Finite Group Databasebeta

← #149709

Subgroups — #149709

Order 512.

271 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 68 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#21658C211yesmaximal, characteristic
2#11485C212yesmaximal
3#7368C212yesmaximal, complemented
4#7361C212yesmaximal, complemented
5#3131C411yescharacteristic
6#2960C411yescharacteristic
7#2931C2211yescharacteristic
8#2115—22no—
9#1994C2211yescharacteristic
10#1962C411yescharacteristic
11#1937—22no—
12#1837C2211yescharacteristic
13#1764—22no—
14C42×C8C411yescharacteristic
15#1327C2214yes—
16#1321—22no—
17C2×#131D412yes—
18#533C2×C411yescharacteristic
19#532C2×C411yescharacteristic
20C2×M32—42no—
21#488C814yes—
22#487—21no—
23#486—21no—
24#446—41no—
25C8×Q8—41no—
26#436—41no—
27C8×D4—41no—
28#424—21no—
29#423C2×C411yescharacteristic
30#423C2311yescharacteristic
31C2×#104—22no—
32#416—22no—
33#408D412yes—
34#405—21no—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39C43C2×C411yescharacteristic
40#349—24no—
41#349—42no—
42#341—22no—
43#339—22no—
44C2×#41C22⋊C412yes—
45#131—42no—
46#131—42no—
47C2×M16—22no—
48C2×M16—21no—
49C2×M16C2×D412yes—
50C22×C8—21no—
51C22×C8—22no—
52C22×C8C22⋊C411yescharacteristic
53C22×C8C22⋊C411yescharacteristic
54C4⋊Q8—21no—
55C4⋊Q8M1611yescharacteristic
56#127M1611yescharacteristic
57#125C2×C812yes—
58#124C2×C812yes—
59C22⋊Q8—41no—
60#121—41no—
61C4×Q8—41no—
62C4×D4—41no—
63#117—21no—
64C2×C42—21no—
65C2×C42—41no—
66C2×C42C22×C411yescharacteristic
67M32—44no—
68M32—82no—
69C2×C16—82no—
70C4⋊C8—41no—
71C4⋊C8—41no—
72C4⋊C8—42no—
73#104—44no—
74#103—24no—
75#102—24no—
76C22⋊C8—42no—
77#97—41no—
78C4×C8—21no—
79C4×C8—21no—
80C4×C8—21no—
81C4×C8—42no—
82C4×C8—41no—
83C4×C8—41no—
84C4×C8—41no—
85C4×C8C2×C812yes—
86#41—42no—
87#41—42no—
88C2×Q8—41no—
89C2×Q8C22⋊C812yes—
90C2×D4—22no—
91C2×D4—41no—
92C2×D4C22⋊C812yes—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—41no—
96C22×C4C2×C22⋊C411yescharacteristic
97M16—42no—
98M16—42no—
99M16—82no—
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—22no—
103C2×C8—21no—
104C2×C8—22no—
105C2×C8—41no—
106C2×C8—41no—
107C2×C8—42no—
108C2×C8—42no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8#9911yescharacteristic
112C2×C8#9911yescharacteristic
113C4⋊C4—22no—
114C4⋊C4—41no—
115C4⋊C4—42no—
116C4⋊C4—41no—
117C22⋊C4—42no—
118C22⋊C4—41no—
119C42—21no—
120C42—22no—
121C42—41no—
122C42—44no—
123C42—41no—
124C42C22×C811yescharacteristic
125C42C2×M1611yescharacteristic
126C16—84no—
127C23—22no—
128C23#41311yescharacteristic
129Q8—42no—
130Q8—82no—
131D4—42no—
132D4—82no—
133D4—82no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—44no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4C2×C22⋊C811yescharacteristic
146C2×C4C2×#9911yescharacteristic
147C8—21no—
148C8—42no—
149C8—42no—
150C8—41no—
151C8—81no—
152C8#35411yescharacteristic
153C8#35411yescharacteristic
154C22—21no—
155C22—41no—
156C22—42no—
157C22—82no—
158C22#145911yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—44no—
163C4—42no—
164C4—82no—
165C4C2×#35411yescharacteristic
166C4#213211yescharacteristic
167C2—21no—
168C2—41no—
169C2—82no—
170C2#1147711yescharacteristic
subgroupG/H
centerC8#354
commutatorC42C22×C8
Frattini, cosocle#423C23
socleC2#11477
prime radical#532C2×C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G
LECSG ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #7368C2semidirect
4. #7361C2semidirect

Maximal subgroups

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

subgroupindex
#216582
#114852
#73682
#73612