic
32#405—21no—
33#405D412yes—
34C2×#97—21no—
35C2×#97—21no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C43C2×C411yescharacteristic
39#364—42no—
40#349—24no—
41#347—42no—
42#339—22no—
43#339—22no—
44C2×M16—21no—
45C2×M16C2×D412yes—
46C22×C8—21no—
47C22×C8C22⋊C411yescharacteristic
48C22×C8C22⋊C411yescharacteristic
49C4⋊Q8—21no—
50C4⋊Q8—21no—
51#125—21no—
52#125M1612yes—
53C22⋊Q8—41no—
54C22⋊Q8—41no—
55C4×Q8—41no—
56C4×Q8—41no—
57#117—21no—
58#117C22⋊C412yes—
59C2×C42—21no—
60C2×C42—41no—
61C2×C42C22×C411yescharacteristic
62M32—44no—
63#108—44no—
64#107—24no—
65#106—24no—
66C4⋊C8—41no—
67C4⋊C8—42no—
68C4⋊C8—41no—
69C4⋊C8—42no—
70#97—24no—
71#97—22no—
72#97—42no—
73#97—41no—
74C4×C8—22no—
75C4×C8—21no—
76C4×C8—21no—
77C4×C8—21no—
78C4×C8—41no—
79C4×C8M1611yescharacteristic
80C4×C8M1611yescharacteristic
81C2×Q8—41no—
82C2×Q8—41no—
83C22×C4—21no—
84C22×C4—41no—
85C22×C4C2×C22⋊C411yescharacteristic
86M16—42no—
87M16—42no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—21no—
91C2×C8—21no—
92C2×C8—22no—
93C2×C8—21no—
94C2×C8—22no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8#9911yescharacteristic
98C2×C8#9911yescharacteristic
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—42no—
102C4⋊C4—42no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C22⋊C4—22no—
106C22⋊C4—41no—
107C42—21no—
108C42—22no—
109C42—22no—
110C42—41no—
111C42—41no—
112C42—82no—
113C42C2×M1611yescharacteristic
114C42C2×M1611yescharacteristic
115C16—84no—
116C23#41311yescharacteristic
117Q8—82no—
118Q8—82no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—82no—
128C2×C4#40711yescharacteristic
129C2×C4C2×#9911yescharacteristic
130C8—22no—
131C8—22no—
132C8—21no—
133C8—42no—
134C8—41no—
135C8—81no—
136C8#35612yes—
137C22—21no—
138C22—41no—
139C22#146511yescharacteristic
140C4—22no—
141C4—21no—
142C4—41no—
143C4—82no—
144C4—82no—
145C4—82no—
146C4#212611yescharacteristic
147C4C2×#35611yescharacteristic
148C2—21no—
149C2—41no—
150C2#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
#217342
#115392
#73732
#73612